In math ratios, we will mainly learn about the introduction or the basic of ratio, ratio in the simplest form, comparison of ratios, conversion of fraction ratio into a whole number ratio and also dividing given quantity in the given ration.

We come across certain situations in everyday life where we need to compare the two quantities. This comparison is done by means of ratio and proportion. We will review the same and learn new ways to compare quantities.

The method of comparing two quantities of the same kind and in the same units by division is known as a ratio.

• The symbol to denote the ratio is **:**

• If a and b are two quantities, they can be expressed as a : b.

Here, **a** is called *antecedent* and **b** is called *consequent*.

• Ratio has no units.

• It can be expressed as a fraction. 2 : 3 can be expressed as 2/3.

• The two quantities that are compared should be of the same kind. 3 liters and 2 grams cannot be compared.

• The two quantities must have the same units. The ratio between 10 g and 15 g is 10 : 15.

• The ratio must be expressed in the simplest form. 3 : 9 can be expressed as 1 : 3.

**If a and b are two quantities.**

• The ratio a : b is said to be in the simplest form if the H.C.F. of a and b is 1.

• If the H.C.F. of 'a' and 'b' is not 1, then divide 'a' and 'b' by the H.C.F. of 'a' and 'b', the ratio will be reduced to the lowest form.

*Example:*

Express the ratio 16 : 20 in the simplest form.

**Solution:**

We write the given ratio as a fraction. i.e., 16/20

Now, divide numerator and denominator of the fraction by 4

(Highest Common Factor of 16 and 20)

(16 ÷ 4)/(20 ÷ 4)

= 4/5

= 4 : 5

The process, in which the two quantities having the same units are compared by division, is called the *comparison by ratio*.

As the ratios can be expressed as fractions, therefore, we can compare the ratios as we compare the fractions.

*Example:*

Compare 3¹/₂ : 1²/₅

**Solution:**

3¹/₂ : 1²/₅

= 7/2 : 7/5

Convert them into equivalent ratios.

7/2 and 7/5

= (7 × 5)/(2 × 5) and (7 × 2)/(2 × 2)

= 35/10 and = 14/10

Now, we have 35/10 : 14/10

Therefore, 35/10 > 14/10

So, 3¹/₂ > 1²/₅

i.e., 7 : 2 > 7 : 5

We know that (a/b) ÷ (c/d) = a/b × d/c

*Example:*

Convert 1/6 : 1/8 into a whole number ratio.

**Solution:**

1/6 : 1/8

= 1/6 ÷ 1/8

= 1/6 × 8/1

= 8̶/6̶

= 4/3

= 4 : 3

Let the given quantity be 'p'. It is to be divided in the ratio a : b.

• Add 'a' and 'b'

• 1ˢᵗ part = a/(a + b) × p

• 2ⁿᵈ part = b/(a + b) × p

*Example:*

1. Divide $60 in the ratio 3 : 2.

**Solution:**

The two parts are 3 and 2

The sum of the parts = 3 + 2 = 5

Therefore, 1ˢᵗ part = 3/5̶ × 6̶0̶ = $36

2ⁿᵈ part = 2/5̶ × 6̶0̶ = $24.

2. Divide 94 columns among A, B and C in the ratio 1/3 : 1/4 : 1/5.

**Solution:**

The least common multiple of 3, 4, 5 is 60.

Therefore, 1/3 : 1/4 : 1/5

= 1/3 × 60 ∶ 1/4 × 60 ∶ 1/5 × 60

= 20 ∶ 15 ∶ 12

So, the total part = 20 + 15 + 12 = 47

Therefore, 1ˢᵗ part = 20/47 × 94 = 40

2ⁿᵈ part = 15/47 × 94 = 30

3ʳᵈ part = 12/47 × 94 = 24

*Worked-out problems on ratios with the detailed explanation showing the step-by-step are discussed below to show you how do you do a ratio in different examples. *

**1.** If a : b = 7 : 12 and b : c = 3/14 find a/c.

**Solution:**

a/b = 7/12 ……………. (1)

b/c = 3/14 ……………. (2)

Multiplying (1) and (2) we get;

a/b × b/c

= 7/12 × 3/14

= 1/8

Therefore, a/c = 1/8

or, a : c = 1 : 8

**2.** If a : b = 3 : 5 and b : c = 6 : 7, find a : b : c.

**Solution:**

We have,

a : b = 3 : 5

i.e., a : b = 3/5 : 1

Also, b : c = 6 : 7

i.e., b : c = 1 : 7/6

Therefore, a : b : c

= 3/5 ∶ 1 ∶ 7/6

Taking the L.C.M. of 5 and 6, we get 3

Therefore, a : b : c

= 3/5 × 30 ∶ 1 × 30 ∶ 7/6 × 30

= 18 : 30 : 35

**3.** A certain amount is divided into 2 parts in the ratio 2 : 3. If the first part is 210, find the total amount.

**Solution:**

The sum of the parts = 2 + 3 = 5

When first part is 2, then total parts are 5.

When first part is 1, then total parts are 5/2

When first part is 210, then total parts are 5/2̶ × 2̶1̶0̶ = 525

**4.** Divide $105 into three parts such that the first part is 4/5 of the second and the ratios between the second and third part is 5 : 6.

**Solution:**

Let the ratio of the three parts be a : b : c

a = ⁴/₅b

Therefore, a/b = 4/5

i.e., a : b = 4/5 : 1

Again, b/c = 5/6

Therefore, b/c = 1/(6/5)

i.e., b : c = 1 : 6/5

Therefore, a : b : c = 4/5 : 1 : 6/5

The L.C.M of the denomination is 5

Therefore, a : b : c=

= 4 : 5 : 6

Now, total number of parts = 4 + 5 + 6 = 15

Therefore, first part = 4/15 × 105 = 28

Therefore, second part = 5/15 × 105 = 35

Therefore, third part = 6/15 × 105 = 42

**5.** Two numbers are in the ratio 1 : 4. Their difference is 30. Find the numbers. **Solution:**

Let the common ratio be x. So, the smaller number is 1x.

And the greater number is 4x.

Their difference is 30.

i.e., 4x - x = 30

3x = 30

x = 30/3

x = 10

Therefore, 1x = 1 × 10 = 10

4x = 4 × 10 = 40

Therefore, the two numbers are 10 and 40. **6.** The ratio of number of boys and girls in a class is 9 : S. If the number of boys is 27, find the number of girls. **Solution:**

(No. of boys)/(No. of girls) = 9/5

Then, 27/(No. of girls) = 9/5

Therefore, No. of girls = (27 × 5)/9

The number of girls in the class is 15.

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