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/480m

= 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 ³/₈.

Solution:

Given ratios = ³/₂ : ⁵/₄ : ³/₈.

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

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

Therefore, Share of A = 12/29 × 290 = $120

Share of B = 10/29 × 290 = $100

Share of C = 3/29 × 290 = $30

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.


 Ratio and Proportion

What is Ratio and Proportion?

Worked out Problems on Ratio and Proportion

Practice Test on Ratio and Proportion


 Ratio and Proportion - Worksheets

Worksheet on Ratio and Proportion










8th Grade Math Practice

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