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 β‰  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 β‰  0) So, a : b = xa : xb

● a/b = (a/x)/(b/x), (x β‰  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 Γ· 5)/(10 Γ· 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 Γ— 100) cm

= 64 cm/480 m

= 64/480

= 2/15

= 2 : 15

(b) Required ratio = 36 minutes/36 seconds

= (36 Γ— 60 seconds)/(36 seconds)

= 60/1

= 60 ∢ 1

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

= (30 Γ— 12)/(2 Γ— 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) ⁡/β‚‚ ∢ Β³/β‚ˆ ∢ ⁴/₉

(b) 2ΒΉ/₇ ∢ 3Β²/β‚…

Solution:

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

                                        = 8 Γ— 9

                                        = 72

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

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

So, the ratio becomes 160 : 27 : 32

(b) 2ΒΉ/₇ ∢ 3Β²/β‚…

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

= 15/7 Γ— 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?

2ΒΉ/₃ ∢ 3ΒΉ/β‚‚, 2.5 : 3.5, 4/5 ∢ 3/2

Solution:

Simplifying the given 3 ratios

2ΒΉ/₃ ∢ 3ΒΉ/β‚‚ = ⁷/₃ ∢ ⁷/β‚‚ = ⁷/₃ Γ· ⁷/β‚‚ = ⁷/₃ Γ— Β²/₇ = Β²/₃

2.5 : 3.5 = ²⁡/₃₅ = ⁡/₇

⁴/β‚… : Β³/β‚‚ = ⁴/β‚… Γ— Β²/₃ = ⁸/₁₅

Β²/₃, ⁡/₇, ⁸/₁₅

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

Β²/₃ = (2 Γ— 35)/(3 Γ— 35) = ⁷<span style='font-size: 50%'>/₁₀₅,

⁡/₇ = (5 Γ— 15)/(7 Γ— 15) = ⁴⁡/₁₀₅,

⁸/₁₅ = (8 Γ— 7)/(15 Γ— 7) = ⁡⁢/₁₀₅

\(\frac{70}{105}\) > \(\frac{56}{105}\) > \(\frac{45}{105}\)


Therefore, Β²/₃ > ⁸/₁₅ > ⁡/₇

Therefore, 2ΒΉ/₃ ∢ 3ΒΉ/β‚‚ > 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 1Λ’α΅— part = {a/(a + b)} Γ— p and 2ⁿᡈ part {b/(a + b)} Γ— p

Example:

Divide $290 among A, B, C in the ratio 1ΒΉ/β‚‚, 1ΒΉ/β‚„ and 7/8.

Solution:

Given ratios = Β³/β‚‚ : ⁡/β‚„ : 7/8.

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

So we have Β³/β‚‚ Γ— 8 : ⁡/β‚„ Γ— 8 ∢ 7/8 Γ— 8 = 12 ∢ 10 : 7

Therefore, Share of A = \(\frac{12}{29}\) Γ— 290 = $120

Share of B = \(\frac{10}{29}\) Γ— 290 = $100

Share of C = \(\frac{7}{29}\) Γ— 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).

β‡’ \(\frac{a}{b}\) = \(\frac{c}{d}\)

β‡’ a Γ— d = b Γ— c

β‡’ 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

β‡’ \(\frac{a}{b}\) = \(\frac{b}{c}\)

β‡’ a Γ— c = bΒ²

β‡’ bΒ² = ac

β‡’ b = √ac

Here, b is called the mean proportional of a and c. The square of middle term is equal to the product of 1Λ’α΅— term and 3ʳᡈ 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) 1Β²/₃, 6ΒΉ/β‚„, ⁴/₉, ⁡/₃

Solution:

(a) Here, product of first term and third term = 6 Γ— 24 = 144 and square of middle term = (12) Β² = 12 Γ— 12 = 144

(b) 1Β²/₃, 6ΒΉ/β‚„, ⁴/₉, ⁡/₃

Here, a = 1Β²/₃   b = 6ΒΉ/β‚„   c = ⁴/₉   d = ⁡/₃

a : b = 1Β²/₃ : 6ΒΉ/β‚„         c : d = ⁴/₉ : ⁡/₃

        = ⁡/₃ ∢ ²⁡/β‚„                   = (4/9)/(5/3)

        = (5/3)/(25/4)              = 4/9 Γ— 3/5

        = 5/3 Γ— 4/25                = 4/3 Γ— 1/5

        = 4/15                         = 4/15

Since, a : b = c : d

Therefore, 1Β²/₃, 6ΒΉ/β‚„, ⁴/₉, ⁡/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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