Class 7 Ratio and Proportion

Introduction to Ratio

Properties of Ratio

Ratio in the Simplest Form

Comparison of Ratios

Equivalent Ratio

Proportion

Continued Proportion

Ratio and Proportion Test

Ratio and Proportion Worksheet

Answer Sheet

Introduction to Ratio

Ratio is a comparison of measures of two or more quantities of the same kind by division. For example, if 'p' and 'q' are two quantities of the same kind and having same units, then the ^p⁄_q is called the ratio of 'p' to 'q'.

The ratio of p to q can be denoted as ^p⁄_q or p : q.

Here 'p' and 'q' are known as terms of the ratio. The 'p' is known as first term or antecedent and 'q' is known as the second term or consequent.

Properties of Ratio

A ratio does not alter, if it's first and second terms are multiplied or divided by the same non-zero number.

^p⁄_q = ^ap⁄_aq (p : q = ap : aq, where a ≠ 0)
^p⁄_q = ^p÷a⁄_q÷a (p : q = ^p⁄_q : ^p⁄_q, where a ≠ 0)

Ratio in the Simplest Form

A ratio p : q is said to be in the simplest form if it's first term 'p' and second term 'q' have no common factor other than 1. Steps to convert a ratio into it's simplest form.

Step ‐ 1. Convert both the terms of the ratio into same unit.

Step ‐ 2. Find out the HCF of first and second terms.

Step ‐ 3. Divide the first and second terms by their HCF

Let's see some examples.

Example 1. Find the ratio 5 m to 75 cm.

Solution. First, we must convert both the terms into same unit.

5 m = 500 cm

New ratio is 500 : 75.

Find HCF of 500 and 75.

500 = 2 × 2 × 5 × 5 × 5

75 = 3 × 5 × 5

HCF of 500 and 75 = 25

⁵⁰⁰⁄₇₅ = ^500÷25⁄_75÷25 = ²⁰⁄₃

Example 2. Express the 56 : 160 in its simplest form.

Solution. First find out the HCF of 56 and 160.

56 = 2 × 2 × 2 × 7

160 = 2 × 2 × 2 × 2 × 2 × 5

So, HCF of 56 and 160 = 2 × 2 × 2 = 8

Now, divide 56 and 160 by their HCF to get its simplest form.

56 : 160 = ⁵⁶⁄₁₆₀ = ^56÷8⁄_160÷8 = ⁷⁄₂₀ = 7 : 20

Example 3. Divide Rs. 2500 between Julie and Viona in the ration 2 : 3.

Solution. Sum of both the terms of the ratio = 2 + 3 = 5

Julie's share would be = ²⁄₅ × 2500 = Rs. 1000

Viona's share would be = ³⁄₅ × 2500 = Rs. 1500

So, Julie and Viona would get Rs. 1000 and Rs. 1500 respectively.

Example 4. Divide 15000 tons of rice among Odisha, west Bengal and Bihar in the ratio of 4 : 5 : 6.

Solution. Sum of three terms of the ratio = 4 + 5 + 6 = 15

Odisha's share would be = ⁴⁄₁₅ × 15000 = 4000 tons

West Bengal's share would be = ⁵⁄₁₅ × 15000 = 5000 tons

Bihar's share would be = ⁶⁄₁₅ × 15000 = 6000 tons

So, Odisha, West Bengal and Bihar would get 4000 tons, 5000 tons and 6000 tons respectively.

Example 5. Ratio of the boys and girls in a school is 3 : 5. If there are 120 boys in the school, then find out the girls strength.

Solution. Let's assume total students in the school is 'm'.

        ³⁄₈ × m = 120

      => m = 120 × ⁸⁄₃

      => m = 40 × 8

      => m = 320

Number of students in the school is 320.

Number of girls in the school = 320 − 120 = 200

Example 6. What must be added to each term of the ratio 2 : 3 so that it may become equal to 5 : 9?

Solution. Let the number to be added is 'a'.

        (2 + a) : (3 + a) = 5 : 9

      => ^2+a⁄_3+a = ⁵⁄₉

      => 9 × (2 + a) = 5 × (3 + a)

      => 18 + 9a = 15 + 5a

      => 4a = -3

      => a = ^-3⁄₄

Comparison of Ratios

To compare two given ratios, we must follow the following steps.

Step ‐ 1. Convert both the ratios into fraction in the simplest form.

Step ‐ 2. Find the LCM of the denominators of the fractions generated from step 1.

Step ‐ 3. Obtain both the fractions and their denominators. Divide the LCM by the denominators of both the fractions. Multiply the individual results with it's corresponding numerator and denominators.

Step ‐ 4. Compare the numerators of both the fractions obtained in step 3, fraction having larger numerator will be greater than the other.

Let's see some examples.

Example 1. Compare the ratios 5 : 9 and 2 : 3.

Solution. Write both the ratios in fraction form.

5 : 9 = ⁵⁄₉ and 2 : 3 = ²⁄₃

Now, LCM of 9 and 3 is equal to 9.

Here, we must make both the fraction's denominator as 9.

⁵⁄₉ has denominator 9 already, so we do not have to do anything for it.

²⁄₃ = ^2×3⁄_3×3 = ⁶⁄₉

⁵⁄₉ < ⁶⁄₉

Hence ⁵⁄₉ < ²⁄₃

Example 2. Compare the ratios 4 : 9 and 5 : 24.

Solution. Fraction form of both the ratios are

4 : 9 = ⁴⁄₉ and 5 : 24 = ⁵⁄₂₄

LCM of 8 and 24 is equal to 72

Here, we must make both the fraction's denominator as 72.

      ⁴⁄₉ = ^4×8⁄_9×8 = ³²⁄₇₂

      ⁵⁄₂₄ = ^5×3⁄_24×3 = ¹⁵⁄₇₂

      ³²⁄₇₂ > ¹⁵⁄₇₂

Hence, ⁴⁄₉ > ⁵⁄₂₄

Equivalent Ratio

A ratio obtained by multiplying or dividing the numerator and denominator of a ratio by same number is known as equivalent ratio. Let's see one example to understand it better.

Example 1. Find the equivalent ratio of 3 : 5.

Solution. First convert the ratio into fraction.

3 : 5 = ³⁄₅

³⁄₅ = ^3×2⁄_5×2 = ^3×3⁄_5×3 = ^3×4⁄_5×4 and so on.

³⁄₅ = ⁶⁄₁₀ = ⁹⁄₁₅ = ¹²⁄₂₀

Proportion

Four numbers 'p', 'q', 'r' and 's' are said to be in proportion, if the ratio of 'p' and 'q' is equal to the ratio of 'r' and 's'.

p : q = r : s

In other words p : q = r : s if and only if ps = qr

Let's see some examples.

Example 1. Check if 5 : 15 and 7 : 21 are in proportion.

Solution. 5 : 15 = 1 : 3 and 7 : 21 = 1 : 3

Hence, 5 : 15 = 7 : 21 is a proportion.

Example 2. Check if 6 : 9 and 30 : 45 are in proportion.

Solution. Let's convert both the ratios into simplest form.

6 : 9 = 2 : 3 and 30 : 45 = 2 : 3

Hence, 6 : 9 = 30 : 45 is a proportion.

Continued Proportion

Three numbers 'p', 'q', and 'r' are said to be in continued proportion if 'p', 'q', 'q', 'r' in proportion.

      P : q = q : r

If 'p', 'q', 'q', 'r' are in proportion i.e. p : q = q : r

        ^p⁄_q = ^q⁄_r

      => pr = q²

      => q² = pr

Let's see some examples on proportion.

Example 1. Check if 30, 40, 60, 80 are in proportion.

Solution. 30 : 40 = ³⁰⁄₄₀ = ³⁄₄

60 : 80 = ⁶⁰⁄₈₀ = ³⁄₄

Hence, 30 : 40 = 60 : 80 are in proportion.

Example 2. Check if 4, 16, 36 are in continued proportion?

Solution. We know that three numbers p, q, r in continued proportion, if p, q, q, r are in proportion.

So, in this case 4, 16, 36 will be in continued proportion if 4, 16, 16, 36 are in proportion.

We have to prove that ⁴⁄₁₆ should be equal to ¹⁶⁄₃₆.

⁴⁄₁₆ = ¹⁄₄ and ¹⁶⁄₃₆ = ⁴⁄₉

It is proved that ⁴⁄₁₆ ≠ ¹⁶⁄₃₆

So, 4, 16, 36 are not in continued proportion.

Example 3. The first three terms of a proportion are 2, 5, and 32 respectively. Find the fourth term.

Solution. Let's assume the fourth term to be 'y'.

        ²⁄₅ = ³²⁄_y

      => 2y = 32 × 5

      => y = ^32×5⁄₂

      => y = 16 × 5

      => y = 80

So, the fourth number is 80.

Example 4. What must be added to the numbers 25, 35, 55, 75 so that they are in proportion?

Solution. Let's assume the required number is 'p'.

Then 25 + p, 35 + p, 55 + p, 75 + p are in proportion.

        ^25+p⁄_35+p = ^55+p⁄_75+p

      => (25 + p)(75 + p) = (55 + p)(35 + p)

      => 25 × 75 + 25p + 75p + p² = 55 × 35 + 55p + 35p + p²

      => 1875 + 100p + p² = 1925 + 90p + p²

      => 100p − 90p = 1925 − 1875

      => 10p = 50

      => p = ⁵⁰⁄₁₀

      => p = 5

So, the number is 5.