Session 1: What is a Ratio? – Definition, Simplest Form, Equivalent Ratio, Comparison,Interactives and Examples

Galileo Galilei, the father of observational astronomy, modern physics and even the father of science, says, “the book of nature is written in the language of mathematics”. Mathematics has an impact on our day-to-day life, including architecture. It plays a vital role in design as well as construction. Architects apply these math forms to plan their blueprints or initial sketch designs. Architects and the construction team also calculate the ratio to bring the design vision to life in three dimensions.        

Every building you spend time in–schools, libraries, houses, apartment complexes, movie theatres, and even your favourite ice cream shop–calls for careful ratio calculations.

In this lesson, we will learn to compare quantities of the same kind using a ratio.

We can compare two quantities of the same kind in two ways – by subtraction or division. For example, let us take the two numbers 3 and 12. 

When comparing by subtraction, we may say that 3 is 9 less than 12. Here we find the difference of the magnitudes to see how much more (or less) one quantity is than the other.

However, when comparing by division, we may say that 3 is \(\frac{1}{4}\) of 12. When we compare two quantities of the same kind by division, we form a ratio of the two quantities.

What is a Ratio?

The ratio of two quantities a and b of the same kind and in the same units is the fraction \(\frac{a}{b}\) (where b≠0) and is represented as a:b. It is read as a is to b.

Thus, the ratio is a fraction that expresses how many times or how many parts one quantity is of another of the same kind.

The two numbers which form a ratio are called terms.

In the ratio a:b, a is called the first term or the antecedent and b is called the second term or consequent. 

Simplest Form of a Ratio

A ratio is said to be in its simplest form if its terms (antecedent and consequent) have no common factor other than 1. In other words, the ratio a:b is said to be in simplest form if H.C.F. of a and b is 1. 

For example, to find the simplest form of 32 : 96, we divide each of the terms of the ratio by the H.C.F. of 32 and 96, i.e., 32.

Thus, the simplest form of 32 : 96 is 1 : 3.

A ratio is always expressed in the lowest terms or the simplest form.

Equivalent Ratios

We know that a fraction remains unchanged when its numerator and denominator are multiplied or divided by the same non-zero number.

Similarly, a ratio remains unchanged when both its terms (antecedent and consequent) are multiplied or divided by the same non-zero number.

Thus, a ratio obtained by multiplying or dividing the numerator and denominator of a given ratio by the same non-zero number is called an equivalent ratio.

Consider the ratio 8 : 12. 

We have, 

Also, 

and so on

Thus, 4 : 6, 2 : 3, 16 : 24, 24 : 36… are ratios equivalent to the ratio 8 : 12.

If a:b and c:d are two equivalent ratios, we write \(\frac{a}{b}\)=\(\frac{c}{c}\). 

Comparison of Ratios

We can also compare ratios to find out whether they are equivalent or not. To compare ratios, first, write them as fractions. Then, compare by converting them into like fractions or by cross multiplying them. 

For example, let us find out which is greater, 2:3 or 4:5.

2:3=\(\frac{2}{3}\) and 4:5=\(\frac{4}{5}\).

L.C.M. of 3 and 5=15.

 and .

10<12

⇒\(\frac{10}{15}\)<\(\frac{12}{15}\) so \(\frac{2}{3}\)<\(\frac{4}{5}\).

Thus, 2:3<4:5.

Steps To Compare Two Ratios

We can use the following steps to compare two given ratios:

Step 1: Express each of the ratios in the fractional form.

Step 2: Find the L.C.M. of the denominators of the obtained fractions.

Step 3: Convert each fraction to its equivalent fraction with the denominator equal to L.C.M. obtained in the above step. 

Step 4: Compare the numerators of the fractions obtained. The fractions having the larger numerator will be greater than the other.

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What is a Ratio? – Examples

Example 1

Express 125 and 140 as a ratio in the lowest term.

We have a ratio 125:140.

Since both are divisible by 5 (i.e., 5 is their H.C.F.), divide the terms by 5.

125:140=25:28

The ratio cannot be further reduced.

Therefore, the lowest term is 25:28.

Example 2

Find the ratio of 4 metres and 15 centimetres.

Since both the given terms are in different units, they must be converted to the same unit.

Note: It is always easier to convert a higher unit to a lower unit.

By changing 4 metres to centimetres, we get

400:15

Since 5 is their H.C.F., divide both terms by 5.

400:15=80:3

Therefore, 

4m:15cm=80:3

Example 3

Compare \(1\frac{1}{4}\):2 and \(\frac{3}{4}\):4.

First, we change \(1\frac{1}{4}\) to a whole number.

To change \(1\frac{1}{4}\) to a whole number, we multiply by 4. We also perform the same operation on the second term to keep the value unchanged.

In the second ratio, to change \(\frac{3}{4}\) to a whole number, we multiply by 4. We multiply the second term also by the same number to keep the value unchanged.

L.C.M. of 8 and 16 = 16

Thus,

5:8>3:16

Example 4

Two baskets are filled with apples in a ratio of 4:9. If there are 30 more apples in the second basket than the first, how many apples are there in total?

Let the number of apples in the first basket =4x

Then, the number of apples in the second basket =9x

From the above figure, we form the equation

Total number of apples=(4×6)+(9×6)
=24+54
=78

Example 5

Find the measures of the angles of a triangle if the ratio of their measures is 5 : 12 : 28. 

The sum of measures of all the angles of a triangle=180∘.

Sum of the terms of ratio=5+12+28=45.

Thus, 

Example 6

Find a:b:c, given that a:b=24:30and b:c=42:50.

The term b is represented by 30 and 42 in the two ratios, respectively. 

L.C.M.(30,42)=210

Multiplying the first ratio by 7, we get its equivalent ratio 

Multiplying the second ratio by 5, we get its equivalent ratio 

The term b is now represented by the same values in the two ratios.

∴ a:b=168:210 and b:c=210:250
or a:b:c=168:210:250

Example 7 

An alloy contains copper and zinc in the ratio 7 : 3. If it contains 13.3 g of copper, find the weight of zinc in the alloy. 

We have, 

Weight of copper : Weight of zinc = 7 : 3. 

So, let the weight of copper in the alloy be 7x g, and the weight of zinc in the alloy be 3x g. 

It is given that the weight of copper in the alloy is 13.3 g. 

∴ Weight of zinc in the alloy .

Example 8

Find the numbers if they are in the ratio 7 : 8 and their difference is 6. 

The ratio 7 : 8 can be written as \(\frac{7}{8}\).

We know, 

First numberSecond number=78Since the difference between the two numbers is 6, so if the first number is x, the other is x+6.

Thus, 

 Cross-multiplying, we get

∴ The second number=x+6=42+6=48.

Thus the two numbers are 42 and 48.  

 Remember this!

• A ratio is a way of comparing two numbers or quantities by division. 
• A ratio is an abstract quantity, i.e., it has no unit. 
• Words such as “for every”, “for each”, and “per” are used to describe ratios. 
• Equivalent ratios have the same value. 
• Multiply or divide the numerator and denominator by the same number to find an equivalent ratio.