Session 2: Proportion – Definition, Continued Proportions, Mean Proportional and Examples

In everyday life, we use proportions all the time. When we bake, we need to know how to add the right amount of sugar according to the size of the cake. When we construct a building, it is important to get the proportions right between the height and width of the building.

At the supermarket, suppose you see someone buy five boxes of strawberries. The grocery clerk charges ₹125 for them. Now imagine you wanted two boxes of strawberries for yourself. How much would you expect to pay for them?

In the example above, the price of strawberries and the number of boxes have a proportional relationship. If you buy two-fifths of the number of boxes of strawberries, you will pay two-fifths of ₹125, i.e., ₹50 for your strawberries. The two ratios i.e. the number of boxes and the price of strawberries are equal. These are also known as equivalent ratios.

This equality relation between two ratios is called the proportion.

Proportions

The two ratios, 2 : 3 and 8 : 12, are equivalent. The statement showing the equivalence of two ratios is called a proportion. The above two ratios can be written in proportion as 

2:3::8:12

In general, if a:b::c:d, then a, b, c and d are said to be in proportion. Here, a, b, c and d are respectively known as first, second, third and fourth terms.

• The first and the fourth terms are called extremes, whereas the second and the third terms are called means. 
• The product of extremes is always equal to the product of means.
  ad=bc
• If \(\frac{a}{b}\)=\(\frac{c}{d}\), then the given proportion can be written as \(\frac{b}{a}\)=\(\frac{d}{c}\), by taking reciprocals of terms on both sides. This relationship is known as invertendo. 
• If \(\frac{a}{b}\)=\(\frac{c}{d}\), then multiplying both sides of the proportion by \(\frac{b}{c}\), we get \(\frac{a}{c}\)=\(\frac{b}{d}\). This relationship is known as alternendo. 

1) Are 80, 64, 30 and 24 in proportion?

Four numbers are proportional if the product of extremes equals the product of means. 

We have, 

Product of extremes=80×24=1920.

Product of means=64×30=1920.

∴ Product of extremes = Product of means. 

Hence, 80, 64, 30 and 24 are in proportion. 

Continued Proportions

A continued proportion is one in which the ratio between the first and second term is equal to the ratio between the second and the third term. If a, b and c are in continued proportion, then 

Here, a and c are respectively known as the first, and the third terms and b is known as the mean proportional.

We know, 

Product of extremes=Product of means

 ∴ The mean proportion of a and c is .

2) Find the mean proportional between 4 and 36. 

Let the mean proportion be x.

We have, 

Thus, the continued proportion is 4 : 12 : 36 or the mean proportional between 4 and 36 is 12.  

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Proportion – Examples

Example 1

Check whether the following numbers are in proportion or not.

i. 2,3,4,5

ii. 80,60,120,90

Four numbers are proportional if the product of extremes equals the product of means. 

i. We have, 

    Product of extremes=2×5=10.

    Product of means=3×4=12.

   ∴ Product of extremes ≠ Product of means. 

   Hence, 2,3,4 and 5 are not in proportion. 

ii. We have, 

   Product of extremes=80×90=7200.

   Product of means=60×120=7200.

   ∴ Product of extremes = Product of means. 

   Hence, 80, 60, 120 and 90 are in proportion. 

Example 2

The first, third and fourth terms of a proportion are 2, 5 and 40 respectively. Find the second term.

Let the second term be x. Then 2, x, 5 and 40 are in proportion.

Product of extremes=Product of means

Hence, the second term of the proportion is 16.

Example 3

If 36, x,x,64 are in proportion, find the value of x.

Given that 36, x,x and 64 are in proportion,

Product of extremes=Product of means

Example 4

If 16, 20, x  are in continued proportion, find the value of x.

Since 16, 20, and x are in continued proportion, 16, 20, 20 and x  are in proportion.

Product of extremes=Product of means

Example 5

Find the mean proportional between 25 and 49.

Let the mean proportional between 25 and 49 be x. Then,

Product of means=Product of extremes

Hence, the mean proportional between 25 and 49 is 35.

Example 6

An aeroplane covers a distance of 2020 km in 5 hours. How much distance will it cover in 7 hours?

Let the distance covered by the aeroplane in 7 hours be x km. Then the ratio of distance covered is proportional to the ratio of time taken. 

Hence, the aeroplane will cover 2828 km in 7 hours.

 Remember this!

• The equality of two ratios is called proportion. 
• The first and the fourth terms are called extremes of the proportion, whereas the second and third terms are called means of the proportion. 
• Four numbers are proportional if the product of extremes equals the product of means. 
• If a:b::b:c, then a, b and c are said to be in continued proportion, where b is the mean proportional between a and c.