Summary
# Mathematics problems
This section addresses a variety of mathematical problems spanning arithmetic, algebra, geometry, and number theory, providing practical applications and theoretical concepts.
### 1.1 Arithmetic and Basic Operations
This subsection covers fundamental calculations and their applications in financial and real-world scenarios.
#### 1.1.1 Simple and Compound Interest
* **Simple Interest:** Calculated on the principal amount only.
The formula for simple interest ($SI$) is:
$$SI = P \times R \times T$$
where $P$ is the principal amount, $R$ is the annual interest rate, and $T$ is the time in years.
* **Compound Interest:** Calculated on the principal amount and also on the accumulated interest of previous periods.
The formula for the final amount ($A$) with compound interest is:
$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$
where $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times that interest is compounded per year, and $t$ is the number of years.
> **Example:** A certain amount of money was invested at compound interest. It amounted to 15,000 dollars at the end of the 4th year, and 18,000 dollars at the end of the 8th year. To find the rate of interest per annum, one would typically use the compound interest formula and solve for $r$.
* **Interest Deducted in Advance:** When interest is deducted at the time of borrowing, the actual amount received is less than the borrowed amount, leading to a higher effective interest rate.
> **Example:** A man borrows 10,000 dollars. The rate of simple interest is 15%, but the interest is deducted at the time of borrowing. At the end of one year, he has to pay back 10,000 dollars. The actual rate of interest is higher than 15% because he received less than 10,000 dollars initially.
#### 1.1.2 Percentage and Proportions
Problems involving percentages are common, including calculating gains, losses, and proportions.
* **Profit and Loss:**
* Selling Price ($SP$) = Cost Price ($CP$) + Profit
* Selling Price ($SP$) = Cost Price ($CP$) - Loss
> **Example:** A distributor sold the latest apple phone for 75,000 dollars at a loss of 10%. If the distributor wanted to gain 15% instead, the new selling price would need to be calculated based on the original cost price.
* **Ratios:** Used to compare quantities.
> **Example:** If the ratio of the two legs of a right triangle is 5:12, the ratio of the hypotenuse to the shorter side can be determined using the Pythagorean theorem.
#### 1.1.3 Time, Work, and Distance
These problems often involve rates of work, speeds, and travel times.
* **Work Rate:** If a job can be done by $N$ workers in $D$ days, the total work is $N \times D$. If workers leave or join, the remaining work and the work rate change.
> **Example:** A job could be done by 25 workers in a target time of 90 days. If workers quit after certain periods, the total number of days the project was delayed needs to be calculated by determining the work done in each phase and the remaining work.
* **Speed, Distance, Time:** The fundamental relationship is $Distance = Speed \times Time$.
> **Example:** A cyclist travels 10 km with the wind in 1.3 hours, and 1.6 hours in travelling back against the wind. The wind velocity can be found by setting up equations for speed with and against the wind.
#### 1.1.4 Averages
Averages are calculated by summing all values and dividing by the number of values.
> **Example:** A retailer has 10 items with an average price of 120 dollars each. If one item is removed, and the new average is 115 dollars, the price of the removed item can be calculated.
#### 1.1.5 Rate of Change and Motion
Problems involving acceleration, velocity, and displacement.
* **Acceleration:** The rate of change of velocity.
$$a = \frac{\Delta v}{\Delta t}$$
where $a$ is acceleration, $\Delta v$ is the change in velocity, and $\Delta t$ is the change in time.
> **Example:** What is the acceleration of a body that increases in velocity from 20 m/s to 40 m/s in 3 seconds?
#### 1.1.6 Energy and Power
These problems often involve mechanical work and the rate at which it is done.
* **Power:** The rate at which work is done or energy is transferred.
$$P = \frac{W}{t}$$
where $P$ is power, $W$ is work done, and $t$ is time. Work done against gravity is $W = mgh$, where $m$ is mass, $g$ is acceleration due to gravity, and $h$ is height.
> **Example:** What is the power required to lift 170,000 pounds of water to a height of 150 feet in 2 hours?
#### 1.1.7 Unit Conversions
Converting between different units of measurement is frequently required.
* **Area:**
* Circular Mil: A unit of area equal to the area of a circle with a diameter of one mil (0.001 inch). $1 \text{ circular mil} = \frac{\pi}{4} (\text{diameter in mils})^2$.
> **Example:** A wire has a diameter of 0.125 inch. Its cross-sectional area in circular mils needs to be calculated.
* **Volume:**
* US Gallons: Conversions between cubic feet or cubic meters and US gallons are often necessary.
* **Temperature:**
* Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$
* Fahrenheit to Celsius: $C = \frac{5}{9}(F - 32)$
> **Example:** At what temperature are the degrees Fahrenheit and degrees Celsius equal?
#### 1.1.8 Currency and Financial Calculations
Problems involving currency amounts, interest rates, and investments.
> **Tip:** Always write currency amounts in full letters (e.g., "dollars," "euros," "pounds") and never use currency symbols ($ , €, £$).
> **Example:** Carol and Mike invested 18,000 dollars each at simple interest. Carol's bank offers 4% per annum, while Mike's bank offers 6% per annum. After how many years will Mike's interest be 5,400 dollars more than Carol's?
### 1.2 Algebra
This subsection covers problems involving variables, equations, and algebraic expressions.
#### 1.2.1 Linear Equations
* **Solving for Variables:** Problems may involve solving for one or more variables in linear equations.
> **Example:** If 4 more than x is 2 times y, what is the value of y in terms of x? This can be written as $x + 4 = 2y$, and then solved for $y$.
#### 1.2.2 Polynomials and Roots
* **Finding Roots:** Determining the values of a variable that make a polynomial equation equal to zero.
> **Example:** Find the roots of $x^2 - 6x + 8 = 0$.
#### 1.2.3 Sequences and Series
* **Geometric Sequences:** A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
The $n$-th term of a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$.
> **Example:** Find the 10th term of the geometric sequence if the 6th term is 1458 and the 8th term is 13122.
* **Consecutive Integers:** Problems involving integers that follow each other in order.
> **Example:** Let a, b, c, d, e be five consecutive integers in increasing order. Which of the following expressions is always even?
#### 1.2.4 Binomial Expansion
* **Binomial Theorem:** Used to expand expressions of the form $(x+y)^n$. The general term in the expansion of $(x+y)^n$ is $\binom{n}{k} x^{n-k} y^k$.
> **Example:** Find the constant term in the binomial expansion of $(x - \frac{2}{x})^6$.
#### 1.2.5 Logarithms
* **Logarithmic Equations:** Solving equations involving logarithms.
> **Example:** Solve for B in $\ln(A) – \ln(B) = 4x$.
### 1.3 Geometry
This subsection deals with shapes, their properties, areas, volumes, and spatial relationships.
#### 1.3.1 Basic Geometric Shapes
* **Triangles:**
* **Area:** For a triangle with sides $a, b$ and included angle $C$, the area is $\frac{1}{2}ab\sin(C)$. Using coordinates of vertices $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$, the area can be calculated as $\frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.
* **Sides and Angles:** Problems may involve using the Law of Sines or Cosines.
* Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
* Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$
> **Example:** In triangle ABC, BC = 13.4 m, AC = 9 m, and ∠A = 60°. Find angle C.
> **Example:** A triangle has sides 8 units and 12 units. The included angle is 93°. Find the area of the triangle.
> **Example:** Side A of a triangle is 4 times side B, and Side C is 2 cm less than the sum of sides A and B. If side B is 5.2 cm, find the area of the triangle.
* **Circles:**
* **Arc Length:** The length of a portion of the circumference of a circle. Arc length $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians.
> **Example:** Identify the arc length of a circle with a central angle of 120° and radius of 8 cm.
* **Segment Area:** The area of a region of a circle which is "cut off" from the rest of the circle by a secant or chord. Area of a segment = Area of sector - Area of triangle.
> **Example:** A circle of radius 5.7 cm has a segment subtended by a central angle of 70°. Compute the segment area.
* **Concentric Circles:** Regions between two circles that share the same center.
> **Example:** Which of the following terms refers to the region between two concentric circles?
* **Cones:**
* **Frustum Volume:** The volume of a portion of a cone between two parallel planes.
$$V = \frac{1}{3}\pi h (R^2 + Rr + r^2)$$
where $h$ is the altitude, $R$ is the radius of the lower base, and $r$ is the radius of the upper base.
> **Example:** What is the volume of a frustum of a cone whose upper base is 15 cm in diameter and lower base 10 cm in diameter with an altitude of 25 cm?
* **Lateral Area:** The area of the side surface of a cone (excluding the base).
$$A_{lateral} = \pi r l$$
where $r$ is the radius of the base and $l$ is the slant height.
> **Example:** The slant height of a right circular cone is 5 m long. The base diameter is 6 m. What is the lateral area?
* **Cylinders:**
* **Volume:** $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height.
> **Example:** What is the volume of a cylinder in US gallons if the cylinder is 8 ft tall and has a 23-inch radius?
* **Pyramids:**
* **Total Surface Area:** The sum of the areas of all faces, including the base. For a square pyramid, $A_{total} = s^2 + 2sl$, where $s$ is the base edge length and $l$ is the slant height.
> **Example:** What is the total surface area of a square pyramid with a slant height of 12 cm and a base edge of 7 cm?
* **Polygons:**
* **Regular Dodecagon:** A polygon with 12 equal sides and 12 equal angles. The perimeter of a regular n-gon inscribed in a circle of radius R is $P = 2nR \sin(\frac{\pi}{n})$.
> **Example:** A regular dodecagon is inscribed in a circle of radius 24. Find the perimeter of the dodecagon.
* **Regular Pentagon:** A polygon with 5 equal sides and 5 equal angles. The measure of one interior angle is $(n-2) \times 180^\circ / n$.
> **Example:** What is the measure of one interior angle of a regular pentagon?
#### 1.3.2 Coordinate Geometry
* **Distance Formula:** The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
> **Example:** A ship travels from point A (1,5) to point B (6,8). What is the total distance traveled?
#### 1.3.3 Trigonometry
* **Angles of Elevation and Depression:** Used in problems involving heights and distances.
> **Example:** A man finds the angle of elevation of the top of a tower to be 30°. He walks 85 m nearer the tower and finds its angle of elevation to be 60°. What is the height of the tower?
#### 1.3.4 Solid Geometry and Volumes
* **Cylindrical Tanks:** Problems involving filling or determining the volume of fluid in tanks.
> **Example:** A closed cylindrical tank is 8 ft long and 3 ft in diameter. When lying horizontally, the water is 2 ft deep. If the tank is in the vertical position, what is the depth of the water?
* **Rectangular Containers:** Calculating volume and pressure.
> **Example:** A closed rectangular container with dimensions l = 4 m, w = 5 m, and h = 6 m is filled with water. What is the pressure exerted by the water on the bottom face of the container?
#### 1.3.5 Fluid Mechanics
* **Pressure:** Force per unit area. In fluids, pressure increases with depth.
* Pressure ($P$) = density ($\rho$) $\times$ acceleration due to gravity ($g$) $\times$ height ($h$).
* Specific Gravity ($SG$) = density of substance / density of water.
* Pressure Head: The height of a fluid column that would exert a certain pressure.
> **Example:** A closed cylindrical container is filled with a fluid that has a specific gravity of 1.7. What is the pressure exerted by this fluid at a depth of 15 m?
> **Example:** A pump is required to move oil (SG = 0.8) through a pipeline. The pump must deliver the oil at a pressure of 150 kPa (gauge). The frictional losses in the pipe are 3 meters of oil, and the discharge point is 5 meters above the pump. Calculate the total dynamic head.
> **Example:** A tank that is open above contains a 3-m head of water. Above the water is oil with a head of 2-m and a specific gravity of 0.8. Calculate the total pressure exerted at the bottom of the tank.
> **Example:** The static water pressure at the water meter is 80 psi. At a supply outlet 40 feet above the meter, what is the approximate static water pressure?
* **Velocity of Efflux:** The speed at which a fluid exits an opening. Torricelli's law states that the velocity of efflux from a sharp-edged hole at the bottom of a tank filled to a depth $h$ is $v = \sqrt{2gh}$.
> **Example:** A 3-m high tank is open to the atmosphere and is filled with water. The tank has a spout at the bottom. What is the velocity of the water coming from the spout?
* **Flow Rate:** The volume of fluid passing a point per unit time. $Q = Av$, where $A$ is the cross-sectional area and $v$ is the velocity. The principle of continuity states that for incompressible fluids, $A_1v_1 = A_2v_2$.
> **Example:** Water flows into a horizontal pipe that has an area of 3 sq.cm at point A, and an area of 7 sq.cm at point B. The velocity at point A is 2 m/s, and the pressure is 120,000 Pa. Calculate the pressure at point B.
> **Example:** Water with velocity of 7 fps flows through a 10-inch pipe. What is the flow rate?
> **Example:** An 8-m high tank is full of water and is open to the atmosphere. The water seeps through a 1 sq.cm hole in the bottom. What is the rate of the water leak from the hole?
* **Buoyancy and Specific Gravity:**
* Archimedes' Principle: A body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body.
* Apparent Weight = Weight in Air - Buoyant Force.
* Specific Gravity ($SG$) of an object = Weight in air / (Weight in air - Weight in water) = Density of object / Density of water.
> **Example:** An object's weight in air is 1.5 N. When submerged, it is 1.1 N. What is the SG of the object?
> **Example:** Find the density of an object if 2/3 of it is floating in water.
#### 1.3.6 Mechanics and Forces
* **Centripetal Force:** The force required to keep an object moving in a circular path. $F_c = \frac{mv^2}{r}$, where $m$ is mass, $v$ is velocity, and $r$ is the radius of the circular path.
> **Example:** A 1000 kg vehicle turns around at a rotunda of 200 m radius. If the centripetal force is 700 N, calculate the vehicle's speed.
* **Levers:** Problems involving the use of a lever to lift an object. The principle of moments states that for equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. Moment = Force $\times$ Distance from fulcrum.
> **Example:** Philip uses a 72-inch metal rod to lift a 500 lbs engine. The fulcrum is 12 inches from the load. How much force does he need to exert to lift the engine?
* **Work and Energy:**
* **Kinetic Energy:** The energy of motion. $KE = \frac{1}{2}mv^2$.
> **Example:** What happens to the kinetic energy of a body if its velocity is doubled?
* **Energy Dissipated:** In an electrical circuit, energy dissipated in a resistor is given by $E = VIt = \frac{V^2}{R}t = I^2Rt$, where $V$ is voltage, $I$ is current, $R$ is resistance, and $t$ is time.
> **Example:** A 13 V device is connected across a 7-ohm resistor for 8 seconds. How much energy in Joules is dissipated in the resistor?
* **Forces:** Vector addition of forces.
> **Example:** Three forces act on a single point. The magnitudes of Forces A, B, and C are 90, 45, and 120 respectively. Force A is 80° from the x-axis, Force B acts west, and Force C is -30° from the x-axis. Find the resultant.
> **Example:** Forces 12 N east and 27 N north are perpendicular to each other. Find the direction of the resultant.
* **Mass and Density:** Density ($\rho$) = Mass ($m$) / Volume ($V$).
> **Example:** An asphalt block with density of 2360 kg/m³ weighs 100 N in air. What is the apparent weight of the block in water?
* **Atomic Mass Unit:**
> **Example:** One atom mass unit is _____ in grams.
#### 1.3.7 Property Damage and Legal Terms
* **Wrongful Act:** An action that causes damage to another person's property or reputation.
### 1.4 Number Theory
This subsection deals with properties of integers.
#### 1.4.1 Divisibility
* **Divisibility Rules:** Determining if a number is divisible by another number.
> **Example:** If p – 10 is divisible by 6, then which one of the following must also be divisible by 6?
> **Example:** How many 3-digit numbers are divisible by 5, have no repeating digits, and have no zero digits?
#### 1.4.2 Least Common Multiple (LCM)
* **LCM of Prime Numbers:** The least common multiple of two prime numbers is their product.
> **Example:** Which one is true about the LCM of two prime numbers?
### 1.5 Probability and Statistics
This subsection covers the study of chance and data.
#### 1.5.1 Probability
* **Independent and Dependent Events:** Calculating the probability of multiple events occurring.
> **Example:** If you roll two fair six-sided dice, what is the probability that the sum of the numbers rolled is either 7 or 11?
> **Example:** What is the probability of drawing two consecutive face cards from a deck without replacement?
#### 1.5.2 Set Theory and Surveys
* **Venn Diagrams:** Used to represent relationships between sets and solve problems involving overlapping groups.
> **Example:** In a survey of 500 people, 300 play the piano, 200 play the drums, and 100 play both instruments. How many respondents play neither piano nor drums?
### 1.6 Other Mathematical Concepts
#### 1.6.1 Conic Sections
* **Eccentricity:** A parameter that describes the shape of a conic section. For a hyperbola, the eccentricity is greater than 1.
> **Example:** Determine the conic section if the eccentricity of the curve is more than 1.
#### 1.6.2 Calculus
* **Derivatives:** The rate of change of a function.
> **Example:** Given $f(x) = \frac{x^2 - 3x + 2}{x - 1}$, compute the derivative at $x = 2$.
* **Imaginary Numbers:** Numbers involving the imaginary unit $i$, where $i^2 = -1$.
> **Example:** When an imaginary number is raised to an even exponent, it becomes what?
#### 1.6.3 Inequalities
* **Range of Values:** Describing the possible range of values for an expression involving variables with given constraints.
> **Example:** If 2 < x < 5 and 3 < y < 5, what best describes x – y?
#### 1.6.4 Trigonometric Identities
* **Double Angle Formulas:** Identities involving trigonometric functions of twice an angle.
> **Example:** What is equal to sin(2x)?
#### 1.6.5 Physics Concepts Related to Mathematics
* **Free Fall:** The motion of an object under the sole influence of gravity. When two objects are dropped from the same height, neglecting air resistance, they will hit the ground at the same time.
> **Example:** Ball A is dropped vertically, while Ball B is launched horizontally from the same height at the same time. Neglect air resistance. What happens to Ball A?
* **Similar Triangles:** Triangles with the same shape but possibly different sizes. Their corresponding angles are equal, and the ratios of their corresponding sides are equal.
> **Example:** A vertical pole 6 ft high casts a shadow 4 ft long. At the same time a tree casts a shadow 64 ft long. What is the height of the tree?
#### 1.6.6 Business and Economics Terms
* **Funds for Operations:** The money required to maintain a business's ongoing activities.
* **Partnership:** A business association involving two or more individuals.
* **Performance Bonds:** A type of guarantee in construction contracts ensuring work is completed as agreed.
* **Illegal Payments:** Payments made for work not performed or using excessive labor.
* **Business Organization:** An entity created to provide goods and services, contribute to the economy, and create jobs.
* **Marketplace:** A venue where buyers and sellers meet.
#### 1.6.7 Photogrammetry
* **Relief Displacement:** The apparent shift in the position of an object in a photograph due to its elevation. This is used to determine the height of objects in aerial photographs.
> **Example:** A vertical photograph of a chimney was taken from an elevation of 500 m above M.S.L. The elevation of the base of the chimney was 250 m. If the relief displacement of the chimney was 51.4 mm and the radial distance of the image of the top of the chimney was 110 mm, what is the height of the chimney?
#### 1.6.8 Map Scales
* **Real Length Calculation:** Using a map scale to determine the actual distance of an object. A scale of 1:X means 1 unit on the map represents X units in reality.
> **Example:** A pipeline is shown as 0.20 m on a map with a scale of 1:150,000 mm. What is the real length of the pipeline?
#### 1.6.9 Clock Problems
* **Relative Speed of Clock Hands:** Problems involving the positions of the hour and minute hands of a clock.
> **Example:** In how many minutes after 9 pm will the hands of the clock be together for the first time?
#### 1.6.10 General Properties
* **Scaling Effects:** Understanding how changes in dimensions affect area and perimeter.
> **Example:** If the perimeter of a square doubles, what happens to its area?
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# Business and Finance terms
This section covers essential terminology related to business organizations, financial investments, profit, and economic concepts, providing definitions and examples to solidify understanding.
### 3.1 Business organizations and concepts
Various forms of business structures exist, each with distinct characteristics and purposes.
#### 3.1.1 Partnership
An association of two or more persons for the purpose of engaging in a profitable business is called a partnership.
#### 3.1.2 General Business Entity
A business organization that is formed and provides goods and services, creates jobs, contributes to national income, imports, exports, and sustainable economic development is a general business entity.
#### 3.1.3 Situations of Inefficiency
* **Situation of inefficiency:** A situation whereby a payment is made for work not done. It also applies to a case where more workers are used than reasonably required for efficient operation.
### 3.2 Financial terms and investments
Understanding financial terms is crucial for comprehending investment strategies, profit calculations, and economic principles.
#### 3.2.1 Investment and Interest
* **Simple Interest:** Interest calculated only on the initial principal amount.
* Example: Carol and Mike invested 18,000 dollars each at simple interest. Carol's bank offers 4% per annum, while Mike's bank offers 6% per annum. After how many years will Mike's interest be 5,400 dollars more than Carol's?
* Let $P$ be the principal amount (18,000 dollars), $r_C$ be Carol's interest rate (0.04), and $r_M$ be Mike's interest rate (0.06). Let $t$ be the number of years.
* Carol's interest: $I_C = P \cdot r_C \cdot t = 18000 \cdot 0.04 \cdot t$
* Mike's interest: $I_M = P \cdot r_M \cdot t = 18000 \cdot 0.06 \cdot t$
* We want to find $t$ such that $I_M = I_C + 5400$.
* $18000 \cdot 0.06 \cdot t = (18000 \cdot 0.04 \cdot t) + 5400$
* $1080t = 720t + 5400$
* $360t = 5400$
* $t = \frac{5400}{360} = 15$ years.
* **Simple Interest with Deducted Interest:** When interest is deducted from the loan amount at the time of borrowing.
* Example: A man borrows 10,000 dollars from a loan firm. The rate of simple interest is 15%, but the interest is to be deducted from the loan at the time the money is borrowed. At the end of one year, he has to pay back 10,000 dollars. What is the actual rate of interest?
* Interest deducted: $10000 \cdot 0.15 = 1500$ dollars.
* Actual amount borrowed: $10000 - 1500 = 8500$ dollars.
* The man pays back 10,000 dollars, so the interest paid is $10000 - 8500 = 1500$ dollars.
* Actual rate of interest: $\frac{1500}{8500} \cdot 100\% \approx 17.65\%$.
* **Compound Interest:** Interest calculated on the initial principal and also on the accumulated interest from previous periods.
* Example: A certain amount of money was invested at compound interest. It amounted to 15,000 dollars at the end of the 4th year, and 18,000 dollars at the end of the 8th year. Find the rate of interest per annum.
* Let the principal be $P$ and the rate of interest be $r$.
* Amount after 4 years: $P(1+r)^4 = 15000$
* Amount after 8 years: $P(1+r)^8 = 18000$
* Dividing the second equation by the first: $\frac{P(1+r)^8}{P(1+r)^4} = \frac{18000}{15000}$
* $(1+r)^4 = 1.2$
* $1+r = (1.2)^{1/4}$
* $r = (1.2)^{1/4} - 1 \approx 1.0466 - 1 = 0.0466$ or 4.66%.
* **Final Amount Calculation:** The total amount after a period, including principal and interest.
* Example: Compute the final amount at the end of the year if you loaned 120,000 dollars with 6.5% annual simple interest.
* Interest: $120000 \cdot 0.065 \cdot 1 = 7800$ dollars.
* Final Amount: $120000 + 7800 = 127800$ dollars.
* Example: After 10 years, how much will be the final amount of an initial deposit of 58,000 dollars if it is compounding monthly at a rate of 8%?
* Here, we would need to use the compound interest formula for monthly compounding. The annual rate is 8%, so the monthly rate is $\frac{0.08}{12}$. The number of periods is $10 \times 12 = 120$.
* The formula is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times that interest is compounded per year, and $t$ is the number of years.
* $A = 58000 \left(1 + \frac{0.08}{12}\right)^{120}$
* $A \approx 58000 (1.006667)^{120} \approx 58000 \cdot 2.2196 \approx 128736.80$ dollars.
#### 3.2.2 Profit and Loss
* **Loss:** Selling an item for less than its cost.
* Example: A distributor sold the latest apple phone for 75,000 dollars at a loss of 10%. If the distributor wanted to gain 15% instead, what selling price should the distributor have set?
* Cost Price ($CP$): If selling price ($SP$) is 75,000 dollars and there is a 10% loss, then $SP = CP(1 - 0.10)$.
* $75000 = CP(0.90)$
* $CP = \frac{75000}{0.90} = 83333.33$ dollars.
* Desired Selling Price for 15% gain: $SP_{new} = CP(1 + 0.15) = 83333.33 \cdot 1.15 = 95833.33$ dollars.
#### 3.2.3 Financial Needs
* **Working capital:** The funds that are required to make the enterprise or project a going concern.
#### 3.2.4 Market Concepts
* **Market:** Refers to the place where buyers and sellers gather to facilitate the exchange of goods and services.
### 3.3 Economic concepts
Economic principles govern the production, distribution, and consumption of goods and services.
#### 3.3.1 Inflation and Currency
* **Inflation:** A general increase in prices and a fall in the purchasing value of money. (Implied through interest rate discussions where purchasing power diminishes over time).
#### 3.3.2 Economic Development
* **Sustainable economic development:** Economic growth that meets the needs of the present without compromising the ability of future generations to meet their own needs.
### 3.4 Basic Mathematics in Business and Finance
Several mathematical concepts are fundamental to business and finance applications.
#### 3.4.1 Ratios and Proportions
* **Ratio of legs in a right triangle:** Used to determine the ratio of the hypotenuse to the shorter side.
* Example: If the ratio of the two legs of a right triangle is 5:12, what is the ratio of the hypotenuse to the shorter side?
* Let the legs be $5x$ and $12x$. The hypotenuse $h$ can be found using the Pythagorean theorem: $h^2 = (5x)^2 + (12x)^2 = 25x^2 + 144x^2 = 169x^2$.
* $h = \sqrt{169x^2} = 13x$.
* The ratio of the hypotenuse to the shorter side ($5x$) is $\frac{13x}{5x} = \frac{13}{5}$.
#### 3.4.2 Averages and Statistics
* **Average price:** The sum of prices divided by the number of items.
* Example: A retailer has 10 items with an average price of 120 dollars each. He threw one item, and the new average is 115 dollars. What was the price of the removed item?
* Total price of 10 items: $10 \times 120 = 1200$ dollars.
* Total price of 9 items: $9 \times 115 = 1035$ dollars.
* Price of the removed item: $1200 - 1035 = 165$ dollars.
#### 3.4.3 Set Theory in Business Contexts
* **Venn Diagrams:** Used to represent overlapping sets, such as customer preferences or product ownership.
* Example: In a survey of 500 people, 300 play the piano, 200 play the drums, and 100 play both instruments. How many respondents play neither piano nor drums?
* Let $P$ be the set of people who play piano, and $D$ be the set of people who play drums.
* $|P| = 300$, $|D| = 200$, $|P \cap D| = 100$.
* The number of people who play at least one instrument is $|P \cup D| = |P| + |D| - |P \cap D| = 300 + 200 - 100 = 400$.
* The number of people who play neither is the total number of people minus those who play at least one instrument: $500 - 400 = 100$.
#### 3.4.4 Combinatorics in Business
* **Permutations:** Used for arrangements where order matters, such as awarding prizes.
* Example: 50 people joined a body-building competition. In how many ways can gold, silver, and bronze be awarded?
* This is a permutation problem as the order of awarding matters.
* The number of permutations of $n$ items taken $k$ at a time is given by $P(n, k) = \frac{n!}{(n-k)!}$.
* Here, $n=50$ and $k=3$.
* $P(50, 3) = \frac{50!}{(50-3)!} = \frac{50!}{47!} = 50 \times 49 \times 48 = 117,600$ ways.
#### 3.4.5 Probability in Business
* **Probability of events:** The likelihood of specific outcomes occurring.
* Example: If you roll two fair six-sided dice, what is the probability that the sum of the numbers rolled is either 7 or 11?
* Total possible outcomes when rolling two dice: $6 \times 6 = 36$.
* Outcomes that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 outcomes.
* Outcomes that sum to 11: (5,6), (6,5) - 2 outcomes.
* The probability of the sum being 7 is $\frac{6}{36}$. The probability of the sum being 11 is $\frac{2}{36}$.
* Since these are mutually exclusive events, the probability of either occurring is the sum of their probabilities: $\frac{6}{36} + \frac{2}{36} = \frac{8}{36} = \frac{2}{9}$.
* Example: What is the probability of drawing two consecutive face cards from a deck without replacement?
* A standard deck has 52 cards, with 12 face cards (Jack, Queen, King in each of the 4 suits).
* Probability of drawing the first face card: $\frac{12}{52}$.
* After drawing one face card, there are 11 face cards left and 51 total cards.
* Probability of drawing the second face card (given the first was a face card): $\frac{11}{51}$.
* The probability of both events occurring is the product of their probabilities: $\frac{12}{52} \times \frac{11}{51} = \frac{3}{13} \times \frac{11}{51} = \frac{1}{13} \times \frac{11}{17} = \frac{11}{221}$.
#### 3.4.6 Algebraic Expressions and Equations
* **Consecutive Integers:** Integers that follow each other in order.
* Example: Let a, b, c, d, e be five consecutive integers in increasing order. Which of the following expressions is always even?
* Let the integers be $x, x+1, x+2, x+3, x+4$.
* Consider expressions like their sum: $x + (x+1) + (x+2) + (x+3) + (x+4) = 5x + 10 = 5(x+2)$. This expression is always divisible by 5.
* Consider the difference between two consecutive terms: $(x+1) - x = 1$ (odd).
* Consider the sum of two consecutive integers: $x + (x+1) = 2x+1$ (always odd).
* Consider the sum of three consecutive integers: $x + (x+1) + (x+2) = 3x+3 = 3(x+1)$. This is always divisible by 3.
* Consider the sum of four consecutive integers: $x + (x+1) + (x+2) + (x+3) = 4x+6 = 2(2x+3)$. This is always even.
* Consider the sum of five consecutive integers: $x + (x+1) + (x+2) + (x+3) + (x+4) = 5x+10$. If $x$ is even, $5(\text{even})+10 = \text{even} + \text{even} = \text{even}$. If $x$ is odd, $5(\text{odd})+10 = \text{odd} + \text{even} = \text{odd}$. So the sum of five consecutive integers is not always even.
* The expression $a+c+e$ would be $x + (x+2) + (x+4) = 3x+6 = 3(x+2)$. This is always divisible by 3.
* The expression $a+e$ would be $x + (x+4) = 2x+4 = 2(x+2)$. This is always even.
* **Divisibility:** An integer $a$ is divisible by an integer $b$ if the remainder of the division $a/b$ is zero.
* Example: If $p - 10$ is divisible by 6, then which one of the following must also be divisible by 6?
* If $p - 10 = 6k$ for some integer $k$, then $p = 6k + 10$.
* Consider an expression like $p - 4$: $(6k + 10) - 4 = 6k + 6 = 6(k+1)$, which is divisible by 6.
* **Relationship between variables:** Expressing one variable in terms of another.
* Example: If 4 more than x is 2 times y, what is the value of y in terms of x?
* "4 more than x" can be written as $x+4$.
* "2 times y" can be written as $2y$.
* So, $x+4 = 2y$.
* To find $y$ in terms of $x$, divide both sides by 2: $y = \frac{x+4}{2}$.
* **Roots of a quadratic equation:** The values of the variable that satisfy the equation.
* Example: Find the roots of .
* We can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
* Here, $a=1$, $b=-7$, $c=12$.
* $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(12)}}{2(1)}$
* $x = \frac{7 \pm \sqrt{49 - 48}}{2}$
* $x = \frac{7 \pm \sqrt{1}}{2}$
* $x_1 = \frac{7+1}{2} = \frac{8}{2} = 4$
* $x_2 = \frac{7-1}{2} = \frac{6}{2} = 3$.
* The roots are 3 and 4.
#### 3.4.7 Financial Planning and Annuities
* **Deposits and Withdrawals:** Planning for future financial needs by saving and withdrawing money.
* Example: A student plans to deposit 1500 dollars in the bank now and another 3000 dollars for the next two years. If he plans to withdraw 5000 dollars three years after his last deposit, what will be the amount of money left in the bank after one year of his withdrawal? (Effective annual interest rate is 10%.)
* Initial deposit: 1500 dollars (at time 0).
* Deposit 1: 3000 dollars (at time 1).
* Deposit 2: 3000 dollars (at time 2).
* Withdrawal: 5000 dollars (at time $2+3 = 5$).
* We need to find the amount at time $5+1 = 6$. The interest rate is 10% per year.
* Value of initial deposit at time 6: $1500 \times (1.10)^6$
* Value of deposit 1 at time 6: $3000 \times (1.10)^5$
* Value of deposit 2 at time 6: $3000 \times (1.10)^4$
* Value of withdrawal at time 6: $5000 \times (1.10)^1$
* Amount left at time 6 = $(1500 \times (1.10)^6) + (3000 \times (1.10)^5) + (3000 \times (1.10)^4) - (5000 \times (1.10)^1)$
* Calculate the powers of 1.10: $(1.10)^1 = 1.1$, $(1.10)^4 \approx 1.4641$, $(1.10)^5 \approx 1.61051$, $(1.10)^6 \approx 1.771561$.
* Amount left $\approx (1500 \times 1.771561) + (3000 \times 1.61051) + (3000 \times 1.4641) - (5000 \times 1.1)$
* Amount left $\approx 2667.34 + 4831.53 + 4392.3 - 5500$
* Amount left $\approx 6391.17$ dollars.
#### 3.4.8 Annuity Calculations
* **Annuity:** A series of equal payments made at regular intervals.
* Example: Pipe A fills a tank in 5 hours, Pipe B in 8 hours, and Pipe C in 12 hours. Pipe C is closed for the first hour, then all three work together. How long will it take to fill the tank?
* Rate of Pipe A: $\frac{1}{5}$ tank per hour.
* Rate of Pipe B: $\frac{1}{8}$ tank per hour.
* Rate of Pipe C: $\frac{1}{12}$ tank per hour.
* In the first hour, only Pipe C works, filling $\frac{1}{12}$ of the tank.
* Remaining capacity to fill: $1 - \frac{1}{12} = \frac{11}{12}$ of the tank.
* When all three work together, their combined rate is $\frac{1}{5} + \frac{1}{8} + \frac{1}{12} = \frac{24+15+10}{120} = \frac{49}{120}$ tank per hour.
* Time to fill the remaining capacity: $\frac{\text{Remaining Capacity}}{\text{Combined Rate}} = \frac{11/12}{49/120} = \frac{11}{12} \times \frac{120}{49} = \frac{11 \times 10}{49} = \frac{110}{49}$ hours.
* Total time = 1 hour (first hour) + $\frac{110}{49}$ hours.
* Total time $= \frac{49}{49} + \frac{110}{49} = \frac{159}{49} \approx 3.24$ hours.
### 3.5 Economic Terms
Key economic terms define the principles and practices of economies.
#### 3.5.1 Goods and Services Exchange
* **Market:** The place where buyers and sellers meet to exchange goods and services.
#### 3.5.2 Business Growth and Development
* **Business organization:** A structured entity designed for profitable operations, job creation, and economic contribution.
#### 3.5.3 Economic Impact
* **Contribution to national income:** A measure of a business's role in the overall economy, including its impact on imports, exports, and development.
#### 3.5.4 Unsustainable Practices
* **Work for no output:** A situation where payment is made for work that is not performed or for labor in excess of what is reasonably required. This can lead to economic inefficiency.
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# Abstract and conceptual questions
This section addresses questions requiring definitions of abstract concepts, logical reasoning, or application of principles in word problem scenarios, covering topics from basic arithmetic and geometry to physics and finance.
### 4.1 Fundamental Concepts and Definitions
* **Least Common Multiple (LCM) of prime numbers:** The LCM of two prime numbers is their product.
* **Evil wrong:** An evil wrong is an act committed by a person that damages another person's property or reputation.
* **Funds for going concern:** The funds required to make an enterprise or project a going concern are called working capital.
* **Association for profit:** An association of two or more persons for the purpose of engaging in a profitable business is called a partnership.
* **Bond guaranteeing performance:** This bond guarantees that a contractor will perform the work according to the conditions and requirements of the construction contract.
* **Payment for work not done:** A situation whereby a payment is made for work not done. It also applies to a case where more workers are used than reasonably required for efficient operation. This is known as featherbedding.
* **Place for exchange:** Refers to the place where buyers and sellers gather to facilitate the exchange of goods and services. This is a market.
* **Business organization contributing to economy:** A business organization that is formed and provides goods and services, creates jobs, contributes to national income, imports, exports, and sustainable economic development is called a firm.
* **An imaginary number raised to an even exponent:** When an imaginary number is raised to an even exponent, it becomes a real number.
### 4.2 Mathematical and Geometric Problems
#### 4.2.1 Arithmetic and Algebra
* **Simple Interest Calculations:**
* **Finding time for interest difference:** If Carol and Mike invested the same amount at simple interest with different rates (Carol at 4% per annum, Mike at 6% per annum), the time ($t$) in years for Mike's interest to be a specific amount more than Carol's can be calculated. Let $P$ be the principal amount each invested.
* Carol's interest: $I_C = P \times 0.04 \times t$
* Mike's interest: $I_M = P \times 0.06 \times t$
* We need $I_M - I_C = 5400$.
* $P \times 0.06 \times t - P \times 0.04 \times t = 5400$
* $P \times t \times (0.06 - 0.04) = 5400$
* $P \times t \times 0.02 = 5400$
* **Calculating actual interest rate with deduction:** If a loan of a principal amount ($P$) is taken, and the simple interest is deducted upfront at a given rate ($R$), the actual rate of interest ($R_{actual}$) is higher. If the interest is deducted at the time of borrowing, and the full principal is to be paid back at the end of one year, the effective principal received is $P - (P \times R \times 1)$. The actual interest paid is $P \times R \times 1$.
* $R_{actual} = \frac{\text{Actual Interest Paid}}{\text{Actual Principal Received}} \times 100\%$
* $R_{actual} = \frac{P \times R}{P - (P \times R)} \times 100\% = \frac{P \times R}{P(1-R)} \times 100\% = \frac{R}{1-R} \times 100\%$
* **Work Problems:**
* **Delayed project with worker changes:** For a job done by a certain number of workers in a target time, if workers leave at different stages, the total work done can be calculated as the sum of work done by the initial group and the subsequent groups.
* Total work = (Number of workers $\times$ Days worked) for each phase.
* Work done = Rate of work $\times$ Time. If 1 unit of work is done by $N$ workers in $D$ days, then the rate of one worker is $\frac{1}{N \times D}$.
* If $W$ workers complete a job in $D$ days, the total work is $W \times D$ worker-days.
* When the number of workers changes, the total work is the sum of (workers $\times$ days) for each segment.
* Delayed days = Actual days to complete - Target days.
* **Age Problems:**
* **Ratio of ages:** Problems involving the ratio of ages at different times can be solved by setting up equations based on the given ratios and time intervals.
* Let the son's current age be $s$ and the father's current age be $f$.
* If the father is aged four times more than his son, this implies $f = 5s$. (This phrasing can be ambiguous; sometimes it means $f=4s$. Assuming "four times more" means the father's age is the son's age plus four times the son's age, i.e., $f = s + 4s = 5s$).
* After 8 years: Father's age = $f+8$, Son's age = $s+8$.
* If $f+8 = 2.5(s+8)$.
* Substitute $f=5s$: $5s+8 = 2.5s + 20 \implies 2.5s = 12 \implies s = 4.8$ years. So, father's age is $5 \times 4.8 = 24$ years.
* After further 8 years (16 years from the start): Father's age = $f+16$, Son's age = $s+16$. Calculate the new ratio. $24+16 = 40$, $4.8+16 = 20.8$. Ratio is $40:20.8 = 1.92...$ This suggests the initial interpretation of "four times more" might need adjustment based on the expected outcome. If it means $f=4s$:
* $4s+8 = 2.5(s+8) \implies 4s+8 = 2.5s + 20 \implies 1.5s = 12 \implies s=8$. Father's age = $4 \times 8 = 32$.
* After further 8 years: Father's age = $32+16 = 48$, Son's age = $8+16 = 24$. Ratio = $48:24 = 2$. The father would be 2 times his son's age.
* **Fractions and Proportions:**
* **Cake division:** If Tommy ate 1/3 of the cake, the remaining cake is $1 - \frac{1}{3} = \frac{2}{3}$. If Sam ate 3/4 of the remaining cake, Sam ate $\frac{3}{4} \times \frac{2}{3} = \frac{2}{4} = \frac{1}{2}$ of the whole cake. The fraction eaten by Paulie is the remaining portion: $\frac{2}{3} - \frac{1}{2} = \frac{4-3}{6} = \frac{1}{6}$ of the whole cake.
* **Consecutive Integers:**
* Let five consecutive integers in increasing order be $a, b, c, d, e$. We can represent them as $n, n+1, n+2, n+3, n+4$.
* Consider sums/differences of these. For example, $a+b = n + (n+1) = 2n+1$ (odd). $a+c = n + (n+2) = 2n+2$ (even).
* A sum of two consecutive integers is always odd.
* A sum of an even number of consecutive integers is even. A sum of an odd number of consecutive integers has the same parity as the sum of the first and last integer.
* Consider $a+e = n + (n+4) = 2n+4$ (even).
* Consider $b+d = (n+1) + (n+3) = 2n+4$ (even).
* The sum of all five integers: $n + (n+1) + (n+2) + (n+3) + (n+4) = 5n + 10$. If $n$ is even, $5n$ is even, so $5n+10$ is even. If $n$ is odd, $5n$ is odd, so $5n+10$ is odd. This expression is not always even.
* Consider the sum of two numbers with opposite parity: odd. The sum of two numbers with the same parity: even.
* Among five consecutive integers, there are either 2 even and 3 odd, or 3 even and 2 odd.
* Any expression of the form $2k$ is always even. Expressions like $a+e$, $b+d$ are always even.
* **Number Theory (Divisibility):**
* If $p-10$ is divisible by 6, then $p-10 = 6k$ for some integer $k$. This means $p = 6k + 10$.
* We need to check which expression involving $p$ must also be divisible by 6.
* Consider $p$: $6k+10$ is not always divisible by 6 (e.g., if $k=1$, $p=16$).
* Consider $p+1$: $6k+11$ is not divisible by 6.
* Consider $p+2$: $6k+12 = 6(k+2)$. This is always divisible by 6.
#### 4.2.2 Geometry and Trigonometry
* **Perimeter and Area of Geometric Shapes:**
* **Football field perimeter:** A regulation football field measures 120 yards in length and approximately 53.3 yards in width. The perimeter is $2 \times (\text{length} + \text{width})$.
* Perimeter = $2 \times (120 \text{ yards} + 53.3 \text{ yards}) = 2 \times 173.3 \text{ yards} = 346.6 \text{ yards}$.
* **Square and Triangle rope problem:** A child uses a rope of 1.2 meters to make a square and a triangle. The rope used for the square is 2 times more than that for the triangle. Let the length of the rope for the triangle be $L_T$ and for the square be $L_S$.
* $L_T + L_S = 1.2$ m
* $L_S = 2 L_T$
* Substitute the second equation into the first: $L_T + 2 L_T = 1.2 \implies 3 L_T = 1.2 \implies L_T = 0.4$ m.
* Then $L_S = 2 \times 0.4 = 0.8$ m.
* If the square is made from 0.8 m of rope, the length of one side of the square is $\frac{L_S}{4}$.
* Side of square = $\frac{0.8 \text{ m}}{4} = 0.2$ m.
* **Arc length of a circle:** The arc length ($s$) of a circle with radius ($r$) and a central angle ($\theta$) in degrees is given by $s = \frac{\theta}{360^\circ} \times 2\pi r$.
* Given $\theta = 120^\circ$ and $r = 8$ cm.
* $s = \frac{120^\circ}{360^\circ} \times 2\pi (8 \text{ cm}) = \frac{1}{3} \times 16\pi \text{ cm} = \frac{16\pi}{3}$ cm.
* **Volume of a frustum of a cone:** The volume ($V$) of a frustum of a cone with upper base radius ($r_1$), lower base radius ($r_2$), and height ($h$) is given by $V = \frac{1}{3}\pi h (r_1^2 + r_1 r_2 + r_2^2)$.
* Upper base diameter = 15 cm $\implies r_1 = 7.5$ cm.
* Lower base diameter = 10 cm $\implies r_2 = 5$ cm.
* Altitude $h = 25$ cm.
* $V = \frac{1}{3}\pi (25 \text{ cm}) ((7.5 \text{ cm})^2 + (7.5 \text{ cm})(5 \text{ cm}) + (5 \text{ cm})^2)$
* $V = \frac{25\pi}{3} (56.25 + 37.5 + 25) \text{ cm}^3 = \frac{25\pi}{3} (118.75) \text{ cm}^3$.
* **Area of a triangle with given vertices:** The area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ can be calculated using the formula: Area = $\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.
* Vertices A(2,5), B(5,10), C(5,-2).
* Area = $\frac{1}{2} |2(10 - (-2)) + 5(-2 - 5) + 5(5 - 10)|$
* Area = $\frac{1}{2} |2(12) + 5(-7) + 5(-5)| = \frac{1}{2} |24 - 35 - 25| = \frac{1}{2} |-36| = 18$.
* **Area of a triangle with two sides and included angle:** The area ($A$) of a triangle with sides $a, b$ and included angle $C$ is $A = \frac{1}{2}ab \sin C$.
* Sides 8 units and 12 units, included angle 93°.
* Area = $\frac{1}{2} \times 8 \times 12 \times \sin(93^\circ) = 48 \times \sin(93^\circ)$.
* **Area of a triangle (ambiguous case):** In triangle ABC, BC = 13.4 m, AC = 9 m, and $\angle A = 60^\circ$. To find $\angle C$, we can use the Law of Sines: $\frac{a}{\sin A} = \frac{c}{\sin C}$.
* $\frac{13.4}{\sin 60^\circ} = \frac{9}{\sin C}$
* $\sin C = \frac{9 \times \sin 60^\circ}{13.4} = \frac{9 \times \frac{\sqrt{3}}{2}}{13.4} \approx \frac{9 \times 0.866}{13.4} \approx \frac{7.794}{13.4} \approx 0.5816$.
* $C = \arcsin(0.5816) \approx 35.56^\circ$.
* **Side of an equilateral triangle inscribed in a circle:** If an equilateral triangle is inscribed in a circle with circumference $C_{circle}$, we first find the radius. $C_{circle} = 2\pi r$. Then, the side length ($s$) of an equilateral triangle inscribed in a circle of radius $r$ is $s = r\sqrt{3}$.
* Given circumference = 2086 mm.
* $2\pi r = 2086 \text{ mm} \implies r = \frac{2086}{2\pi} \text{ mm}$.
* Side of triangle = $\frac{2086}{2\pi} \times \sqrt{3} \text{ mm} = \frac{1043\sqrt{3}}{\pi}$ mm.
* **Area of a square given vertex coordinates:** The vertices are (0,6), (6,0), (0,-6), and (-6,0). These points form a square rotated by 45 degrees. The distance between (0,6) and (6,0) is $\sqrt{(6-0)^2 + (0-6)^2} = \sqrt{36+36} = \sqrt{72}$. The side length is $\sqrt{72}$. Area = $(\sqrt{72})^2 = 72$. Alternatively, the diagonals connect (0,6) to (0,-6) and (6,0) to (-6,0). The lengths of the diagonals are 12 and 12 respectively. The area of a rhombus (and thus a square) is $\frac{1}{2}d_1 d_2$. Area = $\frac{1}{2} \times 12 \times 12 = 72$.
* **Lateral area of a right circular cone:** The lateral area ($L.A.$) of a right circular cone with radius ($r$) and slant height ($l$) is $L.A. = \pi r l$.
* Base diameter = 6 m $\implies r = 3$ m.
* Slant height $l = 5$ m.
* $L.A. = \pi \times (3 \text{ m}) \times (5 \text{ m}) = 15\pi \text{ m}^2$.
* **Perimeter of a regular dodecagon:** A regular dodecagon has 12 equal sides. If it is inscribed in a circle of radius $R$, the side length ($s$) is given by $s = 2R \sin(\frac{180^\circ}{12}) = 2R \sin(15^\circ)$. The perimeter is $12s$.
* Radius $R = 24$.
* $s = 2 \times 24 \times \sin(15^\circ) = 48 \times \sin(15^\circ)$. Using $\sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \frac{1}{2} = \frac{\sqrt{6}-\sqrt{2}}{4}$.
* $s = 48 \times \frac{\sqrt{6}-\sqrt{2}}{4} = 12(\sqrt{6}-\sqrt{2})$.
* Perimeter = $12 \times 12(\sqrt{6}-\sqrt{2}) = 144(\sqrt{6}-\sqrt{2})$.
* **Region between two concentric circles:** The region between two concentric circles is called an annulus.
* **Total surface area of a square pyramid:** The total surface area ($T.S.A.$) of a square pyramid with base edge ($b$) and slant height ($l$) is $T.S.A. = \text{base area} + \text{lateral area}$.
* Base area = $b^2$.
* Lateral area = $4 \times (\frac{1}{2} \times b \times l) = 2bl$.
* Base edge $b = 7$ cm, slant height $l = 12$ cm.
* $T.S.A. = (7 \text{ cm})^2 + 2 \times (7 \text{ cm}) \times (12 \text{ cm}) = 49 \text{ cm}^2 + 168 \text{ cm}^2 = 217 \text{ cm}^2$.
* **Trigonometric Identities and Equations:**
* **Double angle identity:** $\sin(2x) = 2\sin x \cos x$.
* **Solving for angle:** (Covered under "Area of a triangle (ambiguous case)").
#### 4.2.3 Calculus
* **Derivatives:**
* **Compute the derivative at a point:** Given a function, find its derivative and evaluate it at a specific point. For example, if $f(x) = x^3 - 2x^2 + 5$.
* The derivative $f'(x) = 3x^2 - 4x$.
* If we need to compute the derivative at $x=2$, $f'(2) = 3(2)^2 - 4(2) = 3(4) - 8 = 12 - 8 = 4$.
* **Binomial Expansion:**
* **Find the constant term:** For a binomial expansion of the form $(ax+b)^n$ or $(ax - b/x)^n$, the constant term is the term that does not contain any variable. For $(x + \frac{2}{x})^6$.
* The general term in the expansion of $(a+b)^n$ is $\binom{n}{k} a^{n-k} b^k$.
* Here, $a=x$, $b=\frac{2}{x}$, $n=6$.
* The general term is $\binom{6}{k} x^{6-k} (\frac{2}{x})^k = \binom{6}{k} x^{6-k} \frac{2^k}{x^k} = \binom{6}{k} 2^k x^{6-2k}$.
* For the constant term, the exponent of $x$ must be 0: $6-2k = 0 \implies 2k = 6 \implies k=3$.
* The constant term is $\binom{6}{3} 2^3 = \frac{6!}{3!3!} \times 8 = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} \times 8 = 20 \times 8 = 160$.
#### 4.2.4 Combinatorics and Probability
* **Permutations for awarding prizes:** The number of ways to award gold, silver, and bronze medals to a group of people is a permutation problem, as the order matters. The formula is $P(n, k) = \frac{n!}{(n-k)!}$.
* 50 people, awarding 3 prizes.
* Number of ways = $P(50, 3) = \frac{50!}{(50-3)!} = \frac{50!}{47!} = 50 \times 49 \times 48$.
* **Probability of dice rolls:** The probability of rolling a sum of 7 or 11 with two fair six-sided dice.
* Total possible outcomes = $6 \times 6 = 36$.
* Outcomes for sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 outcomes.
* Outcomes for sum 11: (5,6), (6,5) - 2 outcomes.
* Probability (sum is 7 or 11) = Probability (sum is 7) + Probability (sum is 11) = $\frac{6}{36} + \frac{2}{36} = \frac{8}{36} = \frac{2}{9}$.
* **Probability of drawing cards without replacement:** The probability of drawing two consecutive face cards from a deck without replacement.
* A standard deck has 52 cards. There are 12 face cards (J, Q, K in 4 suits).
* Probability of drawing the first face card = $\frac{12}{52}$.
* After drawing one face card, there are 11 face cards left and 51 total cards.
* Probability of drawing the second face card = $\frac{11}{51}$.
* Probability of drawing two consecutive face cards = $\frac{12}{52} \times \frac{11}{51} = \frac{3}{13} \times \frac{11}{51} = \frac{1}{13} \times \frac{11}{17} = \frac{11}{221}$.
#### 4.2.5 Algebra and Functions
* **Expressing one variable in terms of another:** If 4 more than $x$ is 2 times $y$, we can write this as $x+4 = 2y$. To find $y$ in terms of $x$, divide by 2: $y = \frac{x+4}{2}$.
* **Relationship between variables with inequalities:** If $2 < x < 5$ and $3 < y < 5$, what best describes $x-y$?
* To find the range of $x-y$, we need to consider the minimum and maximum possible values.
* Maximum value of $x-y$: Occurs when $x$ is maximum and $y$ is minimum. $x_{max} = 5$, $y_{min} = 3$. So, $x-y < 5 - 3 = 2$.
* Minimum value of $x-y$: Occurs when $x$ is minimum and $y$ is maximum. $x_{min} = 2$, $y_{max} = 5$. So, $x-y > 2 - 5 = -3$.
* Therefore, $-3 < x-y < 2$.
* **Roots of a quadratic equation:** To find the roots of $x^2 - 2x + 4 = 0$, we use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
* Here, $a=1, b=-2, c=4$.
* $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)} = \frac{2 \pm \sqrt{4 - 16}}{2} = \frac{2 \pm \sqrt{-12}}{2}$.
* Since the discriminant is negative, the roots are complex. $\sqrt{-12} = \sqrt{12}i = 2\sqrt{3}i$.
* $x = \frac{2 \pm 2\sqrt{3}i}{2} = 1 \pm \sqrt{3}i$.
#### 4.2.6 Number Systems and Units
* **Conversions:**
* **Atom mass unit to grams:** One atom mass unit (amu) is approximately $1.660539 \times 10^{-24}$ grams.
* **Cylindrical volume to US gallons:** Volume of a cylinder is $V = \pi r^2 h$.
* Radius $r = 23$ inches, height $h = 8$ ft = $8 \times 12 = 96$ inches.
* $V = \pi (23 \text{ in})^2 (96 \text{ in}) = \pi \times 529 \times 96 \text{ in}^3 = 50784\pi \text{ in}^3$.
* 1 US gallon = 231 cubic inches.
* Volume in gallons = $\frac{50784\pi \text{ in}^3}{231 \text{ in}^3/\text{gallon}} \approx \frac{50784 \times 3.14159}{231}$ gallons $\approx 691.8$ US gallons.
* **Wire diameter to circular mils:** A wire with a diameter of $d$ in mils (thousandths of an inch) has a cross-sectional area of $d^2$ circular mils. 1 inch = 1000 mils.
* Diameter = 0.125 inch = $0.125 \times 1000$ mils = 125 mils.
* Cross-sectional area = $(125 \text{ mils})^2 = 15625$ circular mils.
#### 4.2.7 Financial Mathematics
* **Compound Interest:**
* **Finding the rate of interest:** If an amount invested at compound interest amounted to $A_1$ at the end of year $t_1$ and $A_2$ at the end of year $t_2$, the interest rate ($r$) can be found. The growth factor over $t_2 - t_1$ years is $\frac{A_2}{A_1}$.
* Let the principal be $P$. Amount after 4 years: $P(1+r)^4 = 15000$. Amount after 8 years: $P(1+r)^8 = 18000$.
* Divide the second equation by the first: $\frac{P(1+r)^8}{P(1+r)^4} = \frac{18000}{15000}$.
* $(1+r)^4 = \frac{18}{15} = \frac{6}{5} = 1.2$.
* $1+r = (1.2)^{1/4}$.
* $r = (1.2)^{1/4} - 1$. Using a calculator, $(1.2)^{1/4} \approx 1.0466$.
* $r \approx 1.0466 - 1 = 0.0466$, or 4.66% per annum.
* **Future Value of deposits and withdrawals:** Problems involving multiple deposits and withdrawals require calculating the future value of each transaction at a specific point in time.
* Deposit PhP 1500 now, PhP 3000 for the next two years. Withdraw PhP 5000 three years after the last deposit. Effective annual interest rate is 10%.
* Assume deposits are at the end of each year.
* Deposit 1 (now): PhP 1500. At the end of year 3 (one year after withdrawal), its value will be $1500(1.10)^3$.
* Deposit 2 (end of year 1): PhP 3000. At the end of year 3, its value will be $3000(1.10)^2$.
* Deposit 3 (end of year 2): PhP 3000. At the end of year 3, its value will be $3000(1.10)^1$.
* Withdrawal (end of year 3): PhP 5000. Its value at the end of year 3 is PhP 5000.
* Amount left after one year of withdrawal (end of year 4):
* Value of deposits at end of year 3: $1500(1.10)^3 + 3000(1.10)^2 + 3000(1.10)^1 = 1500(1.331) + 3000(1.21) + 3000(1.1) = 1996.5 + 3630 + 3300 = 8926.5$.
* At end of year 3, after withdrawal: $8926.5 - 5000 = 3926.5$.
* At end of year 4 (one year after withdrawal): $3926.5 \times 1.10 = 4319.15$.
* **Future Value with monthly compounding:** For an initial deposit $P$ compounding monthly at an annual rate $r$ for $t$ years, the final amount $A$ is given by $A = P \left(1 + \frac{r}{12}\right)^{12t}$.
* Initial deposit = 58,000. Rate = 8% per annum = 0.08. Time = 10 years.
* $A = 58000 \left(1 + \frac{0.08}{12}\right)^{12 \times 10} = 58000 \left(1 + \frac{0.02}{3}\right)^{120}$.
* **Simple Interest (final amount):**
* Compute the final amount at the end of the year if you loaned ₱120,000 with 6.5% annual simple interest.
* Interest = $P \times R \times t = 120000 \times 0.065 \times 1 = 7800$.
* Final Amount = Principal + Interest = $120000 + 7800 = 127800$.
#### 4.2.8 Conic Sections
* **Identifying conic section by eccentricity:**
* If eccentricity $e > 1$, the conic section is a hyperbola.
* If $e = 1$, it's a parabola.
* If $0 < e < 1$, it's an ellipse.
* If $e = 0$, it's a circle.
#### 4.2.9 Set Theory
* **Venn Diagrams (Union/Intersection):** In a survey of 500 people, 300 play piano, 200 play drums, and 100 play both. Number playing neither piano nor drums = Total - (Number playing piano only + Number playing drums only + Number playing both).
* Number playing piano only = 300 - 100 = 200.
* Number playing drums only = 200 - 100 = 100.
* Number playing at least one instrument = 200 (piano only) + 100 (drums only) + 100 (both) = 400.
* Number playing neither = 500 - 400 = 100.
* Alternatively, using the principle of inclusion-exclusion: $|P \cup D| = |P| + |D| - |P \cap D| = 300 + 200 - 100 = 400$.
* Neither = Total - $|P \cup D| = 500 - 400 = 100$.
### 4.3 Physics and Engineering Problems
#### 4.3.1 Mechanics
* **Centripetal Force and Speed:** Centripetal force ($F_c$) is given by $F_c = \frac{mv^2}{r}$, where $m$ is mass, $v$ is speed, and $r$ is the radius of the circular path.
* Mass $m = 1000$ kg, radius $r = 200$ m, centripetal force $F_c = 700$ N.
* $700 \text{ N} = \frac{1000 \text{ kg} \times v^2}{200 \text{ m}}$.
* $v^2 = \frac{700 \text{ N} \times 200 \text{ m}}{1000 \text{ kg}} = \frac{140000}{1000} \frac{\text{N} \cdot \text{m}}{\text{kg}} = 140 \frac{\text{J}}{\text{kg}} = 140 \frac{\text{m}^2}{\text{s}^2}$.
* $v = \sqrt{140} \approx 11.83$ m/s.
* **Work, Power, and Energy:**
* **Power required to lift a weight:** Power ($P$) is the rate at which work is done. Work done ($W$) to lift an object of weight $F$ to a height $h$ is $W = Fh$. Power is $P = \frac{W}{t}$.
* Weight = 170,000 pounds, height = 150 feet, time = 2 hours = $2 \times 3600 = 7200$ seconds.
* Work = $170,000 \text{ lb} \times 150 \text{ ft} = 25,500,000 \text{ ft-lb}$.
* Power = $\frac{25,500,000 \text{ ft-lb}}{7200 \text{ s}} \approx 3541.67 \text{ ft-lb/s}$.
* To convert to horsepower (1 hp = 550 ft-lb/s): Power in hp = $\frac{3541.67}{550} \approx 6.44$ hp.
* **Required horsepower:**
* Weight = 40,000 lbs, height = 200 feet, time = 10 minutes = $10 \times 60 = 600$ seconds.
* Work = $40,000 \text{ lbs} \times 200 \text{ ft} = 8,000,000 \text{ ft-lb}$.
* Power = $\frac{8,000,000 \text{ ft-lb}}{600 \text{ s}} \approx 13333.33 \text{ ft-lb/s}$.
* Power in hp = $\frac{13333.33}{550} \approx 24.24$ hp.
* **Energy dissipated in a resistor:** Energy ($E$) dissipated in a resistor is given by $E = VIt = \frac{V^2}{R}t = I^2Rt$.
* Voltage $V = 13$ V, resistance $R = 7$ ohms, time $t = 8$ seconds.
* $E = \frac{V^2}{R}t = \frac{(13 \text{ V})^2}{7 \text{ ohms}} \times 8 \text{ s} = \frac{169}{7} \times 8 \text{ J} = \frac{1352}{7} \text{ J} \approx 193.14$ J.
* **Kinetic energy change with velocity:** Kinetic energy ($KE$) is given by $KE = \frac{1}{2}mv^2$. If velocity ($v$) is doubled, the new kinetic energy is $KE' = \frac{1}{2}m(2v)^2 = \frac{1}{2}m(4v^2) = 4 \times (\frac{1}{2}mv^2) = 4 \times KE$. The kinetic energy quadruples.
* **Heat energy and furnace run time:** Heat required = 50,000 BTUs. Furnace rated input = 90,000 BTU/hr. Efficiency = 68%.
* Effective heat output per hour = $90,000 \text{ BTU/hr} \times 0.68 = 61,200 \text{ BTU/hr}$.
* Time required = $\frac{\text{Heat required}}{\text{Effective heat output}} = \frac{50,000 \text{ BTU}}{61,200 \text{ BTU/hr}} \approx 0.817$ hours.
* Time in minutes = $0.817 \text{ hours} \times 60 \text{ minutes/hour} \approx 49.02$ minutes.
* **Energy used by electric heater:** Power ($P$) = $VI$. Energy ($E$) = $P \times t$.
* Current $I = 12$ A, Voltage $V = 240$ V.
* Power $P = 12 \text{ A} \times 240 \text{ V} = 2880$ W = 2880 J/s.
* 1 BTU $\approx 1055$ J.
* 4000 BTU $= 4000 \times 1055$ J $= 4,220,000$ J.
* Time $t = \frac{E}{P} = \frac{4,220,000 \text{ J}}{2880 \text{ J/s}} \approx 1465.28$ seconds.
* **Torque and Levers (Mechanical Advantage):** Using a lever, Force $\times$ Distance from fulcrum = Load $\times$ Distance from fulcrum.
* Metal rod length = 72 inches. Fulcrum is 12 inches from the load. This means the distance from the fulcrum to the load is 12 inches, and the distance from the fulcrum to where the force is applied is $72 - 12 = 60$ inches.
* Force (effort) $\times 60$ inches = Load (engine weight) $\times 12$ inches.
* Force $\times 60 = 500 \text{ lbs} \times 12$.
* Force $= \frac{500 \times 12}{60} = \frac{6000}{60} = 100$ lbs.
* **Fluid Mechanics:**
* **Pressure at a depth:** Pressure ($P$) at a depth ($h$) in a fluid is $P = \rho g h$, where $\rho$ is density and $g$ is acceleration due to gravity. Pressure gauge is often used to measure gauge pressure.
* **Container filled with water:** A closed rectangular container with dimensions $l=4$ m, $w=5$ m, $h=6$ m filled with water. Pressure on the bottom face. $P = \rho_{water} g h$.
* $\rho_{water} \approx 1000$ kg/m³. $g \approx 9.81$ m/s². $h = 6$ m.
* $P = 1000 \text{ kg/m³} \times 9.81 \text{ m/s²} \times 6 \text{ m} = 58860$ Pa.
* **Pressure exerted by oil:** A closed cylindrical container filled with a fluid with specific gravity ($SG$) of 1.7. Depth $h = 15$ m.
* Density of fluid $\rho = SG \times \rho_{water} = 1.7 \times 1000$ kg/m³ $= 1700$ kg/m³.
* $P = \rho g h = 1700 \text{ kg/m³} \times 9.81 \text{ m/s²} \times 15 \text{ m} = 249,945$ Pa.
* **Pressure head:** Pressure head is the height of a column of fluid that would produce a given pressure. $P = \rho g h_{head}$.
* Pressure at outlet = 80 psi. $SG = 0.8$.
* First, convert psi to Pascals: 1 psi $\approx 6894.76$ Pa. So, $P = 80 \times 6894.76 \text{ Pa} = 551580.8$ Pa.
* Density of oil $\rho = 0.8 \times 1000$ kg/m³ $= 800$ kg/m³.
* $h_{head} = \frac{P}{\rho g} = \frac{551580.8 \text{ Pa}}{800 \text{ kg/m³} \times 9.81 \text{ m/s²}} \approx 70.37$ meters of oil.
* **Total dynamic head:** This includes pressure head, velocity head, and elevation head, minus friction losses.
* Pump must deliver at 150 kPa gauge. Frictional losses = 3 meters of oil. Discharge point is 5 meters above pump. $SG = 0.8$.
* Pressure head = $\frac{150 \text{ kPa}}{0.8 \times 9.81 \text{ kPa/m}} \approx 19.21$ m of oil. (Using $\rho g$ in kPa/m for water and then dividing by SG).
* Elevation head = 5 m of oil.
* Friction head loss = 3 m of oil.
* Total Dynamic Head (TDH) = Pressure head + Elevation head + Velocity head - Friction head loss. Assuming velocity head is negligible or already accounted for in pressure at discharge:
* TDH $\approx 19.21 \text{ m} + 5 \text{ m} - 3 \text{ m} = 21.21$ m of oil.
* **Velocity of water from a spout:** Using Torricelli's Law, the velocity of efflux ($v$) from a hole at depth $h$ is $v = \sqrt{2gh}$.
* Tank height = 3 m. Assuming the spout is at the bottom and the tank is full of water.
* $v = \sqrt{2 \times 9.81 \text{ m/s²} \times 3 \text{ m}} = \sqrt{58.86} \approx 7.67$ m/s.
* **Rate of water leak:** Similar to efflux velocity, flow rate ($Q$) = Area $\times$ Velocity.
* Tank height = 8 m. Hole area = 1 sq.cm = $1 \times 10^{-4}$ m².
* Velocity $v = \sqrt{2gh} = \sqrt{2 \times 9.81 \text{ m/s²} \times 8 \text{ m}} = \sqrt{156.96} \approx 12.53$ m/s.
* Flow rate $Q = (1 \times 10^{-4} \text{ m²}) \times (12.53 \text{ m/s}) = 1.253 \times 10^{-3}$ m³/s.
* **Continuity Equation:** For an incompressible fluid, the flow rate is constant. $A_1 v_1 = A_2 v_2$.
* Area at point A ($A_A$) = 3 sq.cm, velocity at A ($v_A$) = 2 m/s. Pressure at A ($P_A$) = 120,000 Pa.
* Area at point B ($A_B$) = 7 sq.cm. Find pressure at B ($P_B$).
* $A_A v_A = A_B v_B \implies (3 \text{ cm}^2) \times (2 \text{ m/s}) = (7 \text{ cm}^2) \times v_B$.
* $v_B = \frac{3 \times 2}{7} \text{ m/s} = \frac{6}{7}$ m/s.
* Use Bernoulli's equation: $P_A + \frac{1}{2}\rho v_A^2 + \rho g h_A = P_B + \frac{1}{2}\rho v_B^2 + \rho g h_B$. Assuming horizontal pipe, $h_A = h_B$.
* $P_B = P_A + \frac{1}{2}\rho (v_A^2 - v_B^2)$. Density of water $\rho = 1000$ kg/m³.
* $P_B = 120000 \text{ Pa} + \frac{1}{2}(1000 \text{ kg/m³}) ((2 \text{ m/s})^2 - (\frac{6}{7} \text{ m/s})^2)$
* $P_B = 120000 + 500 (4 - \frac{36}{49}) = 120000 + 500 (\frac{196-36}{49}) = 120000 + 500 (\frac{160}{49}) \approx 120000 + 1632.65 \approx 121633$ Pa.
* **Flow rate:** Water velocity = 7 fps. Pipe diameter = 10 inches.
* Radius $r = 5$ inches. Convert to feet: $r = \frac{5}{12}$ ft.
* Area $A = \pi r^2 = \pi (\frac{5}{12} \text{ ft})^2 = \frac{25\pi}{144}$ sq ft.
* Flow rate $Q = A \times v = \frac{25\pi}{144} \text{ sq ft} \times 7 \text{ fps} = \frac{175\pi}{144}$ cubic feet per second.
* **Buoyancy and Specific Gravity:** Specific Gravity ($SG$) of an object is the ratio of its density to the density of water, or the ratio of its weight to the buoyant force when fully submerged.
* Weight in air = 1.5 N. Apparent weight when submerged = 1.1 N.
* Buoyant force ($F_B$) = Weight in air - Apparent weight = $1.5 \text{ N} - 1.1 \text{ N} = 0.4$ N.
* $SG = \frac{\text{Weight of object}}{\text{Buoyant force}} = \frac{1.5 \text{ N}}{0.4 \text{ N}} = 3.75$.
* **Density of a floating object:** If 2/3 of an object is floating in water, then the buoyant force equals the weight of the object. The buoyant force is equal to the weight of the displaced fluid.
* Weight of object = Weight of (2/3 of object's volume) of water.
* $\rho_{object} V g = \frac{2}{3} V \rho_{water} g$.
* $\rho_{object} = \frac{2}{3} \rho_{water}$.
* Density of object = $\frac{2}{3} \times 1000$ kg/m³ $= \frac{2000}{3} \approx 666.67$ kg/m³.
* **Pressure head with varying fluids:** A tank open above contains a 3-m head of water. Above the water is oil with a head of 2-m and SG = 0.8.
* Pressure from water at the bottom: $P_{water} = \rho_{water} g h_{water} = 1000 \times 9.81 \times 3$ Pa.
* Pressure from oil at the bottom: $P_{oil} = \rho_{oil} g h_{oil} = (0.8 \times 1000) \times 9.81 \times 2$ Pa.
* Total pressure at the bottom = $P_{water} + P_{oil} = (1000 \times 9.81 \times 3) + (800 \times 9.81 \times 2)$ Pa.
* Total pressure $= 9.81 \times (3000 + 1600) = 9.81 \times 4600 = 45126$ Pa.
* **Thermodynamics:**
* **Furnace efficiency:** (Covered under "Heat energy and furnace run time").
#### 4.3.2 Electrical Engineering
* **Ohm's Law and Circuits:**
* **Parallel resistors:** For resistors in parallel, the total current is the sum of currents through each resistor. The equivalent resistance $R_{eq}$ is given by $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$.
* Power supply $V = 8$ V. Parallel resistors: $R_1 = 10$ ohms, $R_2 = 15$ ohms, $R_3 = 20$ ohms.
* $\frac{1}{R_{eq}} = \frac{1}{10} + \frac{1}{15} + \frac{1}{20} = \frac{6+4+3}{60} = \frac{13}{60}$.
* $R_{eq} = \frac{60}{13}$ ohms.
* Total current $I = \frac{V}{R_{eq}} = \frac{8 \text{ V}}{\frac{60}{13} \text{ ohms}} = 8 \times \frac{13}{60} = \frac{104}{60} = \frac{26}{15}$ A $\approx 1.73$ A.
* **Power and energy dissipation:** (Covered under "Work, Power, and Energy").
#### 4.3.3 Kinematics and Dynamics
* **Acceleration:** Acceleration ($a$) is the rate of change of velocity ($v$). If velocity changes from $v_i$ to $v_f$ in time $t$, then $a = \frac{v_f - v_i}{t}$.
* Initial velocity $v_i = 20$ m/s, final velocity $v_f = 40$ m/s, time $t = 3$ seconds.
* $a = \frac{40 \text{ m/s} - 20 \text{ m/s}}{3 \text{ s}} = \frac{20 \text{ m/s}}{3 \text{ s}} = \frac{20}{3}$ m/s² $\approx 6.67$ m/s².
* **Distance Calculation (Coordinate Geometry):** The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
* Point A (1,5) to Point B (6,8).
* Distance = $\sqrt{(6-1)^2 + (8-5)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}$.
* **Projectile Motion:**
* **Comparison of dropped vs. launched horizontally:** Ball A is dropped vertically, Ball B is launched horizontally from the same height at the same time. Neglecting air resistance, both balls will hit the ground at the same time. The horizontal motion of Ball B does not affect its vertical motion, which is governed by gravity alone.
* **Relative Motion (Wind Velocity):** Cyclist travels with the wind and against the wind.
* Let $v_c$ be the cyclist's speed in still air, and $v_w$ be the wind velocity.
* With wind: Distance = $(v_c + v_w) \times t_1$.
* Against wind: Distance = $(v_c - v_w) \times t_2$.
* Distance = 10 km. $t_1 = 1.3$ hours, $t_2 = 1.6$ hours.
* $10 = (v_c + v_w) \times 1.3 \implies v_c + v_w = \frac{10}{1.3} \approx 7.69$.
* $10 = (v_c - v_w) \times 1.6 \implies v_c - v_w = \frac{10}{1.6} = 6.25$.
* Add the two equations: $2v_c = 7.69 + 6.25 = 13.94 \implies v_c = 6.97$ km/h.
* Subtract the second from the first: $2v_w = 7.69 - 6.25 = 1.44 \implies v_w = 0.72$ km/h.
* **Height of tree from shadow:** Using similar triangles.
* Pole height = 6 ft, shadow = 4 ft. Ratio of height to shadow = $\frac{6}{4} = 1.5$.
* Tree shadow = 64 ft. Height of tree = $1.5 \times 64$ ft $= 96$ ft.
* **Rate of water leak from a hole:** (Covered under "Fluid Mechanics").
#### 4.3.4 Surveying and Photogrammetry
* **Height of a chimney from photograph:** Given photo taken from an elevation above Mean Sea Level (MSL), elevation of chimney base, relief displacement, and radial distance.
* Let $H$ be the flying height above ground (not MSL), $h$ be the height of the object, $r$ be the radial distance of the top of the object from the photo center, and $d$ be the relief displacement.
* Flying height above ground = Elevation of camera - Elevation of object base = 500 m - 250 m = 250 m. So, $H = 250$ m.
* Relief displacement formula: $d = \frac{rh}{H}$.
* $r$ is the radial distance of the image of the top of the chimney from the principal point. The radial distance of the image of the top of the chimney was 110 mm. $r = 110$ mm $= 0.110$ m.
* Relief displacement of the chimney $d = 51.4$ mm $= 0.0514$ m.
* $0.0514 \text{ m} = \frac{0.110 \text{ m} \times h}{250 \text{ m}}$.
* $h = \frac{0.0514 \text{ m} \times 250 \text{ m}}{0.110 \text{ m}} = \frac{12.85}{0.110} \text{ m} \approx 116.82$ m.
#### 4.3.5 Engineering Materials and Properties
* **Density and Weight:**
* **Apparent weight of block:** Density of asphalt block = 2360 kg/m³. Weight in air = 100 N.
* Volume $V = \frac{\text{Mass}}{\text{Density}}$. Mass $m = \frac{W_{air}}{g} = \frac{100 \text{ N}}{9.81 \text{ m/s²}} \approx 10.19$ kg.
* $V = \frac{10.19 \text{ kg}}{2360 \text{ kg/m³}} \approx 0.004318$ m³.
* Buoyant force in water ($F_B$) = $\rho_{water} V g = 1000 \text{ kg/m³} \times 0.004318 \text{ m³} \times 9.81 \text{ m/s²} \approx 42.36$ N.
* Apparent weight = Weight in air - $F_B = 100 \text{ N} - 42.36 \text{ N} = 57.64$ N.
#### 4.3.6 Mechanical Engineering
* **Pumps and Head:** (Covered under "Fluid Mechanics").
* **Cylindrical Tank Volume:**
* A closed cylindrical tank is 8 ft long and 3 ft in diameter. When lying horizontally, the water is 2 ft deep. If the tank is in the vertical position, what is the depth of the water?
* First, calculate the volume of water. The cross-section of the water is a segment of a circle. The circle has radius $r=1.5$ ft. The depth is 2 ft. The distance from the center to the water surface is $1.5 - 2 = -0.5$ ft. This means the water level is 0.5 ft below the center of the circular cross-section.
* The chord length $l$ at a distance $d$ from the center is $l = 2\sqrt{r^2 - d^2}$. Here $d = -0.5$ ft (or distance from center can be thought of as 0.5 if we consider the geometry symmetrically from the center). Let's measure from the bottom, so the water level is at 2 ft from the bottom. The center is at 1.5 ft from the bottom. The distance of the water surface from the center is $1.5 - 2 = -0.5$.
* Let's find the angle subtended by the water surface. The distance from the center to the water surface is $d = |r-h_{depth}| = |1.5-2| = 0.5$ ft.
* Using trigonometry, if $\theta$ is half the angle subtended at the center by the water surface chord, $\cos \theta = \frac{d}{r} = \frac{0.5}{1.5} = \frac{1}{3}$. So $\theta = \arccos(\frac{1}{3})$.
* The area of the segment is $A_{segment} = r^2 \arccos(\frac{d}{r}) - d\sqrt{r^2-d^2}$.
* $A_{segment} = (1.5)^2 \arccos(\frac{0.5}{1.5}) - 0.5 \sqrt{(1.5)^2 - (0.5)^2} = 2.25 \arccos(\frac{1}{3}) - 0.5 \sqrt{2.25 - 0.25} = 2.25 \arccos(\frac{1}{3}) - 0.5 \sqrt{2}$.
* $\arccos(\frac{1}{3})$ is in radians. $\arccos(1/3) \approx 1.231$ radians.
* $A_{segment} \approx 2.25 \times 1.231 - 0.5 \times 1.414 = 2.76975 - 0.707 = 2.06275$ sq ft.
* Volume of water = Area of segment $\times$ length of tank = $2.06275 \text{ sq ft} \times 8 \text{ ft} = 16.502$ cubic feet.
* Now, if the tank is vertical, the radius of the base is 1.5 ft. Let the depth of the water be $h_{vertical}$. The volume is $\pi r^2 h_{vertical}$.
* $16.502 \text{ cu ft} = \pi (1.5 \text{ ft})^2 \times h_{vertical} = 2.25\pi \times h_{vertical}$.
* $h_{vertical} = \frac{16.502}{2.25\pi} \approx \frac{16.502}{7.068} \approx 2.33$ ft.
#### 4.3.7 Civil Engineering
* **Bridge construction:** A 580-ft wide river is spanned by a bridge. The bridge constructed across the river banked 1/7 of its length in the east and 1/6 in the west.
* This describes parts of the bridge's span relative to its total length. Let $L$ be the total length of the bridge.
* Banked in the east: $\frac{1}{7}L$. Banked in the west: $\frac{1}{6}L$.
* The total span across the river is 580 ft. If the bridge spans the river, its length relevant to the river's width is likely considered the part that crosses the river directly. The wording "banked" suggests something about supports or approach spans. If we assume the 580 ft is the length of the bridge crossing the river:
* If the bridge banks mean a portion of the length is on the banks, then the span across the river is the total length minus the banked portions.
* Let $L$ be the total length of the bridge. The length spanning the river might be $L - \frac{1}{7}L - \frac{1}{6}L$.
* Or, if the 580 ft is the direct span, and the "banked" portions are additional lengths, the problem is underspecified.
* Assuming the question implies parts of the river span are "banked" in some way, and the 580 ft is a key measurement. A common interpretation for such problems is that these fractions relate to the total length. However, the phrasing "across the river banked 1/7 of its length in the east and 1/6 in the west" is ambiguous. If the entire bridge has length $L$, and a section of length $L/7$ is on the east bank and $L/6$ on the west bank, then the span across the river is $L - L/7 - L/6$. If this span is 580 ft, then $580 = L(1 - 1/7 - 1/6) = L(\frac{42-6-7}{42}) = L(\frac{29}{42})$. So, $L = 580 \times \frac{42}{29} \approx 840$ ft. The length of the bridge would be approximately 840 ft.
#### 4.3.8 Geodetic Engineering / Surveying
* **Map Scale and Real Length:** A pipeline is shown as 0.20 m on a map with a scale of 1:150,000 mm.
* The scale 1:150,000 means 1 unit on the map represents 150,000 of the same units in reality.
* The map distance is 0.20 m.
* Real length = Map distance $\times$ Scale factor.
* Real length = $0.20 \text{ m} \times 150,000 = 30,000$ m $= 30$ km.
* The "mm" in 1:150,000 mm is likely a typo and should represent the unit of measurement for the scale factor. Assuming the scale is 1:150,000 for the given map units.
### 4.4 Other Conceptual Questions
* **Temperature equality:** Fahrenheit and Celsius scales are equal at -40 degrees.
* Formula to convert Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$.
* Formula to convert Fahrenheit to Celsius: $C = \frac{5}{9}(F - 32)$.
* Set $F=C$: $C = \frac{5}{9}(C - 32) \implies 9C = 5C - 160 \implies 4C = -160 \implies C = -40$. So $F = -40$ as well.
* **Conic section eccentricity:** (Covered under "Conic Sections").
* **Arithmetic sequence (10th term):** Find the 10th term of a geometric sequence if the 6th term is 1458 and the 8th term is 13122.
* Let the geometric sequence be $a, ar, ar^2, \dots$
* $T_6 = ar^5 = 1458$.
* $T_8 = ar^7 = 13122$.
* Divide the eighth term by the sixth term: $\frac{ar^7}{ar^5} = \frac{13122}{1458}$.
* $r^2 = 9 \implies r = 3$ (assuming a positive common ratio).
* Substitute $r=3$ into $ar^5 = 1458$: $a(3)^5 = 1458 \implies a(243) = 1458 \implies a = \frac{1458}{243} = 6$.
* The 10th term is $T_{10} = ar^9 = 6 \times (3)^9 = 6 \times 19683 = 118098$.
* **Geometric interpretation of division by zero:** Division by zero is undefined in standard arithmetic.
* **Square pyramid slant height:** (Covered under "Area of a square pyramid").
* **Ratio of hypotenuse to shorter side in a right triangle:** If the ratio of the two legs of a right triangle is 5:12, let the legs be $5x$ and $12x$. By the Pythagorean theorem, the hypotenuse $h$ is $\sqrt{(5x)^2 + (12x)^2} = \sqrt{25x^2 + 144x^2} = \sqrt{169x^2} = 13x$. The shorter side is $5x$. The ratio of the hypotenuse to the shorter side is $\frac{13x}{5x} = \frac{13}{5}$.
* **Area change with perimeter doubling:** If the perimeter of a square doubles, its side length also doubles. If the original side length is $s$, the perimeter is $4s$ and the area is $s^2$. If the new side length is $2s$, the new perimeter is $4(2s) = 8s$ (doubled), and the new area is $(2s)^2 = 4s^2$ (quadrupled). The area quadruples.
* **Average price calculation:** A retailer has 10 items with an average price of 120 each. Total price = $10 \times 120 = 1200$. One item is removed, and the new average is 115 for 9 items. New total price = $9 \times 115 = 1035$. Price of removed item = Original total price - New total price = $1200 - 1035 = 165$.
* **Clock hands together:** For the hands of a clock to be together for the first time after 9 pm. The hands are together every approximately 65.45 minutes. At 9:00, the hour hand is at 9 and the minute hand is at 12. The minute hand has to catch up. The relative speed of the minute hand to the hour hand is $11/2$ divisions per minute (where a full circle is 60 divisions). At 9 pm, the minute hand is at 0 divisions, the hour hand is at 45 divisions. The minute hand needs to cover 45 divisions plus the distance the hour hand moves. Let $t$ be minutes past 9. Minute hand position: $6t$. Hour hand position: $45 + 0.5t$. Set them equal: $6t = 45 + 0.5t \implies 5.5t = 45 \implies t = \frac{45}{5.5} = \frac{90}{11} \approx 8.18$ minutes. So, it will be about 8.18 minutes past 9 pm.
* **Three-digit numbers with conditions:** Divisible by 5, no repeating digits, no zero digits.
* Divisible by 5 means the last digit must be 5 (since no zero digits).
* So, the number is of the form _ _ 5.
* The digits available are 1, 2, 3, 4, 5, 6, 7, 8, 9.
* The last digit is 5.
* For the first digit, we can choose any of the remaining 8 digits (excluding 5).
* For the second digit, we can choose any of the remaining 7 digits (excluding the first digit and 5).
* Number of such numbers = $8 \times 7 \times 1 = 56$.
* **Comparison of speeds (Vertical vs. Horizontal Launch):** (Covered under "Projectile Motion").
* **Divisibility rule:** If $p-10$ is divisible by 6, then $p-10 = 6k$. $p = 6k+10$. $p+2 = 6k+12 = 6(k+2)$, which is always divisible by 6.
#### 4.4.1 Business and Finance Terms
* **Bond guaranteeing performance:** (Covered under "Fundamental Concepts and Definitions").
* **Funds for going concern:** (Covered under "Fundamental Concepts and Definitions").
* **Association for profit:** (Covered under "Fundamental Concepts and Definitions").
* **Business organization contributing to economy:** (Covered under "Fundamental Concepts and Definitions").
### 4.5 General Principles and Relationships
* **Relationship between perimeter and area of squares:** If the perimeter of a square doubles, its area quadruples.
* **Relationship between velocity and kinetic energy:** Kinetic energy is proportional to the square of the velocity. If velocity doubles, kinetic energy quadruples.
* **Relationship between temperature scales:** Fahrenheit and Celsius scales are equal at -40 degrees.
* **Effect of wind on travel:** Wind increases speed when traveling in the same direction and decreases speed when traveling against it.
* **Work and power:** Power is the rate at which work is done.
* **Pressure in fluids:** Pressure increases with depth and depends on the fluid's density and gravity.
* **Parallel circuits:** In parallel circuits, the voltage is the same across each component, and the total current is the sum of currents through each component.
* **Geometric sequences:** Each term is found by multiplying the previous term by a constant factor (the common ratio).
* **Properties of consecutive integers:** Sums of consecutive integers often exhibit predictable parity (even or odd).
* **Mathematical inequalities:** The range of a difference between two variables can be determined from the ranges of the individual variables.
### 4.6 Abstract Reasoning
* **Conditional statements:** If $p-10$ is divisible by 6, then $p+2$ must also be divisible by 6.
* **Logical deduction:** Understanding the implications of given conditions in word problems.
* **Conceptual understanding of physical laws:** Knowing how changing one variable affects another (e.g., velocity and kinetic energy, perimeter and area).
* **Understanding abstract mathematical concepts:** Definitions of terms like LCM, annulus, eccentricity, and the properties of different number types (prime, consecutive integers).
#### 4.6.1 Conceptual Physics Principles
* **Gravity and motion:** Objects dropped and launched horizontally from the same height experience the same vertical acceleration due to gravity.
* **Work and energy conservation:** Energy can be transformed between different forms (e.g., potential to kinetic).
* **Fluid dynamics:** Principles like continuity and Bernoulli's equation describe fluid behavior.
* **Thermodynamics:** Efficiency is a key concept in energy conversion processes.
#### 4.6.2 Abstract Mathematical Concepts
* **Parity:** Understanding whether a number is even or odd.
* **Roots of equations:** Distinguishing between real and complex roots.
* **Ratios and proportions:** Applying these to geometric and algebraic problems.
#### 4.6.3 Conceptual Business/Economic Terms
* **Working capital:** Funds needed for ongoing operations.
* **Partnership:** A business structure involving two or more individuals.
* **Market:** A place or system for exchange of goods and services.
* **Featherbedding:** Inefficient labor practices.
> **Tip:** When encountering word problems, carefully identify what is being asked, list all given information, and then determine the relevant formulas or concepts needed to solve the problem. Break down complex problems into smaller, manageable steps.
---
## Common mistakes to avoid
- Review all topics thoroughly before exams
- Pay attention to formulas and key definitions
- Practice with examples provided in each section
- Don't memorize without understanding the underlying concepts
Glossary
| Term | Definition |
|------|------------|
| LCM (Least Common Multiple) | The smallest positive integer that is a multiple of two or more integers. For two prime numbers, their LCM is their product. |
| Regulation Football Field | A standard-sized field for playing football, typically measuring 120 yards in length (including end zones) and approximately 53.3 yards in width. |
| Power | The rate at which work is done or energy is transferred. It is measured in units like watts or horsepower. |
| Simple Interest | A method of calculating interest on a loan or deposit based on the principal amount only, not on any accumulated interest. The formula is $I = P \times R \times T$, where $I$ is interest, $P$ is principal, $R$ is rate, and $T$ is time. |
| Actual Rate of Interest | The true or effective interest rate after accounting for all fees and charges, especially when interest is deducted upfront. |
| Centripetal Force | The force that acts on a body moving in a circular path and is directed towards the center around which the body is moving. The formula is $F_c = \frac{mv^2}{r}$, where $F_c$ is centripetal force, $m$ is mass, $v$ is velocity, and $r$ is radius. |
| Arc Length | The distance along the curved line that forms part of the circumference of a circle. The formula is $s = r\theta$, where $s$ is arc length, $r$ is radius, and $\theta$ is the central angle in radians. |
| Central Angle | An angle whose vertex is the center of a circle and whose sides are radii intersecting the circle at two distinct points. |
| Selling Price | The price at which a product or service is offered for sale to customers. |
| Loss | The financial situation where the expenses exceed the revenue. It is often expressed as a percentage of the cost price. |
| Efficiency | A measure of how well a system converts input energy or resources into useful output. It is typically expressed as a percentage. |
| Acceleration | The rate at which the velocity of an object changes over time. It is a vector quantity, meaning it has both magnitude and direction. The formula is $a = \frac{\Delta v}{\Delta t}$, where $a$ is acceleration, $\Delta v$ is the change in velocity, and $\Delta t$ is the change in time. |
| Distance Traveled | The total length of the path covered by a moving object, regardless of its direction. |
| Frustum of a Cone | A portion of a cone obtained by cutting off the top part with a plane parallel to the base. |
| Altitude | The perpendicular distance from the apex or vertex of a geometric figure to the base. In a frustum of a cone, it is the perpendicular distance between the two bases. |
| Simple Interest Rate | The percentage charged on a principal amount for a specified period, calculated only on the original principal. |
| Actual Rate of Interest | The effective interest rate experienced by a borrower, which may differ from the stated rate due to how interest is calculated or applied, such as being deducted upfront. |
| Workers | Individuals employed to perform a specific task or job. |
| Resigned | Voluntarily left a job or position. |
| Centripetal Force | The force that keeps an object moving in a circular path by pulling it towards the center of the circle. |
| Vehicle's Speed | The rate at which a vehicle covers distance, typically measured in meters per second or kilometers per hour. |
| Rope | A strong cord made of twisted strands of fiber or wire, used for tying, pulling, or supporting. |
| Square | A plane figure with four equal straight sides and four right angles. |
| Triangle | A plane figure with three straight sides and three angles. |
| Arc Length | The distance along a curved line segment of a circle's circumference. |
| Central Angle | An angle formed by two radii of a circle with the vertex at the center. |
| Radius | The distance from the center of a circle to any point on its circumference. |
| Ways (Combinations/Permutations) | The number of different arrangements or selections possible from a set of items, often involving factorials in calculations. |
| Horsepower | A unit of power, equivalent to the power needed to lift 550 pounds one foot in one second. Used to measure the output of engines and motors. |
| Selling Price | The price at which a good or service is sold to a customer. |
| Loss | The decrease in value or profit when expenses exceed revenues. |
| Gain | The increase in value or profit. |
| Distributor | An intermediary who buys goods from manufacturers and sells them to retailers or other businesses. |
| Furnace | An appliance or enclosure for heating, typically used in a building for space heating. |
| Efficiency | The ratio of useful output to total input, often expressed as a percentage, indicating how much energy is effectively used. |
| Space Demand | The amount of heating or cooling required for a particular space to maintain a comfortable temperature. |
| Furnace Run Time | The duration for which a furnace operates to meet a heating demand. |
| Acceleration | The rate at which an object's velocity changes over time. |
| Velocity | The speed of an object in a particular direction. |
| Total Distance Traveled | The sum of the lengths of all the paths taken by an object, irrespective of its starting and ending points. |
| Coordinates | A set of numbers used to locate a point in a coordinate system, such as (x, y) for a 2D plane. |
| Volume | The amount of three-dimensional space occupied by a solid object or contained within a vessel. |
| Frustum of a Cone | A part of a solid cone created by cutting off the top with a plane parallel to the base. |
| Diameter | The distance across a circle through its center, equal to twice the radius. |
| Altitude | The perpendicular height of a geometric figure from its base to its apex or highest point. |
| Evil Wrong | An immoral act or offense that causes harm or damage to another person's property or reputation. |
| Metal Rod | A long, slender piece of metal. |
| Fulcrum | The point on which a lever rests or pivots. |
| Load | The weight or resistance that a lever or other mechanical device must overcome. |
| Force | An influence that can cause an object to change its motion or to deform. |
| Triangle ABC | A triangle with vertices labeled A, B, and C. |
| Sides | The line segments forming the boundary of a polygon. |
| Angle (∠) | The figure formed by two rays sharing a common endpoint, the vertex. |
| Pump | A mechanical device used to move fluids (liquids or gases) by mechanical action. |
| Specific Gravity (SG) | The ratio of the density of a substance to the density of a reference substance, usually water. |
| Pressure (kPa gauge) | The pressure relative to the ambient atmospheric pressure, often referred to as gauge pressure. |
| Frictional Losses | The energy dissipated due to friction as a fluid flows through a pipe. |
| Discharge Point | The location where a fluid exits a system or pipe. |
| Total Dynamic Head (TDH) | The total equivalent height that a fluid is to be pumped, considering elevation changes, pressure differences, and friction losses. |
| Segment Area | The area of a region of a circle bounded by a chord and the arc subtended by the chord. |
| Central Angle | The angle subtended by an arc at the center of a circle. |
| Radius | The distance from the center of a circle to any point on its circumference. |
| Hands of the Clock | The hour, minute, and second indicators on a clock face. |
| Together | In close proximity or contact; coinciding. |
| 3-digit Numbers | Integers ranging from 100 to 999. |
| Divisible by 5 | A number that can be divided by 5 without leaving a remainder. |
| Repeating Digits | Digits that appear more than once in a number. |
| Zero Digits | The numeral '0'. |
| Pipeline | A long pipe, typically underground, used for conveying water, gas, oil, or other fluid substances. |
| Map Scale | The ratio between distances on a map and corresponding distances in reality, often expressed as 1:X. |
| Real Length | The actual physical length of an object or feature. |
| Closed Rectangular Container | A three-dimensional box-like shape with six faces, where all sides are enclosed. |
| Dimensions | The measurements of length, width, and height of an object. |
| Filled with Water | Occupied entirely by water. |
| Pressure Exerted | The force applied per unit area. |
| Bottom Face | The lowest surface of a three-dimensional object. |
| Circuit | A closed path through which electric current flows. |
| Power Supply | A device that provides electrical energy to a circuit. |
| Parallel Resistors | Components connected across the same two points in a circuit, so the current divides among them. |
| Total Current | The sum of the currents flowing through all parallel branches of a circuit. |
| Father and Son Age Problem | A word problem involving the ages of a father and son at different points in time. |
| Consecutive Integers | Integers that follow each other in order, differing by 1. |
| Even | An integer that is divisible by 2. |
| Roots | The values of a variable that satisfy an equation, making the equation true. |
| Binomial Expansion | The algebraic expansion of powers of a binomial $(x+y)^n$. |
| Constant Term | The term in an expansion that does not contain any variables. |
| Geometric Sequence | A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. |
| Term | A single item in a sequence or expression. |
| Common Ratio | The constant factor by which each term in a geometric sequence is multiplied to get the next term. |
| Region Between Two Concentric Circles | The area that lies between the circumference of two circles that share the same center but have different radii. This region is also known as an annulus. |
| Concentric Circles | Circles that share the same center point but have different radii. |
| Cyclist | A person who rides a bicycle. |
| Wind Velocity | The speed and direction of the wind. |
| Kinetic Energy | The energy an object possesses due to its motion. The formula is $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity. |
| Velocity Doubled | The speed of the object is multiplied by two. |
| Regular Pentagon | A polygon with five equal sides and five equal interior angles. |
| Interior Angle | An angle inside a polygon. |
| Vertical Photograph | An aerial photograph taken with the camera axis pointed directly downwards. |
| Chimney | A tall vertical structure, usually made of brick or metal, that allows smoke and gases to escape from a building. |
| Elevation | The height of a point or object above a reference level, such as mean sea level. |
| M.S.L. (Mean Sea Level) | The average height of the sea surface over a period of time, used as a reference point for elevations. |
| Relief Displacement | The apparent shift in the position of an object in an aerial photograph due to its height above the terrain. |
| Radial Distance | The distance measured from the principal point (center of the photograph) to a point on the photograph. |
| Deposit | To place money into a bank account. |
| Withdrawal | To take money out of a bank account. |
| Effective Annual Interest Rate | The actual rate of interest earned on an investment or paid on a loan over one year, taking into account the effect of compounding. |
| Survey | An examination or investigation of a group of people to collect information. |
| Play the Piano | To perform music on a piano instrument. |
| Play the Drums | To perform music on a drum set. |
| Both Instruments | Participating in activities involving both the piano and the drums. |
| Neither Piano nor Drums | Not participating in activities involving the piano or the drums. |
| Pressure at Altitude | The atmospheric pressure decreases as the altitude above sea level increases. |
| Fluid | A substance that flows freely, such as a liquid or gas. |
| Specific Gravity | The ratio of the density of a substance to the density of water at a standard temperature. |
| Head of Water | The height of a column of water that exerts a certain pressure. |
| Total Pressure | The sum of all pressures acting on a point or surface, including atmospheric and hydrostatic pressure. |
| Forces | Pushes or pulls that can cause an object to change its motion. |
| Magnitudes | The size or strength of a force. |
| Resultant | The single force that has the same effect as two or more other forces acting together. |
| Volt (V) | The SI unit of electric potential difference or electromotive force. |
| Resistor | An electrical component that opposes the flow of electric current. |
| Energy Dissipated | The amount of energy converted into heat or other forms of energy due to resistance. |
| Joules (J) | The SI unit of energy. |
| Probability | The measure of the likelihood that an event will occur. |
| Rolling Two Fair Six-Sided Dice | A probability experiment involving the random outcome of throwing two standard dice, each with faces numbered 1 to 6. |
| Sum of Numbers Rolled | The total obtained by adding the numbers showing on the faces of the dice. |
| Cylinder | A three-dimensional solid with two parallel circular bases connected by a curved surface. |
| US Gallons | A unit of volume commonly used in the United States. |
| Tall | Having a great height. |
| Radius | The distance from the center of a circle to its edge. |
| Temperature | The degree or intensity of heat present in a substance or object. |
| Degrees Fahrenheit (°F) | A scale for measuring temperature. |
| Degrees Celsius (°C) | A scale for measuring temperature. |
| Consecutive Integers | Integers that follow each other in order, like 1, 2, 3. |
| Variable | A symbol that represents a quantity that can change. |
| Equation | A mathematical statement that asserts the equality of two expressions. |
| Pipe A, B, C | Identifiers for different pipes used in a system. |
| Fills a Tank | The process of filling a container with a substance. |
| Work Together | To collaborate or cooperate on a task. |
| Ratio | A comparison of two quantities by division. |
| Legs of a Right Triangle | The two sides of a right triangle that form the right angle. |
| Hypotenuse | The side of a right triangle opposite the right angle. |
| Shorter Side | The side of a triangle with the smallest length. |
| Mother and Daughter Age Problem | A word problem related to the ages of a mother and daughter at different times. |
| Mother is 11 times as old as her daughter | The mother's age is 11 multiplied by the daughter's age. |
| Sixteen Years Later | After a period of 16 years has passed. |
| Mother is 3 times as old as the daughter | The mother's age will be three times the daughter's age. |
| Water Seeps | Water slowly leaks or escapes through a small opening. |
| Hole | A small opening or gap. |
| Rate of Water Leak | The volume or mass of water that escapes per unit of time. |
| Interior Angle | An angle formed inside a polygon by two adjacent sides. |
| Regular Pentagon | A polygon with five equal sides and five equal angles. |
| Vertical Pole | An upright pole standing perpendicular to the ground. |
| Shadow | A dark area or shape produced by an object coming between rays of light and a surface. |
| Tree | A tall woody plant. |
| Situation whereby a payment is made for work not done | A fraudulent practice where payment is made for services or labor that were not actually performed. |
| More workers are used than reasonably required for efficient operation | An inefficient staffing practice where the number of employees exceeds what is necessary for optimal productivity. |
| Horizontal Pipe | A pipe oriented parallel to the ground. |
| Area | The extent or measure of a surface. |
| Velocity | The speed and direction of motion. |
| Pressure | Force applied per unit area. |
| Divisible by 6 | A number that can be divided by 6 without leaving a remainder. |
| Investment | The action or process of investing money for profit. |
| Compound Interest | Interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods. |
| Amounted to | Reached a total value of. |
| Rate of Interest per Annum | The percentage of interest charged or earned annually. |
| Side A, B, C | The lengths of the sides of a triangle. |
| Included Angle | The angle between two sides of a triangle. |
| Area of the Triangle | The measure of the space enclosed by the sides of a triangle. |
| Closed Cylindrical Tank | A container in the shape of a cylinder that is sealed at both ends. |
| Lying Horizontally | Positioned parallel to the ground. |
| Vertical Position | Standing upright. |
| Vertices | The corner points of a geometric shape. |
| Equipment | Tools, machinery, or other necessary items for a particular purpose. |
| Height | The measurement from base to top. |
| Seconds | A unit of time. |
| Power Exerted | The rate at which work is done by a device or person. |
| Bond | A written agreement promising to pay a debt or fulfill an obligation. |
| Contractor | A person or company that undertakes a contract to provide materials or labor for a project. |
| Construction Contract | A legally binding agreement between a client and a contractor for the execution of a construction project. |
| Conditions and Requirements | The specific terms and stipulations outlined in an agreement. |
| Tank | A container for storing liquids or gases. |
| Open to the Atmosphere | Not sealed, allowing free exchange of gases with the surrounding air. |
| Spout | A projecting tube or lip from which a liquid is poured. |
| Velocity of Water | The speed at which water is moving. |
| Atom Mass Unit (amu) | A unit of mass used to express the mass of atoms and molecules. |
| Grams (g) | A unit of mass in the metric system. |
| Fluid | A substance that can flow, like a liquid or gas. |
| Specific Gravity (SG) | The ratio of the density of a substance to the density of water. |
| Depth | The distance from the top or surface down to the bottom. |
| Force | A push or pull that can cause an object to move or change shape. |
| Resultant | The single vector that represents the sum of two or more vectors. |
| Electric Water Heater | A device that heats water using electricity. |
| Electric Current (A) | The flow of electric charge, measured in amperes. |
| Power Supply (V) | The voltage provided by an electrical source, measured in volts. |
| BTU (British Thermal Unit) | A unit of energy, often used to measure heating or cooling capacity. |
| Imaginary Number | A number that can be written as a real number multiplied by the imaginary unit $i$, where $i^2 = -1$. |
| Even Exponent | An exponent that is an even integer. |
| Real Number | Any number that can be found on the number line, including rational and irrational numbers. |
| Variable | A symbol representing a quantity that can change. |
| Inequality | A mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥. |
| Described | Characterized or explained. |
| Regular Dodecagon | A polygon with twelve equal sides and twelve equal angles. |
| Inscribed | Drawn inside another figure so as to touch at as many points as possible. |
| Circle | A set of all points in a plane that are equidistant from a central point. |
| Circumference | The distance around the outside of a circle. |
| Perimeter | The total distance around the outside of a polygon. |
| Solid Steel Ball | A sphere made entirely of steel. |
| Immersed | Placed or submerged in a liquid. |
| Displaces Water | Pushes aside a volume of water equal to its own volume. |
| Depth | The measurement from top to bottom. |
| Cylinder | A three-dimensional geometric shape with two parallel circular bases connected by a curved surface. |
| Derivative | The instantaneous rate of change of a function with respect to a variable, found using calculus. |
| Evaluate | To calculate or find the numerical value of an expression. |
| Asphalth block | A rectangular block made of asphalt. |
| Density | Mass per unit volume. |
| Apparent Weight | The weight of an object when it is submerged in a fluid, which is less than its actual weight due to buoyancy. |
| River | A natural flowing watercourse. |
| Spanned by a Bridge | Connected by a bridge. |
| Banked | Sloped or inclined. |
| Male to Female Employees Ratio | The comparison of the number of male employees to the number of female employees. |
| Left the Job | Departed from employment. |
| New Ratio | The updated comparison of male to female employees after some have left. |
| Water Velocity | The speed at which water is flowing. |
| Pipe | A tube used to convey water, gas, oil, or other fluid substances. |
| Flow Rate | The volume of fluid that passes through a given cross-sectional area per unit of time. |
| Square | A plane figure with four equal straight sides and four right angles. |
| Coordinates of the Vertices | The (x, y) positions of the corner points of a shape. |
| Slant Height | The distance from the apex of a cone or pyramid to a point on the edge of the base. |
| Right Circular Cone | A cone with a circular base and its apex directly above the center of the base. |
| Base Diameter | The distance across the circular base of a cone through its center. |
| Lateral Area | The surface area of a cone or pyramid, excluding the area of the base. |
| Geometric Sequence | A sequence of numbers where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. |
| Term | A single element in a sequence. |
| Binomial Expansion | The algebraic expansion of $(a+b)^n$. |
| Constant Term | The term in an expansion that does not have any variables. |
| Region Between Two Concentric Circles | The area enclosed by two circles that share the same center but have different radii; also known as an annulus. |
| Cyclist | A person who rides a bicycle. |
| Wind Velocity | The speed and direction of the wind. |
| KPH (Kilometers Per Hour) | A unit of speed. |
| Kinetic Energy | The energy of motion. |
| Velocity is Doubled | The speed of the object is multiplied by two. |
| Total Surface Area | The sum of the areas of all the faces of a three-dimensional object. |
| Square Pyramid | A pyramid with a square base. |
| Slant Height | The distance from the apex to the midpoint of an edge of the base. |
| Base Edge | The length of one side of the square base. |
| Probability | The likelihood of a specific event occurring. |
| Drawing Two Consecutive Face Cards | Selecting two playing cards that are face cards (Jack, Queen, King) one after the other. |
| Deck | A standard 52-card deck of playing cards. |
| Without Replacement | After a card is drawn, it is not put back into the deck before the next draw. |
| Perimeter of a Square | The total length of all sides of a square. |
| Doubles | Becomes two times larger. |
| Area | The amount of space inside a two-dimensional shape. |
| Retailer | A person or business that sells goods to consumers. |
| Items | Individual products or articles. |
| Average Price | The mean price of a set of items. |
| Removed Item | An item that has been taken away from the set. |
| Final Amount | The total value of a financial investment at the end of a period. |
| Initial Deposit | The starting amount of money placed in an account. |
| Compounding Monthly | Interest is calculated and added to the principal every month. |
| Rate of 8% | An annual interest rate of eight percent. |
| Buyers and Sellers Gather | Individuals or entities come together to engage in commercial transactions. |
| Exchange of Goods and Services | The trade of products and assistance between parties. |
| Business Organization | A legal entity formed to conduct commercial activities. |
| Profitable Business | A business that generates more revenue than its expenses. |
| Creates Jobs | Provides employment opportunities for individuals. |
| Contributes to National Income | Adds to the total economic output of a country. |
| Imports | Goods or services brought into a country from abroad. |
| Exports | Goods or services sent to a country from abroad. |
| Sustainable Economic Development | Economic growth that meets the needs of the present without compromising the ability of future generations to meet their own needs. |
| sin(2x) | The sine function applied to twice the angle x. This is a trigonometric identity related to double angles. |
| Ball A is Dropped Vertically | An object is released to fall straight down due to gravity. |
| Ball B is Launched Horizontally | An object is projected sideways with an initial forward motion. |
| Same Height | Both objects start at the same vertical elevation. |
| Same Time | The events of dropping and launching occur simultaneously. |
| Neglect Air Resistance | Assume that the force of air friction on the objects is negligible. |
| What happens to Ball A? | Asking about the motion or state of Ball A. |
| Vertical Pole | An upright pole. |
| Casts a Shadow | Creates a dark area due to blocking light. |
| At the Same Time | During the same period. |
| Tree | A large perennial plant. |
| Height of the Tree | The vertical dimension of the tree. |
| Payment for Work Not Done | Compensation provided for labor or tasks that were not actually performed. |
| More Workers than Reasonably Required | Employing an excessive number of staff for a given task, leading to inefficiency. |
| Horizontal Pipe | A pipe oriented parallel to the ground. |
| Area | The measure of a two-dimensional surface. |
| Point A, Point B | Specific locations within the pipe system. |
| Velocity | The speed and direction of movement. |
| Pressure | Force exerted per unit area. |
| Calculate | To determine a value or result using mathematical methods. |
| p – 10 is divisible by 6 | An algebraic expression where the result of subtracting 10 from p can be divided by 6 without a remainder. |
| Must also be divisible by 6 | This expression will also yield a result that can be divided by 6 without a remainder. |