Proportion

A proportion is a special form of an algebra equation. It is used to compare two ratios or make equivalent fractions. A ratio is a comparison between two values. Equality of two ratios is called a proportion. When two ratios are equal, then the cross products of the ratios are equal. That is, for the proportion, a:b = c:d , a x d = b x c.

Third and Fourth proportional

i). Third Proportional:

a : b = c : d, then c is called the third proportion to a and b.

ii). Fourth Proportional:

If a : b = c : d, then d is called the fourth proportional to a, b, c.

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.

Question 1:

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.

Question 2:

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.

Question 3:

Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:

Solution:Let the third number be x.

Then, first number = 120% of x =120x/100 = 6x/5

Second number =150% of x = 150x/100 = 3x/2

Ratio of first two numbers = 6x/5 : 3x/2 = 12x : 15x = 4 : 5

Question 4:

A cubical block of metal weighs 6 pounds. How much will another cube of the same metal weigh if its sides are twice as long?

Solution: If you double the sides of a cube, the ratio of the surface areas of the old and new cubes will be 1: 4. The ratio of the volumes of the old and new cubes will be 1: 8.

Weight is proportional to volume. So, If the first weighs 6 pounds, the second weighs 6×8 pounds =48.

Question 5:

Pencils, Pens and Exercise books in a shop are in the ratio of 10: 2 : 3. If there are 120 pencils, the number of exercise books in the shop is:

Solution: Let Pencils = 10x, Pens = 2x & Exercise books = 3x. Now, 10x = 120 hence x = 12.
Number of exercise books = 3x = 36.