Ratio and Proportion: Practice Questions & Notes

A ratio compares two quantities of the same kind by division.

If A and B are two quantities, then their ratio is written as:

\text{A : B} = \frac{A}{B}

Example: If a bag has 2 red and 3 blue balls, the ratio of red to blue is 2:3.

Key Points:

A ratio has no units since it's a comparison.
The order matters: 2:3 ≠ 3:2
Ratios can be simplified like fractions.

Understanding Proportion

When two ratios are equal, they are said to be in proportion.

\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } a : b :: c : d

This is read as: "a is to b as c is to d."

Key Formulas and Concepts

1. Simplifying Ratios

Divide both terms by their HCF.

Example:

24 : 36 = \frac{24}{12} : \frac{36}{12} = 2 : 3

2. Equivalent Ratios

Multiply/divide both terms by the same non-zero number.

Example:

2:3 = \frac{2 \times 4}{3 \times 4} = 8 : 12

3. Compound Ratio

The product of two or more ratios.

If A : B = 2:3 and B : C = 4:5,
then A : C = (2 × 4) : (3 × 5) = 8 : 15

4. Inverse Ratio

Swap the terms.

If A : B = 3 : 5
→ Inverse = 5 : 3

5. Dividing a Quantity in a Given Ratio

If a quantity Q is divided in the ratio $m : n$ , then:

First part = $\frac{m}{m+n} \times Q$
Second part = $\frac{n}{m+n} \times Q$

Example:
Divide ₹300 in 2:3

Total = 2 + 3 = 5
First share = $\frac{2}{5} \times 300 = \text{Rs.~}120$
Second share = ₹180

6. Comparing Ratios

To compare A:B and C:D, convert both to a common second term or decimal.

Example:
Compare 2:5 and 3:7

Decimal values: $\frac{2}{5} = 0.4$ , $\frac{3}{7} \approx 0.428$
Hence, 3:7 > 2:5

Types of Proportion

Type	Rule	Example
Direct	Increase in one → increase in another	Cost ∝ Quantity
Inverse	Increase in one → decrease in another	Speed ∝ 1/Time
Continued	a : b = b : c → a, b, c are in continued proportion	2:4 = 4:8
Mean	If a : x = x : b → x² = ab → x = $\sqrt{ab}$	Mean proportion

Important Tricks & Shortcuts

1. Cross Multiplication (for checking proportion)

If $a : b = c : d$ , then:

a \times d = b \times c

2. Shortcut for Adjusting Ratios

If the same quantity is added/subtracted to both parts of a ratio, the ratio may change. Use algebra to verify:
Let A:B = 4:5
Add 20 to both → New = $\frac{4x + 20}{5x + 20}$

Only if x is known, we can compute.

3. Shortcut for Merging Multiple Ratios

To combine A:B and B:C into A:B:C:

Multiply the middle terms to make them equal.

Example:
A : B = 3 : 4 and B : C = 2 : 5
→ LCM of 4 and 2 = 4
→ Convert: A:B = 3:4, B:C = 4:10
→ A : B : C = 3 : 4 : 10

Conceptual Insights & Common Mistakes

Mistake	Why it happens	Tip
Misinterpreting order	A:B ≠ B:A	Always check who is first
Forgetting total in division	Only uses one part	Add all parts before dividing
Confusing with percentages	Mixing % and ratios	Convert one form fully
Using ratios on different units	Invalid comparison	Ensure same units

Real-Life Applications

Splitting profits in partnerships
Comparing incomes or expenses
Scaling recipes or construction material
Voting and polling data

Visual Summary

Ratio → Comparison of two values
Proportion → Two ratios are equal

Key tools:
- Simplify using HCF
- Cross multiply to check
- Convert ratios to decimals for comparison
- Merge ratios using LCM

Practice Examples

Q1. If ₹720 is divided in the ratio 5:3, what are the shares?

Total parts = 8
First = $\frac{5}{8} \times 720 = \text{Rs.~}450$
Second = ₹270

Q2. A : B = 3 : 4, B : C = 2 : 5. Find A : B : C.

Convert to common B:
A : B = 3 : 4
B : C = 4 : 10
→ A : B : C = 3 : 4 : 10

Q3. If A : B = 2 : 3, B : C = 5 : 6, what is A : C?

A:B = 2:3, B:C = 5:6
→ Convert B terms to LCM (15):
A:B = 10:15, B:C = 15:18
→ A:B:C = 10:15:18
→ A:C = 10:18 = 5:9