4.3 Geometry, Mensuration & Commercial Math (Speed, Distance, Profit/Loss)

Geometry & Mensuration Formulas

Geometry and mensuration test a candidate's comprehension of 2D shapes, 3D solids, area, perimeter, volume, and geometric theorems.

Key 2D Geometric Formulas

Pythagorean Theorem & Triples

For a right-angled triangle with legs $a, b$ and hypotenuse $c$:

$$a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}$$

Standard Pythagorean Triples for Quick Calculation

• $3 - 4 - 5$ (and multiples: $6-8-10$, $9-12-15$)
• $5 - 12 - 13$ (and multiples: $10-24-26$)
• $8 - 15 - 17$
• $7 - 24 - 25$

3D Mensuration Formulas

Worked Example: Cylinder Mensuration

Problem: A cylindrical fuel storage tank at an airbase has a radius of $7\text{ m}$ and a height of $10\text{ m}$. Calculate its total capacity (volume) and total surface area. (Use $\pi = \frac{22}{7}$).

Solution:

1. Volume Calculation: $$V = \pi r^2 h = \frac{22}{7} \times 7^2 \times 10 = \frac{22}{7} \times 49 \times 10 = 22 \times 7 \times 10 = 1,540\text{ m}^3$$
2. Total Surface Area Calculation: $$TSA = 2\pi r (r + h) = 2 \times \frac{22}{7} \times 7 \times (7 + 10) = 44 \times 17 = 748\text{ m}^2$$

Speed, Distance, Time & Commercial Mathematics

Speed, Distance, and Time

The fundamental equation governing uniform motion is:

$$\text{Distance } (d) = \text{Speed } (v) \times \text{Time } (t)$$ $$\text{Speed } (v) = \frac{d}{t}, \qquad \text{Time } (t) = \frac{d}{v}$$

Unit Conversion Rules

• $\text{km/h}$ to $\text{m/s}$: Multiply by $\frac{5}{18}$ (e.g., $90 \text{ km/h} \times \frac{5}{18} = 25 \text{ m/s}$).
• $\text{m/s}$ to $\text{km/h}$: Multiply by $\frac{18}{5}$ (e.g., $20 \text{ m/s} \times \frac{18}{5} = 72 \text{ km/h}$).

Relative Speed & Average Speed

• Objects Moving in Same Direction: Relative Speed $v_{rel} = |v_1 - v_2|$.
• Objects Moving in Opposite Directions: Relative Speed $v_{rel} = v_1 + v_2$.
• Average Speed:

$$\text{Average Speed} = \frac{\text{Total Distance Covered}}{\text{Total Time Taken}}$$

Special Case: If equal distances are covered at speeds $v_1$ and $v_2$, the harmonic mean formula applies:

$$\text{Average Speed} = \frac{2 v_1 v_2}{v_1 + v_2}$$

Worked Example: Relative Speed (Aircraft Interception)

Problem: A PAF jet flies at $720 \text{ km/h}$ from Base A toward Base B ($600 \text{ km}$ apart). Simultaneously, a transport plane flies at $180 \text{ km/h}$ from Base B toward Base A. How long will it take for them to pass each other?

Solution:

• Opposite directions $\implies$ Relative Speed $v_{rel} = 720 + 180 = 900 \text{ km/h}$.
• Distance $d = 600 \text{ km}$.
• Time $t = \frac{d}{v_{rel}} = \frac{600}{900} = \frac{2}{3} \text{ hours} = 40 \text{ minutes}$.

Commercial Mathematics: Profit, Loss & Interest

Profit and Loss Definitions

• Cost Price (CP): The purchasing price of an article.
• Selling Price (SP): The price at which an article is sold.
• Profit (SP > CP): $\text{Profit} = \text{SP} - \text{CP} \implies \text{Profit \%} = \frac{\text{Profit}}{\text{CP}} \times 100\%$.
• Loss (CP > SP): $\text{Loss} = \text{CP} - \text{SP} \implies \text{Loss \%} = \frac{\text{Loss}}{\text{CP}} \times 100\%$.

Simple Interest ($I$) and Compound Interest ($A$)

• Simple Interest: $$I = \frac{P \cdot R \cdot T}{100}$$ $$\text{Total Amount } (A) = P + I = P \left(1 + \frac{R \cdot T}{100}\right)$$
• Compound Interest (Compounded Annually): $$A = P \left(1 + \frac{R}{100}\right)^T, \qquad \text{Compound Interest} = A - P$$

Worked Example: Simple Interest

Problem: An officer invests PKR 50,000 at a simple interest rate of $8\%$ per annum for $3$ years. Find the total interest earned and total final amount.

Solution:

• $P = 50,000$, $R = 8$, $T = 3$.
• $I = \frac{50,000 \times 8 \times 3}{100} = 500 \times 24 = \text{PKR } 12,000$.
• Total Amount $A = 50,000 + 12,000 = \text{PKR } 62,000$.