What is a Ratio?
A ratio is a comparison of two quantities, in which the first quantity is called the antecedent and the second quantity is called the consequent. Ratios can be ‘part to part’, ‘part to whole’, or ‘whole to part’ comparisons. In math, the ratio of a and b is written as a : b and read as a is to b.
The value of the ratio a : b is \(\frac{a}{b}\).
\(\frac{a}{b}\) is the fractional representation of the ratio a : b.
Equivalent Ratios
Equivalent ratios are ratios that express the same comparison or relationship between the numbers. Once we simplify a set of equivalent ratios, we will get the same ratio of values.
Let us understand this with an example.
Consider the ratios \(\frac{12}{20}\), \(\frac{6}{10}\) and \(\frac{18}{30}\).
Simplify each ratio:
\(\frac{12}{20}\) = \(\frac{12~~\div~~4}{20~~\div~~4}\) = \(\frac{3}{5}\)
\(\frac{6}{10}\) = \(\frac{6~~\div~~2}{10~~\div~~2}\) = \(\frac{3}{5}\)
\(\frac{18}{30}\) = \(\frac{18~~\div~~6}{30~~\div~~6}\) = \(\frac{3}{5}\)
Since the simplest form of all the ratios is \(\frac{3}{5}\), therefore, \(\frac{12}{20}\), \(\frac{6}{10}\) and \(\frac{18}{30}\) are equivalent ratios.
How do we find Equivalent Ratios?
To find the equivalent ratios of a specific ratio, multiply both the antecedent and consequent with the same number. Since there are infinite numbers, the number of equivalent ratios of any ratio is infinite. Let us find three equivalent ratios of \(\frac{2}{3}\).
Multiply both the antecedent and the consequent with three different numbers to obtain three equivalent ratios.
\(\frac{2~\times~2}{3~\times~2}\) = \(\frac{4}{6}\)
\(\frac{2~\times~3}{3~\times~3}\) = \(\frac{6}{9}\)
\(\frac{2~\times~4}{3~\times~4}\) = \(\frac{8}{12}\)
So, \(\frac{4}{6}\), \(\frac{6}{9}\) and \(\frac{8}{12}\) are three equivalent ratios of \(\frac{2}{3}\).
Solved Equivalent Ratio Examples
Example 1: Jenny takes 20 seconds to cover 100 meters and Amelia takes 30 seconds to cover 150 meters while jogging. Do both friends have the same speed while jogging?
Solution: Write the ratio to represent the speed of Jenny and Amelia.
Speed of Jenny = 100 : 20
⇒ \(\frac{100}{20}\) = \(\frac{100~\div~20}{20~\div~20}\)
⇒ \(\frac{5}{1}\) [Simplify]
Speed of Amelia = 150 : 30
⇒ \(\frac{150}{30}\) = \(\frac{150~\div~30}{30~\div~30}\)
⇒ \(\frac{5}{1}\) [Simplify]
From the solution the simplest form of both ratios is the same, that is, 5 : 1. So both the ratios are equivalent.
Therefore Jenny and Amelia are jogging at the same speed.
Example 2: Determine whether the ratios \(\frac{5}{3}\) and \(\frac{25}{15}\) are equivalent.
Solution:
The ratios are \(\frac{5}{3}\) and \(\frac{25}{15}\).
Simplify the second ratio:
⇒ \(\frac{25}{15}\) = \(\frac{25~\div~5}{15~\div~5}\)
⇒ \(\frac{5}{3}\)
We get the first ratio after simplifying \(\frac{25}{15}\).
So, the two ratios are equivalent.
[This question can also be solved by multiplying the antecedent and consequent of the first ratio by 5, that is,
\(\frac{5~\times~5}{3~\times~5}\) = \(\frac{25}{15}\)
We get the second ratio.
So, the ratios are equivalent.]
Example 3: Determine whether the ratios \(\frac{3}{7}\) and \(\frac{12}{14}\) are equivalent.
Solution:
Multiplying the antecedent and consequent of the first ratio by 4, that is,
\(\frac{3~\times~4}{7~\times~4}\) = \(\frac{12}{28}\)
We do not get the second ratio or the value of ratios are not equivalent.
So, the ratios are not equivalent.
To find equivalent ratios for a given ratio we need to multiply both numbers of the ratio by the same number.
Four equivalent ratios of 1 : 2 are 2 : 4, 3 : 6, 4 : 8 and 5 : 10.
The simplest form of 27 : 18 is 3 : 2.
The simplest form of 6 : 4 is 3 : 2 and that of 9 : 6 is also 3 : 2
So, the two ratios are equivalent.
The equivalence of two ratios is known as a proportion.