Ratio and proportion

A ratio compares two quantities of the same kind, written $a : b$ . Two ratios are in proportion when they are equal. Ratios are simplified the same way as fractions, by dividing both parts by a common factor.

Formulas

Simplifying a ratio

a : b = (a \div k) : (b \div k)

Sharing a quantity Q in the ratio a : b

first share = \frac{a}{a + b} \times Q second share = \frac{b}{a + b} \times Q

Proportion

\frac{a}{b} = \frac{c}{d} ⟺ a d = b c

Worked examples

Example 1. Simplify the ratio 24 : 36.

The HCF of 24 and 36 is 12.
$24 \div 12 : 36 \div 12 = 2 : 3$ .

Answer: 2 : 3

Example 2. Share ₹1,200 between two friends in the ratio 3 : 5.

Total parts = $3 + 5 = 8$ , so one part is $1200 \div 8 = 150$ .
Shares: $3 \times 150 = 450$ and $5 \times 150 = 750$ .

Answer: ₹450 and ₹750

Example 3. 5 pens cost ₹60. What do 8 pens cost?

One pen costs $60 \div 5 = 12$ .
Eight pens cost $8 \times 12 = 96$ .

Answer: ₹96

Example 4. Are 4 : 6 and 10 : 15 in proportion?

Cross-multiply: $4 \times 15 = 60$ and $6 \times 10 = 60$ .
They are equal, so yes (both simplify to 2 : 3).

Answer: Yes

Common mistakes

Writing the ratio in the wrong order. "Boys to girls" is not the same as "girls to boys".
Comparing quantities in different units. Change them to the same unit first (50 cm : 2 m becomes 50 : 200).
Dividing a quantity by the ratio parts without adding them first. Divide by the total number of parts.

Practice questions with answers

1. The ratio of 4 to 9 is written as:

A4 : 9
B9 : 4
C4 + 9
D4 - 9

Show answer

Answer: A) 4 : 9

A ratio of 4 to 9 is written in the order given, as 4 : 9.

2. Which ratio is equivalent to 1:2?

A2:4
B2:3
C3:2
D1:3

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Answer: A) 2:4

Multiply both terms of 1:2 by 2 to get 2:4. So 1:2 = 2:4.

3. In the ratio 2 : 3, how many equal parts are there in all?

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Answer: B) 5

Total parts = 2 + 3 = 5. (6 multiplies the terms instead of adding them.)

4. Express the ratio 12 : 16 in simplest form.

A3 : 4
B4 : 3
C2 : 4
D3 : 5

Show answer

Answer: A) 3 : 4

HCF of 12 and 16 is 4. Divide both: 12÷4 = 3, 16÷4 = 4. Ratio = 3 : 4.

5. A sum of money is shared between Neha and Pooja in the ratio 4 : 5.
Neha's share is ₹240.
What is the total sum of money shared (in ₹)?

A1080
B540
C595
D485

Show answer

Answer: B) 540

Neha = 4 parts = ₹240, so 1 part = ₹60. Total = (4 + 5) parts =

9 \times 60

= ₹540.

6. A ratio 9:12 is to be written as an equivalent ratio of the form ?:20.
A student notices that 9:12 must first be simplified before scaling so the second term can become 20.
What is the correct first term?

Show answer

Answer: A) 15

9:12 simplifies to 3:4. To make the second term 20, multiply by 5:

4 \times 5 = 20

and

3 \times 5 = 15

. So the first term is 15.

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Frequently asked questions

How do you simplify a ratio?

Divide both parts by their highest common factor, exactly as you simplify a fraction.

How do you divide a quantity in a given ratio?

Add the parts to get the total, divide the quantity by that total to find one part, then multiply by each ratio number.

What is the difference between ratio and proportion?

A ratio compares two quantities. A proportion is a statement that two ratios are equal.

What is direct proportion?

Two quantities are in direct proportion when one increases by the same factor as the other, for example the cost of pens and the number of pens.

Ratio and proportion

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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