Proportion

Four non-zero quantities, a, b, c, and d, are said to be in proportion (or are proportional) if:

a : b = c : d.
The above equation is expressed as a : b :: c : d 

This is read as “a is to b as c is to d.”

1. Proportion format:

• If a:b = c:d, it means two ratios are equal. 
  where a = first term, b = second term, c = third term, d = fourth term

2. Extremes and means:

• a and d are called the extremes (outside terms).
• b and c are called the means (middle terms).

3. Cross-multiplication rule:

• a × d = b × c
  (Product of extremes = Product of means)

4. Fourth proportional:

• In a:b = c:d, the fourth term, d, is called the fourth proportional.

 5. Same kind and units:

• For any ratio or proportion, the compared quantities must be of the same kind and unit.
• Units of $c$ and $d$ must match; units of $a$ and $b$ must match.

Check whether or not the given ratios form a proportion:

1) 15:24 and 35:56

Solution:
Product of extremes = 15 × 56 = 840

and product of means = 24 × 35 = 840

Since product of extremes = product of means

⇒ The given two ratios form a proportion

2) 2`1/4` : 5`2/5` and 3`1/3` : 4`1/6`

Solution:
Product of extremes = 2`1/4` × 4`1/6` = `9/4` × `25/6` = `75/8`

product of means = 5`2/5` × 3`1/3` = `27/5` × `10/3` = `18/1`

⇒ product of extremes `\cancel(=)` product of means

⇒ The given two ratios do not form a proportion. 

1. The numbers 8, x, 9 and 36 are in proportion. Find x.
2. If x : 15 = 8 : 12, find x.

Solution:

i) The numbers 8, x, 9 and 36 are in proportion. 
⇒ 8: x = 9 : 36
⇒ x × 9 = 8 × 36

⇒  x =`"8 × 36 " / 9` = 32

ii) x:15 = 8:12
⇒ x × 12 = 15 × 8

⇒ x = `"15 × 8 " / 9` = 10

The first, third and fourth terms of a proportion are 12, 8 and 14, respectively.
Find the second term.

Solution:
Let the second term be x.
∴ 12, x, 8 and 14 are in proportion, i.e., 12:x = 8:14.

=> x × 8 = 12 × 14

=> x = `"12 × 14 " / 8` = 21

∴  The second term of the proportion is 21.

The ratio of the length and the width of a sheet of paper is 3 : 2. If the length is 12 cm, find the width.

Solution:
Let width = x cm
The ratio of length to width = 12 : x
According to the given statement, 12 : x = 3 : 2

=> x × 3 = 12 × 2

=> x = `"12 × 2 " / 3` = 8

∴ Width = 8 cm 

Situation: If you need 6 erasers for every 3 pencils, how many erasers do you need for 12 pencils?

Let the number of erasers be x.
Set up proportion: 3:6::12:x

• 3 × x = 6 × 12

• 3x = 72 → x = 24

• Answer: You need 24 erasers.

• Ratios compare values of the same kind.

• A proportion means two ratios are the same.

• Use cross-multiplication to check proportion.

• Always write ratios in their simplest form.