Ratio and Proportion: Concepts, Formulas, and Clear Examples

Ratio $a:b$ compares parts; proportion $a:b=c:d$ means $\frac{a}{b}=\frac{c}{d}$ or $ad=bc$ (cross product).

Partnership, mixtures, and scale maps are proportion word problems—keep units aligned.

Combine ratios only through a common middle term or convert to fractions of whole.

Simplifying ratios

Fraction from ratio

$$a:b \Rightarrow \frac{a}{a+b},\ \frac{b}{a+b}$$

• Divide by HCF.
• Same units before comparing (cm vs m trap).

Proportion

Cross product

$$a:b = c:d \;\Rightarrow\; ad = bc$$

Fourth proportional

$$If a:b=c:x,\ x=\frac{bc}{a}$$

Example — Find fourth proportional

2:3 = 8:x → $2x=24$ → $x=12$.

Continued ratio a:b:c

• From a:b and b:c, align b; multiply parts to get a:b:c.
• Total parts = sum of ratio terms.

Inverse proportion

Inverse

$$x_1 y_1 = x_2 y_2$$

• More workers → less days (same work).

Compounding ratios

• Income ratios with capital × time in partnership tie here.

Exam applications

• Divide quantity Q in ratio m:n → $\frac{m}{m+n}Q$, $\frac{n}{m+n}Q$.

Step-by-step method for exam questions

1. Convert to same units; simplify ratio.
2. For proportion, cross-multiply.
3. Merge ratios via common term.
4. Assign total to sum of parts.
5. Check inverse vs direct wording.
6. Select option.

Formula cheat sheet

Frequently asked questions

Add ratios?

Not directly; convert to fractions of whole or find common term.

Ratio vs fraction?

a:(a+b) of total vs part:whole.

Duplicate ratio 2:3:2:3?

Simplify to 2:3 if same quantities.

Unit mismatch?

Convert first; ratio undefined across units.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.