What are Equivalent Fractions?
Alright, the best way to understand about equivalent fractions is to talk about an example and using cakes as the example will be a good place to start. If you cut a whole cake into two equal pieces and eat one of them, you will have consumed half of the cake. Now, it wouldn’t matter even if the cake had been cut into ten equal pieces and you ate five of them, you would still have consumed half of the cake. These are called equivalent fractions.
For example; The fractions \( \frac{1}{2},~\frac{2}{4},~\frac{3}{6},~\frac{4}{8} \) all appear to be unique. However, they all represent the same value. These are called equivalent fractions.
How to Generate Equivalent Fractions?
By multiplying or dividing both the numerator and denominator by the same number creates an equivalent fraction. This is why when these fractions are simplified, they are reduced to the same number.
Generate Equivalent Fractions Using Multiplication Operations
For example, multiply the numerator ‘4’ and the denominator ‘5’ by the same number, say ‘3’, to get an equivalent fraction of \( \frac{4}{5} \). As a result, \( \frac{12}{15} \) is an equivalent fraction of \( \frac{4}{5} \). By multiplying the numerator and denominator of the given fraction by the same number, we can find some other equivalent fractions.
\( \frac{4}{5}=\frac{4\times 4}{5\times 4}=\frac{16}{20} \)
\( \frac{4}{5}=\frac{4\times 5}{5\times 5}=\frac{20}{25} \)
As a result, \( \frac{12}{15},~\frac{16}{20} \) and \( \frac{20}{25} \) are equivalent fractions of \( \frac{4}{5} \).
Let’s take an example using a block diagram to find the equivalent fractions
Let’s find the equivalent fractions for \( \frac{5}{7} \).
Draw a model that shows the whole figure divided into different numbers of parts.
Divide one into seven parts and the other into 14 parts.
When 5 parts are shaded the fraction becomes \( \frac{5}{7} \). To represent the same fraction in a 14 part whole we need to shade 10 parts.
Multiplying the numerator and denominator by 2.
\( \frac{5}{7}=\frac{5\times 2}{7\times 2}=\frac{10}{4} \).
Therefore, \( \frac{10}{4} \) is an equivalent fraction for \( \frac{5}{7} \).
Let’s take an example using a number line diagram to find the equivalent fractions
Let’s find the equivalent fractions for \( \frac{5}{4} \).
Draw the number line diagram as given below from 0 to 2. Divide the number line into fourths and label the dot marks. As we know \( \frac{5}{4} \) lies in between 1 and 2, plot \( \frac{5}{4} \) on the number line.
Multiplying the numerator and denominator by 3 as per the line diagram.
\( \frac{5}{4}=\frac{5\times 3}{4\times 3}=\frac{15}{12} \)
Similarly, each fraction in the division is multiplied by 3 both in the numerator and the denominator. The corresponding equivalent fractions of each fraction below is plotted above it.
Therefore, \( \frac{15}{12} \) is an equivalent fraction for \( \frac{5}{4} \).
Generate Equivalent Fractions Using Division Operations
In mathematics, a factor is a number or algebraic expression that evenly divides another number or expression, leaving no remainder. For instance, 2 and 4 are factors of 8 because 8 ÷ 2 = 4 and 8 ÷ 4 = 2.
In mathematics, common factors are factors that are common in two or more numbers. In other words, a common factor is a number that will divide a group of two or more numbers exactly.
To create equivalent fractions, divide the numerator and denominator by the common factors between the numerator and denominator to find equivalent fractions for any given fraction.
For example, To find an equivalent fraction of \( \frac{24}{64} \)
We must first determine the common factors of the numbers 24 and 64. We know that both 24 and 64 have a common factor in 2. By dividing the numerator and denominator by 2, an equivalent fraction of \( \frac{12}{32} \) can be found. As a result, \( \frac{12}{32} \) is equivalent to \( \frac{24}{64} \).
Let’s look at how the fraction can be simplified even more:
A common factor of 12 and 32 is 2. Thus, \( \frac{12}{32}=\frac{12\div 2}{32\div 2}=\frac{6}{16} \).
2 is a common factor in both 6 and 16. Thus, \( \frac{6}{16}=\frac{6\div 2}{16\div 2}=\frac{3}{8} \)
As a result, \( \frac{6}{16} \) and \( \frac{3}{8} \) are two equivalent fractions of \( \frac{24}{64} \). Because there is no common factor (other than 1) of 2, \( \frac{3}{8} \) is the simplified form of \( \frac{24}{64} \).
Let’s take an example using a block Diagram to find the Equivalent Fractions
Let’s find the equivalent fractions for \( \frac{14}{16} \).
Let’s find the common factors of 14 and 16.
The factors of 14 are 1, 2, 7 and 14.
The factors of 16 are 1, 2, 4, 8 and 16.
So, the common factors are 1 and 2.
The first whole is divided into 16 parts and the other whole is divided into 8 parts.
To represent ‘\( \frac{14}{16} \)’, 14 parts have to be shaded in the first whole.
To find an equivalent fraction of ‘\( \frac{14}{16} \)’, dividing the numerator and denominator by 2 is the step to carry out.
\( \frac{14}{16}=\frac{14\div 2}{16\div 2}=\frac{7}{8} \)
Hence 7 parts need to be shaded in the second whole.
Therefore, \( \frac{7}{8} \) is an equivalent fraction for \( \frac{14}{16} \).
Let’s take an example of using a Number Line to find the Equivalent Fractions
Let’s use a number line to find the equivalent fractions of the fraction \( \frac{6}{12} \).
The first step is to find the common factors of 6 and 12.
The factors of 6 are 1, 2, 3 and 6.
The factors of 12 are 1, 2, 3, 4, and 6.
So, the common factors are 1, 2, 3, 6 and 12.
Draw the number line diagram as given below from 0 to 1. Divide the number line into twelfths and label the dot marks. As we know \( \frac{6}{12} \) lies in between 0 and 1, it should be easy to plot it on the number line.
Dividing the numerator and denominator by 6.
\( \frac{6}{12}=\frac{6\div 6}{12\div 6}=\frac{1}{2} \)
Therefore, \( \frac{1}{2} \) is an equivalent fraction of the fraction \( \frac{6}{12} \).
Solved Examples on Equivalent Fractions Using Division & Multiplication
Examples 1. Write the equivalent fractions for the diagrams given below:
a.
b.
c.
d.
Solution:
a. Let’s find the equivalent fractions for \( \frac{4}{16} \)
First we need to find the common factors of 4 and 16.
The factors of 4 are 1, 2 and 4.
The factors of 16 are 1, 2, 4, 8 and 16.
So, the common factors are 1, 2 and 4.
Dividing the numerator and denominator by 4.
\( \frac{4}{16}=\frac{4\div 4}{16\div 4}=\frac{1}{4} \)
Therefore, \( \frac{1}{4} \) is an equivalent fraction for \( \frac{4}{16} \).
b. Let’s find the equivalent fractions for \( \frac{2}{12} \)
First we find the common factors of 2 and 12.
The factors of 2 are 1, 2.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
So, the common factors are 1, 2.
Dividing the numerator and denominator by 2.
\( \frac{2}{12}=\frac{2\div 2}{12\div 2}=\frac{1}{6} \)
Therefore, \( \frac{1}{6} \) is an equivalent fraction for \( \frac{2}{12} \).
c. Let’s find the equivalent fractions for \( \frac{3}{6} \)
The common factors of 3 and 6 have to be found out first.
The factors of 3 are 1, 3.
The factors of 6 are 1, 2, 3 and 6.
So, the common factors are 1 and 3.
Dividing the numerator and denominator by 3.
\( \frac{3}{6}=\frac{3\div 3}{6\div 3}=\frac{1}{2} \)
Therefore, \( \frac{1}{2} \) is an equivalent fraction for \( \frac{3}{6} \).
d. Let’s find the equivalent fractions for \( \frac{12}{14} \)
First of all, find the common factors of 12 and 14.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
The factors of 14 are 1, 2, 7 and 14.
So, the common factors are 1, 2.
Dividing the numerator and denominator by 2.
\( \frac{12}{14}=\frac{12\div 2}{14\div 2}=\frac{6}{7} \)
Therefore, \( \frac{6}{7} \) is an equivalent fraction for \( \frac{12}{14} \).
Example 2. Mary ate \( \frac{2}{4} \) of a pizza and Angelina ate \( \frac{4}{8} \) of a pizza. Did the girls share the pizza equally? Support your answer with images.
Solution: Mary ate \( \frac{2}{4} \) of a pizza.
Angelina ate \( \frac{4}{8} \) of a pizza
Let’s find the equivalent fractions for \( \frac{2}{4} \)
Multiplying the numerator and denominator by 2.
\( \frac{2}{4}=\frac{2\times 2}{4\times 2}=\frac{4}{8} \)
Therefore, \( \frac{4}{8} \) is an equivalent fraction of \( \frac{2}{4} \).
This means that the girls shared the pizza equally.
Example 3. Mark walks \( \frac{2}{3} \) miles to their school and John walks \( \frac{4}{6} \) miles to their school. Did the boys cover the same amount of distance? Support your answer with images.
Solution: Mark walks \( \frac{2}{3} \) miles to school.
John walks \( \frac{4}{6} \) miles to school.
Let’s find the equivalent fractions for \( \frac{2}{3} \)
Multiplying the numerator and denominator by 2.
\( \frac{2}{3}=\frac{2\times 2}{3\times 2}=\frac{4}{6} \)
Therefore, \( \frac{4}{6} \) is an equivalent fraction for \( \frac{2}{3} \)
Yes the boys covered the same amount of distance.
Fractions that have different numerators and denominators but have the same value are called equivalent fractions. \( \frac{1}{2} \) and \( \frac{4}{8} \) are examples of equivalent fractions. The numerators and denominators are different but the values are the same and equal to 12. We know that a fraction is a portion of something larger and equivalent fractions represent the same amount of the total.
Multiplying or dividing the numerator and denominator by the same number obtains equivalent fractions. To add or subtract fractions or to compare two fractions, fractions are converted to like fractions, hence equivalent fractions are used.