Proportions

In math proportions we will mainly learn about introduction or basic concepts of proportion and also about continued proportion.

What is a proportion?

Equality of two ratios is called a proportion.

We already learnt that —

Statement of equality of ratios is called proportion.

Let us consider the two ratios.

6 : 10 and 48 : 80

The ratio 6 : 10 in the simplest form can be written as 3 : 5 and the ratio 48 : 80 in the simplest form can be written as 3 : 5.

i.e., 6 : 10 = 48 : 80

So, we say that four numbers 6, 10, 48, 80 are in proportion and the numbers are called the terms of the proportion. The symbol used to denote proportion is :: .

We write 6 : 10 :: 48 : 80. It can be read as 6 is to 10 as 48 is to 80.

In general we know, if four quantities a, b, c, d are in proportion, then a : b = c : d

or, a/b = c/d or a × d = b ×c

Here,

• First and fourth terms (a and d) are called extreme terms.

• Second and third terms (b and c) are called mean terms.

• Product of extreme terms = Product of mean terms

• If a : b : : c : d, then d is called the fourth proportional of a, b, c.

Also,

• If a : b :: b : c, then we say that a, b, c are in continued proportion, then c is the third proportional of a and b.

• Also, b is called the mean proportional between a and C.

• In general if a, b, c are in continued proportion then b² = ac or b = √ac.

Worked-out problems on proportions with the detailed explanation showing the step-by-step are discussed below to show how to solve proportions in different examples.

1. Determine if 8, 10, 12, 15 are in proportion.

Solution:

Product of extreme terms = 8 × 15 = 120

Product of mean terms = 10 × 12 = 120

Since, the product of means = product of extremes.

Therefore, 8, 10, 12, 15 are in proportion.

2. Check if 6, 12, 24 are in proportion.

Solution:

Product of first and third terms = 6 × 24 = 144

Square of the middle terms = (12)² = 12 × 12 = 144

Thus, 12² = 6 × 24

So, 6, 12, 24 are in proportion and 12 is called the mean proportional between 6 and 24.

3. Find the fourth Proportional to 12, 18, 20

Solution:

Let the fourth proportional to 12, 18, 20 be x.

Then, 12 : 18 :: 20 : x

⇒ 12 × x = 20 × 18 (Product of Extremes = Product of means)

⇒ x = (20 × 18)/12

⇒ x = 30

Hence, the fourth proportional to 12, 18, 20 is 30.

4. Find the third proportional to 15 and 30.

Solution:

Let the third proportional to 15 and 30 be x.

then 30 × 30 = 15 × x [b² = ac ]

⇒ x = (30 × 30)/15

⇒ x = 60

Therefore, the third proportional to 15 and 30 is 60.

5. The ratio of income to expenditure is 8 : 7. Find the savings if the expenditure is $21,000.

Solution:

Income/Expenditure = 8/7

Therefore, income = $ (8 × 21000)/7 = $24,000

Therefore, Savings = Income - Expenditure

= $(24000 - 21000) = 3000

6. Find the mean proportional between 4 and 9.

Solution:

Let the mean proportional between 4 and 9 be x.

Then, x × x = 4 × 9

⇒ x² = 36

⇒ x = √36

⇒ x = 6 × 6

⇒ x = 6

Therefore, the mean proportional between 4 and 9 is 6.