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# Ratio and Proportions

## Introduction

In our day to day life we come across situations where we need to compare quantities in terms of their magnitude or measurements. For example, at the time of admission in a college, marks obtained by students in the qualifying examination are compared. Similarly, at the time of recruitment in forces measurements of candidates pertaining to their weight, heights etc. are compared. In general, this comparison can be done in two ways –

• The first one is comparison by finding the difference of magnitude of two quantities. This is known as comparison by difference.
• The second one is comparison by division of the magnitude of two quantities. This is known as comparison by division.

When we compare two quantities of the same kind by division, we say that we form a ratio of two quantities. So, how do we define ratio? Let us find out.

## What is Ratio?

The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times a quantity is of another quantity of the same kind. The ratio of two numbers “ a “ and “ b “ where b ≠ 0, is a ÷ b or $\frac{a}{b}$ and is denoted by a : b

In the ratio, a : b, the quantities or numbers a and b are called the terms of the ratio. The former “ a” is called the first term or antecedent and the latter term “ b” is called the second term or consequent.

Let us understand the ratio with the help of an example.

Suppose we have two brothers, Sam and Peter having their weights as 50 kg and d40 kg respectively. Now, if we compare the weight of Sam with the weight of Peter, we will get

$\frac{Weight\:of\:Sam}{Weight\:of\:Peter} = \frac{50}{40} = \frac{5}{4} = 5 : 4$

Hence, we can say that the ratio of the weight of Sam to the weight of Peter is 5 : 4.

### Can Ratios be Equivalent?

We already know that a fraction does not change when its numerator and denominator are multiplied or divided by the same non-zero number. It is important to note here that in ratio as well, there is no change in the ratio if the first and the second term are multiplied or divided by the same non-zero number.

Let us understand this by an example.

Suppose we have the ratio 7 : 3. Now is we multiply both the first and the second term by 5, we will get the ratio 35 : 15. Similarly, if we multiply both the first and the second term by 3, we will get the ratio 21 : 9. So, we have

7 : 3 = 35 : 15 = 21 : 9

Hence, the ratios are equivalent in the same manner as fractions are.

## Ratios in Simplest form

A ratio a :  b is said to be in its simplest form if its antecedent a and consequent b have no common factor other than 1. The ratio is simplest form is also called ratio in lowest terms.

Let us understand this by an example.

Example

Suppose we have the ratio 80 : 32

Solution

We can clearly see that this ratio is not in its simplest form because 16 is a common factor of its antecedent and the consequent. Therefore, upon dividing both the antecedent and the consequent by 16 we will get, 5 : 2 which is now in lowest term.

So, we do we reduce a ratio to its lowest term? Let us find out.

### Important Points to Be Considered when Reducing a Ratio to its Simplest Form

1. It is important to note that in a ratio, we compare two quantities. The comparison becomes meaningless if the quantities being compared are not of the same kind, i.e. they are not measured in the same units. It is just meaningless to compare 20 books with 200 monkeys. Therefore to find the ratio of two quantities there must be expressed in the same units.
2. Since the ratio of two quantities of the same kind determine how many times one quantity is contained by the other so the ratio of any two quantities of the same kind is an abstract quantity. In other words ratio has no unit or is independent of the units heroes in the quantities compared.
3. The order of the terms in a ratio a : b is very important. The ratio 3 : 2 is different from the ratio 2 : 3.

Let us now understand the comparison of ratios

## Comparison of Ratios

What steps should be followed to compare the two ratios? We should follow the following steps for the comparison between two ratios –

1. Obtain the given ratios
2. Express each of them in the form of a fraction in the simplest form.
3. Find the L. C. M of the denominator of the fractions obtained in the previous step.
4. Obtain the first fraction and its denominator. Now, divide the L. C. M of obtained in the previous step by the denominator to get a number, say ( p ).
5. Now, multiply the numerator and denominator of the fraction by p.
6. Repeat the same procedure for the other fraction.
7. Now, we can see that the denominators of the fractions will be the same.
8. Compare the numerators of fractions obtained in the previous step.
9. The fraction having a larger numerator will be greater than the other.

Let us understand the above steps using an example.

Example

Suppose we have two ratios 5 : 12 and 3 : 8 and we want to compare them.

Solution

We have been given ratios 5 : 12 and 3 : 8 for comparison.

Let us start with the first ratio. This ratio in the fraction form will be –

5 : 12 = $\frac{5}{12}$

Now, let us consider the second ratio. This ratio in the fraction form will be –

3 : 8 = $\frac{3}{8}$

Now, we have the denominators as 12 and 8. The L. C. M of 12 and 8 will be 24.

We will now have,

$\frac{5}{12} = \frac{5 x 2}{12 x 2} = \frac{10}{24}$ and

$\frac{3}{8} = = \frac{3 x 3}{8 x 3} = = \frac{9}{24}$

Now, since the denominators of both the fractions are equal, we will compare their numerators. We can see that

The numerator 10 is greater than the numerator 9. Therefore,

10 > 9

⇒  $\frac{5}{12} > \frac{3}{8}$

## Proportion

Proportion is an equality of two ratios. For example, consider two ratios, 6 : 18 and 8 : 24. We can see that

6 : 18 = 1 : 3 and 8 : 24 = 1 : 3

Therefore, 6 : 18 = 8 : 24

Thus the ratios 6 : 18 and 8 : 24 are in proportion.

Therefore, we can say that four numbers a, b, c and d are said to be in proportion if the ratio of the first two is equal to the ratio of the last two. This means, four numbers a, b, c and d are said to be in proportion, if a : b =  c : d

If four numbers a, b, c and d are said to be in proportion, then we write

a : b : : c : d

which is read as “ a is to b as c is to d” or “ a to b as c to d”. Here a, b, c and are the first second, third and fourth terms of the proportion. The first and the fourth terms of the proportion are called extreme terms or extremes. The second and the third terms are called the middle terms or means.

Let us understand this by an example.

Consider four terms 40, 70, 200 and 350. We find that 40 : 70 = 200 : 350. So, the given numbers are in proportion. Clearly, 40 and 350 are extreme terms and 70 and 200 are middle terms. We find that,

Product of extreme terms  = 40 x 350 = 14000

Similarly, product of middle terms = 70 x 20 = 1400

Therefore,

Product of extreme terms = Product of middle terms

Thus, we can say that if four numbers are in proportion then the product of the extreme terms is equal to the product of the middle terms.

### Continued Proportion

Three numbers a b c are said to be in continued proportion if a, b, b, c are in proportion.

Thus, if a, b and c are in proportion, then we have a : b : : b : c

Product of extreme terms = Product of middle terms

⇒ a x c = b  x b

⇒ a c  = b 2

⇒ b 2 = a c

### Mean Proportion

If a, b and c are in continued proportion then b is called the mean proportional between a and c. This means that if b is the mean proportional between a and c then b 2 is equal to  a c.

## Solved Examples

Example 1 Express the following ratios in its simplest form.

a) 150 : 400

b) a dozen to a score

Solution We have been given two ratios and we need to reduce them in the simplest form. Let us do them one by one.

a) 150 : 400

Let us first write the ratio in its fractional form. We will have,

150 : 400 = $\frac{150}{400}$

Now, we will find common factors between 150 and 400. The fraction will be reduced to

$\frac{150}{400} = \frac{150 ÷50}{400 ÷50} = \frac{3}{8}$

Hence, the ratio 150 : 40  in its simplest form will be 3 : 8

b) a dozen to a scor.

Since we have been given the ratio in words, first let us convert it into numbers. We know that a dozen is equal to 12 and a score is equal to 20. Therefore,

a dozen to a score = 12 : 20

Now, let us write the ratio in its fractional form. We will have,

12 : 20 = $\frac{12}{20}$

Now, we will find common factors between 12 and 20. The fraction will be reduced to

$\frac{12}{20} = \frac{12 ÷4}{20 ÷4} = \frac{3}{5}$

Hence, the ratio of a dozen to a score  or 12 : 20  in its simplest form will be 3 : 5

Example 2 In a school library, the ratio of mathematics books to science books is the same as the ratio of science books to English books. If there are 450 books in Science and 300 books in English, find the number of books in mathematics.

Solution We have been given that in a school library, the ratio of mathematics books to science books is the same as the ratio of science books to English books. Also, there are 450 books in Science and 300 books in English and we need to find the number of books in mathematics. Let us first summarise the information given to us.

Number of Science Books in the library = 450

Number of English Books in the library = 300

Also,

Ratio of mathematics books to science books = ratio of science books to English books

Now,

Ratio of science books to English books = 450 : 300

If we reduce this ratio to its simplest form, we will get,

450 : 300 = 3 : 2

Now, let the number of mathematics books in the library be p. therefore, we have,

Ratio of mathematics books to science books = ratio of science books to English books

⇒ p : 450 = 3 : 2

⇒ $\frac{p}{450} = \frac{3}{2}$

⇒ p = $\frac{3}{2}$ x 450

⇒ p = 675

Hence, the number of mathematics books in the library = 675.

Example 3 Two numbers are in the ratio 3 : 4. If the sum of numbers is 63, find the numbers.

Solution We have been given that two numbers are in the ratio 3 : 4. Also the sum of numbers is 63. Let us first summarise the information given to us.

Sum of the terms of the ratio = 3 + 4 = 7

Sum of numbers = 63

Therefore, first number = 3/7 × 63 = 27

Second number = 4/7 × 63 = 36

Therefore, the two numbers are 27 and 36.

Example 4 The first, second and fourth terms of a proportion are 6, 18 and 28 respectively. Find its third term.

Solution We have been given that the first, second and fourth terms of a proportion are 6, 18 and 28 respectively. We are required to find the third term. Let us summarise the information given to us.

The proportion of the first, second and fourth terms = 6, 18 and 28

Let the third term be p. Then,

6, 18 and p and 25 are in proportion.

⇒ Product of extreme terms = Product of means terms

⇒ 6  x 25 = 18 x p

⇒ 150 = 18 p

⇒ p = $\frac{150}{18} = \frac{25}{3}$

Hence, the third term of the proportion is $\frac{25}{3}$.

Example 5 The ratio of the length of a school ground to its width is 5 : 2. Find the length of the width of the ground is 50 m.

Solution We have been given that the ratio of the length of a school ground to its width is 5 : 2. We need to find the length of the width of the ground is 50 m.

Let the length of the school ground be p metres.

Then, the ratio of the length to the width = p : 50

But, the ratio of the length to its width = 5 : 2

Therefore,

p : 50 = 5 : 2

⇒ $\frac{p}{50} = \frac{5}{2}$

⇒ p = $\frac{5}{2}$ x 50

⇒ p = 125

Hence, the length of the school playground = 125 m

## Key Facts and Summary

1. The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times a quantity is of another quantity of the same kind.
2. The ratio of two numbers “ a “ and “ b “ where b ≠ 0, is a ÷ b or $\frac{a}{b}$ and is denoted by a : b
3. In the ratio, a : b, the quantities or numbers a and b are called the terms of the ratio. The former “ a” is called the first term or antecedent and the latter term “ b” is called the second term or consequent.
4. There is no change in the ratio if the first and the second term are multiplied or divided by the same non-zero number.
5. A ratio a :  b is said to be in its simplest form if its antecedent a and consequent b have no common factor other than 1.
6. Proportion is an equality of two ratios.
7. If four numbers are in proportion then the product of the extreme terms is equal to the product of the middle terms.
8. Three numbers a b c are said to be in continued proportion if a, b, b, c are in proportion.
9. If a, b and c are in continued proportion then b is called the mean proportional between a and c.