In this post we will try to understand the concept of equivalent ratios with examples.
The concept requires basic understanding of ratios and fraction manipulation so ensure you have completed the previous chapters.
In this post we will first learn to identify the equivalent ratios and then will learn to create equivalent ratios from given ratio.
What are Equivalent ratios?
Two or more ratios having the same lowest terms are equivalent ratios.
Ratios can be simplified into lowest terms using fraction manipulations.
So the ratios, which after simplification have similar values are called equivalent ratios.
How to find if ratios are equivalent?
To check if the ratios are equivalent or not, do the following steps:
(a) Convert all the ratios into fractions.
(b) For a fraction, find the common factor present in numerator and denominator.
This can be done by finding HCF of numerator and denominator.
(c) Divide the numerator and denominator by HCF number.
(d) Now the fraction has been reduced to lowest terms.
Compare all the fractions and check if the values are equal or not.
Example 01
Are ratios 1 : 2 and 2 : 4 equivalent ?
Solution
Follow the below steps:
(a) Convert the ratios into fraction
1 : 2 ⟹ 1 / 2
2 : 4 ⟹ 2 / 4
(b) Reduce the fraction to lowest terms
Fraction 1/2
1 / 2 is already in lowest form
Fraction 2/4
(i) Take HCF of numerator and denominator
HCF ( 2, 4 ) = 2
(ii) Divide numerator and denominator by 2
\mathtt{\frac{2\ \div \ 2}{4\ \div \ 2} \ \Longrightarrow \frac{1}{2}}
Fraction 2/4 is reduced to 1/2
(c) Compare the fractions
Both the fractions reduced to value 1/2.
Hence the given ratio is equivalent ratio.
Example 02
Check if the below ratios are equivalent or not?
4 : 16 and 12 : 48
Solution
Follow the below steps:
(a) Convert the given ratios into fractions
4 : 16 ⟹ 4 / 16
12 : 48 ⟹ 12 / 48
(b) Reduce all the fractions to its lowest terms
Fraction 4 : 16
(i) Find HCF of numerator and denominator
HCF ( 4, 16 ) = 4
(ii) Divide numerator and denominator by HCF = 4
\mathtt{\frac{4\ \div \ 4}{16\ \div \ 4} \ \Longrightarrow \frac{1}{4} \ }
Fraction 12 : 48
(i) Find HCF of numerator and denominator
HCF ( 12, 48 ) = 12
(ii) Divide numerator and denominator by HCF = 12
\mathtt{\frac{12\ \div \ 12}{48\ \div \ 12} \ \Longrightarrow \frac{1}{4} \ }
(c) Compare the fractions
Here both the ratios have been reduced to value 1/4.
Hence the given ratios are equivalent ratios.
Example 03
Check if the given ratios are equivalent or not
15 : 25 and 30 : 50
Solution
Follow the below steps:
(a) Convert all the ratios into fractions
15 : 25 ⟹ 15 / 25
30 : 50 ⟹ 30 / 50
(b) Reduce the fraction to its lowest terms
Fraction 15/25
(i) Find HCF of numerator and denominator
HCF ( 15, 25 ) = 5
(ii) Divide numerator and denominator by HCF = 5
\mathtt{\frac{15\ \div \ 5}{25\ \div \ 5} \ \Longrightarrow \frac{3}{5} \ }
Fraction 30 / 50
(i) Find HCF of numerator and denominator
HCF (30, 50) = 10
(ii) Divide numerator and denominator by 10
\mathtt{\frac{30\ \div \ 10}{50\ \div \ 10} \ \Longrightarrow \frac{3}{5} \ }
(c) Compare the fractions
Both the ratios are reduced to fraction 3/5.
Hence, the given ratios are equivalent ratios.
Example 04
Check if the below ratios are equivalent or not
5 : 4 and 16 : 12
Solution
Follow the below steps:
(a) Convert the ratios into fraction
5 : 4 ⟹ 5 / 4
16 : 12 ⟹ 16 / 12
(b) Reduce the fraction to its lowest terms
Fraction 5/4
The fraction is already in reduced form.
Fraction 16/12
(i) Find HCF of numerator and denominator
HCF (16, 12 ) = 4
(ii) Divide numerator and denominator by 4
\mathtt{\frac{16\ \div \ 4}{12\ \div \ 4} \ \Longrightarrow \frac{4}{3} \ }
(c) Compare the fractions
Both the fractions have been reduced to its lowest forms.
5 : 4 ⟹ 5 / 4
16 : 12 ⟹ 4 / 3
Both the values are different.
Hence the given ratios are not equivalent.
Example 05
Check if the ratios are equivalent or not
11 : 13 and 33 : 39
Solution
Follow the below steps:
(a) Convert the ratio into fraction
11 : 13 ⟹ 11 / 13
33 : 39 ⟹ 33 / 39
(b) Reduce the fraction into lowest terms
Fraction 11/13
The fraction is already reduced to lowest terms.
Fraction 33/39
(i) Find HCF of numerator and denominator
HCF ( 33, 39 ) = 3
(ii) Divide numerator and denominator by 3
\mathtt{\frac{33\ \div \ 3}{39\ \div \ 3} \ \Longrightarrow \frac{11}{13}}
(c) Compare the fractions
We have converted the ratios into lowest terms.
11 : 13 = 11/13
33 : 39 = 11/13
Both the fractions are same.
Hence, the given ratios are equivalent.
How to create equivalent ratios?
From the given ratio, you can create equivalent ratio by multiplying/dividing the ratio with any possible number.
All the ratios developed by multiplication/division have the same lowest terms, that’s why they are also equivalent ratio.
Given below are some examples for your understanding.
Example 01
Find two equivalent ratio of 2 : 3
Solution
First convert the ration into fraction.
2 : 3 ⟹ 2 / 3
To get equivalent ratio, just multiply the fraction with any possible number.
(i) Multiply the numerator and denominator by 2
\mathtt{\frac{2\ \times \ 2}{3\ \times \ 2} \ \Longrightarrow \frac{4}{6} \ }
Here we get the ratio 4 : 6
(ii) Now multiply numerator and denominator by 3.
\mathtt{\frac{2\ \times \ 3}{3\ \times \ 3} \ \Longrightarrow \frac{6}{9} \ }
Here we get the ratio 6 : 9.
Both the above ratios 4 : 6 and 6 : 9 are equivalent ratios as they are derived from same base ratio 2 : 3.
Also, if desired, both the ratios can be reduced to its same lowest form 2 : 3.
Example 02
Find three equivalent ration of 5 : 3
Solution
First convert the ration into fraction form
5 : 3 ⟹ 5 / 3
(i) Multiply numerator and denominator by 4
\mathtt{\frac{5\ \times \ 4}{3\ \times \ 4} \ \Longrightarrow \frac{20}{12}}
Here we get the ratio 20 : 12.
(ii) Multiply numerator and denominator by 6
\mathtt{\frac{5\ \times \ 6}{3\ \times \ 6} \ \Longrightarrow \frac{30}{18}}
Here we get the ratio 30 : 18.
(iii) Multiply numerator and denominator by 7.
\mathtt{\frac{5\ \times \ 7}{3\ \times \ 7} \ \Longrightarrow \frac{35}{21}}
Here all the ratios 20 : 21, 30 : 18 and 35 : 21 are equivalent ratios since they are derived from the same base ratio 5 : 3.
Also, if desired, all the ratios can be converted back into the base ratio 5 : 3.