Ratio and Proportion

In math ratio and proportion we will elaborate the terms and discuss more about it in detailed explanation.

● Ratio and terms of ratio

● Properties of ratio

● Ratio in the simplest form

● Simplification of ratio

● Comparison of ratio

● Dividing the given quantity in the given ratio

● Proportion

● Continued proportion

● Examples on ratio and proportion

Ratio

The ratio of two quantities 'a' and 'b' of the same kind and in the same units is a fraction $\frac{a}{b}$ which shows that how many times one quantity is of the other and is written as a : b and is read as 'a is to b' where b ≠ 0.

Terms of the ratio

In the ratio a : b, the quantities a and b are called terms of the ratio. Here, 'a' is called the first term or the antecedent and `b' is called the second term or consequent.

Example:

In the ratio 5 : 9, 5 is called the antecedent and 9 is called the consequent.

Properties of ratio

If the first term and the second term of a ratio are multiplied/divided by the same non-zero number, the ratio does not change.

● a/b = xa/xb, (x ≠ 0) So, a : b = xa : xb

● a/b = (a/x)/(b/x), (x ≠ 0) So, a : b = a/x : b/x

Ratio in the simplest form

A ratio a : b is said to be in the simplest form if a and b have no common factor other than 1.

Example:

Express 15 : 10 in the simplest form.

Solution:

15/10

= (15 ÷ 5)/(10 ÷ 5)

= 3/2 (In this we cancelled the common factor 5)

Thus, we have expressed the ratio 15/10 in the simplest form, i.e., 3/2 and the terms 3 and 2 have common factor only 1.

Note:

● In ratio, quantities being compared must be of the same kind, otherwise the comparison becomes meaningless.

For example; comparing 20 pens and 10 apples is meaningless.

● They must be expressed in the same units.

● In a ratio, order of the terms is very important. The ratio a: b is different from b : a.

● The ratio has no units.

For example; Dozen = 12, Gross = 144, Score = 20

Decade = 10, Century = 100, Millennium = 1000

Example:

Express the following ratios in the simplest form.

(a) 64 cm to 4.8 m

(b) 36 minutes to 36 seconds

(c) 30 dozen to 2 hundred

Solution:

(a) Required ratio = 64 cm/4.8 m

= 64 cm/(4.8 × 100) cm

= 64 cm/480 m

= 64/480

= 2/15

= 2 : 15

(b) Required ratio = 36 minutes/36 seconds

= (36 × 60 seconds)/(36 seconds)

= 60/1

= 60 ∶ 1

(c) Required ratio = (30 dozen)/(2 hundred)

= (30 × 12)/(2 × 100 )

= 3/10

= 3 ∶ 10

Simplification of ratio

If the terms of the ratio are expressed in fraction form; then find the Least Common Multiple of the denominators of these fractions. Now, multiply each fraction by the L.C.M. The ratio is simplified.

Example:

Simplify the following ratios.

(a) ⁵/₂ ∶ ³/₈ ∶ ⁴/₉

(b) 2¹/₇ ∶ 3²/₅

Solution:

(a) The L.C.M. of 2, 8, 9 = 2 × 2 × 2 × 3 × 3

= 8 × 9

= 72

Now, multiplying each fraction by the L.C.M.

5/2 × 72 = 160 3/8 × 72 = 27 4/9 × 72 = 32

So, the ratio becomes 160 : 27 : 32

(b) 2¹/₇ ∶ 3²/₅

= 15/7 : 17/5 (Here, we have used (a/b)/(c/d) = $\frac{a}{b}$ × $\frac{d}{c}$)

= 15/7 × 5/17

= 75/119

So, the ratio becomes 75 : 119

Comparison of ratios

Ratios can be compared as fractions. Convert them into equivalent ratios as we convert the given fractions into equivalent fractions and then compare.

Example:

Which ratio is greater?

2¹/₃ ∶ 3¹/₂, 2.5 : 3.5, 4/5 ∶ 3/2

Solution:

Simplifying the given 3 ratios

2¹/₃ ∶ 3¹/₂ = ⁷/₃ ∶ ⁷/₂ = ⁷/₃ ÷ ⁷/₂ = ⁷/₃ × ²/₇ = ²/₃

2.5 : 3.5 = ²⁵/₃₅ = ⁵/₇

⁴/₅ : ³/₂ = ⁴/₅ × ²/₃ = ⁸/₁₅

²/₃, ⁵/₇, ⁸/₁₅

L.C.M. of 3, 7, 15 = 105

²/₃ = (2 × 35)/(3 × 35) = ⁷<span style='font-size: 50%'>/₁₀₅,

⁵/₇ = (5 × 15)/(7 × 15) = ⁴⁵/₁₀₅,

⁸/₁₅ = (8 × 7)/(15 × 7) = ⁵⁶/₁₀₅

$\frac{70}{105}$ > $\frac{56}{105}$ > $\frac{45}{105}$

Therefore, ²/₃ > ⁸/₁₅ > ⁵/₇

Therefore, 2¹/₃ ∶ 3¹/₂ > 4/5 ∶ 3/2 > 2.5 : 3.5

Dividing the given quantity in the given ratio

If 'p’ is the given quantity to be divided in the ratio a : b, then add the terms of the a ratio, i.e., a + b, then the 1ˢᵗ part = {a/(a + b)} × p and 2ⁿᵈ part {b/(a + b)} × p

Example:

Divide $290 among A, B, C in the ratio 1¹/₂, 1¹/₄ and 7/8.

Solution:

Given ratios = ³/₂ : ⁵/₄ : 7/8.

The L.C.M. of 2, 4, 8 is 8.

So we have ³/₂ × 8 : ⁵/₄ × 8 ∶ 7/8 × 8 = 12 ∶ 10 : 7

Therefore, Share of A = $\frac{12}{29}$ × 290 = $120

Share of B = $\frac{10}{29}$ × 290 = $100

Share of C = $\frac{7}{29}$ × 290 = $70

Proportion

We have already learnt that statement of equality of ratios is called proportion, if four quantities a, b, c, d are in proportion, then a : b = c : d or a : b : : c : d (: : is the symbol used to denote proportion).

⇒ $\frac{a}{b}$ = $\frac{c}{d}$

⇒ a × d = b × c

⇒ ad = bc

Here a, d are called the extreme terms in which a is called the first term and d is called the fourth term and b, c are called the mean terms in which b is called the second term and c is called the third term.

Thus, we say, if product of mean terms = the product of extreme terms, then the terms are said to be in proportion.

Also, if a : b :: c : d, then d is called the fourth proportional of a, b, c.

Continued Proportion

The three quantities a, b, c are said to be in continued proportion if a : b :: b : c

⇒ $\frac{a}{b}$ = $\frac{b}{c}$

⇒ a × c = b²

⇒ b² = ac

⇒ b = √ac

Here, b is called the mean proportional of a and c. The square of middle term is equal to the product of 1ˢᵗ term and 3ʳᵈ term.

Also, if a : b :: b : c, then c is called the third proportional of a, b.

Example:

Determine if the following are in proportion.

(a) 6, 12, 24

(b) 1²/₃, 6¹/₄, ⁴/₉, ⁵/₃

Solution:

(a) Here, product of first term and third term = 6 × 24 = 144 and square of middle term = (12) ² = 12 × 12 = 144

(b) 1²/₃, 6¹/₄, ⁴/₉, ⁵/₃

Here, a = 1²/₃ b = 6¹/₄ c = ⁴/₉ d = ⁵/₃

a : b = 1²/₃ : 6¹/₄ c : d = ⁴/₉ : ⁵/₃

= ⁵/₃ ∶ ²⁵/₄ = (4/9)/(5/3)

= (5/3)/(25/4) = 4/9 × 3/5

= 5/3 × 4/25 = 4/3 × 1/5

= 4/15 = 4/15

Since, a : b = c : d

Therefore, 1²/₃, 6¹/₄, ⁴/₉, ⁵/₃ are in proportion.

Follow the examples on ratio and proportion then, practice the problems given in the worksheet.