Ratio In The Simplest Form

Ratio in the simplest form : A ratio a : b is said to be in the simplest form if its antecedent ‘a’ and consequent ‘b’ have no common factor other than 1.

A ratio in the simplest form is also called the ratio in the lowest terms.

**Example : ** The ratio 32 :48 is not in the simplest form, because 16 is a common factor of its antecedent and consequent. The simplest form of this ratio is 2 : 3 ( dividing the 1st term and 2nd term by 16).

**Note 1 : ** To find the ratio of two quantities, they must be expressed in the same units.

**Note 2 : ** As the ratio of two quantities are in same units so ratio has no unit or it is independent of the units used in the quantities compared.

**Note 3 :** The order of the terms in a ratio a : b is very important. The ratio

3 : 2 is different from the ratio 2 : 3.

** Examples : **

1) Express the following ratios in the simplest form.

a) 150 : 400

**Solution :**

The given ratio is 150 : 400 = 150/400

The common factor of 150 and 400 is 50.

(150 ÷ 50)/ (400 ÷ 50) = 3/8 = 3 : 8.

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b) A dozen to score

**Solution :**

One dozen = 12 and one score = 20.

So, a dozen to score = 12/20 = 12 : 20.

(12 ÷ 4)/ (20 ÷ 4) [ As the common factor is 4]

= 3/5 = 3 : 5.

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c) 200 grams to 4 kg

**Solution :**

200 g to 4 kg

As both the quantities are in different units, first make their units same.

1kg = 1000 g

∴ 4 kg = 4000 g.

200 g to 4000 g = 200 / 4000

Common factor of 200 and 4000 is 200.

(200 ÷ 200)/(4000 ÷ 200) = 1/20 = 1 : 20.

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d) Sam earned $4000 and paid $500 as income tax and spent $1500 on household expenses. Find the ratio of : 1) income tax to income 2) household expenses to income 3) savings to earnings

**Solution **

Sam’s income = $4000

Income tax = $500

∴ income tax to income = 500/4000

(500 ÷ 500)/(4000 ÷ 500) = 1/8

Ratio of Income tax to income is 1 : 8

Expenses = $1500

∴ Household expenses to income = 1500/4000

(1500 ÷ 500)/ (4000 ÷ 500) = 3/8

Ratio of Household expenses to income = 3 : 8.

Savings = Income – (expenses + income tax)

Savings = 4000 – (1500 + 500) = $2000.

∴ Savings to earnings = 2000/4000

(2000 ÷ 2000)/(4000 ÷ 2000) = ½

Ratio of savings to expenses = 1 : 2.

**Ratio - Proportion **

• Ratio and Proportion

• Ratio in the simplest form

• Comparison of ratios

• Equivalent ratios

• Proportion

• Continued Proportion

From ratio in the simplest form to number system

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A ratio in the simplest form is also called the ratio in the lowest terms.

3 : 2 is different from the ratio 2 : 3.

1) Express the following ratios in the simplest form.

a) 150 : 400

The given ratio is 150 : 400 = 150/400

The common factor of 150 and 400 is 50.

(150 ÷ 50)/ (400 ÷ 50) = 3/8 = 3 : 8.

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b) A dozen to score

One dozen = 12 and one score = 20.

So, a dozen to score = 12/20 = 12 : 20.

(12 ÷ 4)/ (20 ÷ 4) [ As the common factor is 4]

= 3/5 = 3 : 5.

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c) 200 grams to 4 kg

200 g to 4 kg

As both the quantities are in different units, first make their units same.

1kg = 1000 g

∴ 4 kg = 4000 g.

200 g to 4000 g = 200 / 4000

Common factor of 200 and 4000 is 200.

(200 ÷ 200)/(4000 ÷ 200) = 1/20 = 1 : 20.

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d) Sam earned $4000 and paid $500 as income tax and spent $1500 on household expenses. Find the ratio of : 1) income tax to income 2) household expenses to income 3) savings to earnings

Sam’s income = $4000

Income tax = $500

∴ income tax to income = 500/4000

(500 ÷ 500)/(4000 ÷ 500) = 1/8

Ratio of Income tax to income is 1 : 8

Expenses = $1500

∴ Household expenses to income = 1500/4000

(1500 ÷ 500)/ (4000 ÷ 500) = 3/8

Ratio of Household expenses to income = 3 : 8.

Savings = Income – (expenses + income tax)

Savings = 4000 – (1500 + 500) = $2000.

∴ Savings to earnings = 2000/4000

(2000 ÷ 2000)/(4000 ÷ 2000) = ½

Ratio of savings to expenses = 1 : 2.

• Ratio and Proportion

• Ratio in the simplest form

• Comparison of ratios

• Equivalent ratios

• Proportion

• Continued Proportion

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