What is an Equivalent Fraction?
There are three friends Thomas, Emma and Sarah. Each of them are given a pizza. Thomas cut the pizza into 2 slices only and took 1 of them. So, he has taken \(\frac{1}{2}\)of his pizza.
Emma cut the pizza into 4 slices and took 2 of them. So, she has taken \(\frac{2}{4}\) of her pizza. And Sarah cut her pizza into 6 slices and took 3 of them. So, she has taken \(\frac{3}{6}\) of her pizza.
Now let’s check who has taken more quantity of pizza? On observing this situation we can say that everyone has taken an equal number of slices. This shows that the three fractions \(\frac{1}{2}\) , \(\frac{2}{4}\) and \(\frac{3}{6}\) appear different, but they have the same value. So, we can conclude that these fractions have to be equivalent.
Now, we also know that two or more numbers are said to be equivalent if they have the same numerical value. Similarly, if two or more fractions that show the same part of a whole, are called equivalent fractions.
In the boxes drawn above we can see that, in the first box, out of three equal boxes, two are shaded which gives the fraction as \(\frac{2}{3}\).
In the next box, 4 out of 6 equal boxes are shaded, which gives us the fraction\(\frac{4}{6}\) .
It is evident from the above diagram that the fraction \(\frac{4}{6}\) is the same as \(\frac{2}{3}\) as they show the same shaded region.
Similarly all the fractions \(\frac{2}{3}\) ,\(\frac{4}{6}\) , \(\frac{8}{12}\) , \(\frac{16}{24}\) are equivalent because they all show the same shaded region.
Equivalent Fractions on a Number Line
Equivalent fractions lie on the same positions on the number line.
Let us take an example of two equivalent fractions \(\frac{3}{4}\) and \(\frac{6}{8}\). If we plot \(\frac{3}{4}\) on the number line, we get the following result:
Now, let us plot \(\frac{6}{8}\) on the number line then,
We can verify from the two number lines that equivalent fractions lie on the same position on the number line.
Equivalent Fractions of Whole Numbers
The equivalent fractions of whole numbers are the fractional representations of the whole numbers. We can represent any whole number through a fraction. Let us see an example to understand this better:
Let us consider numbers 1 and 2.
The number line shows 2 wholes here.
Each whole is divided into 1 equal part. 1 whole divided into 1 equal part can be written as \(\frac{1}{1}\)
Similarly, \(1=\frac{1}{1}=\frac{2}{2}=\frac{3}{3}=\frac{4}{4}=\)… ,
Now, 2 wholes each divided into 1 equal part can be written as \(\frac{2}{1}\) .
And, 4 wholes each divided into 2 equal part can be written as \(\frac{4}{2}\).
This shows us that , \(2=\frac{2}{1}=\frac{4}{2}=\frac{6}{3}=\frac{4}{4}=\)….
This is how we can find the equivalent fractions of whole numbers.
Comparing Fractions having the same Denominator (Comparison of like Fractions)
We can compare the fractions of the same denominator easily by comparing the numerators of the fractions. If the fractions have the same denominator then the fraction having a bigger numerator is greater.
Let us understand this better by considering two fractions \(\frac{3}{8}\) and \(\frac{7}{8}\) .
Let’s try to compare them. We can observe that both these fractions have the same denominator. This shows that the whole is divided into the same number of equal parts.
Now let us see how many equal parts are taken by observing the numerators. The numerator that is greater shows a greater number of equal parts, and hence, it is the greater fraction.
Clearly, \(\frac{3}{8}\) is smaller than \(\frac{7}{8}\). And this is represented as,
\(\frac{3}{8}\)<\(\frac{7}{8}\)
Compare Fraction of same Numerator
Let us think of two identical pizzas. You have cut the first one into 4 slices, and the second one into 8 slices. You have taken 3 slices from each pizza. Now, from which pizza you have taken more?
By observing you can say that you got more pizza from the first one. Now, let us analyze the same situation using fractions,
\(Fraction~of~pizza~taken~from~the~first~pizza=\frac{3(Number~of~equal~parts~taken)}{4(Number~of~equal~parts)}\)
\(Fraction~of~pizza~taken~from~the~second~pizza=\frac{3(Number~of~equal~parts~taken)}{8(Number~of~equal~parts)}\)
Clearly, \(\frac{3}{4}\)>\(\frac{3}{8}\)
When the numerators are the same, we look at the denominators of the fractions.
Look at these two fractions, \(\frac{3}{4}\) and \(\frac{3}{8}\)
When we compare such fractions, the greater the numerator, the smaller the fraction. This is because, the more the number of parts the whole is divided into, the smaller the parts are, and lesser the number of parts makes the fraction bigger. Hence, the fraction with the greater denominator is the smaller fraction.
For example, if we compare \(\frac{3}{7}\) and \(\frac{3}{5}\).
Both fractions have the same numerator, 3.
Three times \(\frac{1}{7}\) is \(\frac{3}{7}\) , and three times \(\frac{1}{5}\) is \(\frac{3}{5}\).
So, on comparing the numerators we can see that denominator of \(\frac{3}{7}\) is 7 which is greater than the denominator of \(\frac{3}{5}\) i.e. 5. Since we already know that a fraction with the greater denominator is the smaller fraction, this results in
\(\frac{3}{7}\)being smaller than \(\frac{3}{5}\), or,
\(\frac{3}{7}\)<\(\frac{3}{5}\)
Solved Equivalent Fraction Examples
Example 1:
Order the fractions\(\frac{7}{8}\) ,\(\frac{1}{8}\) and \(\frac{5}{8}\) from least to the greatest.
Solution:
We can find the order of the fractions using a number line.
Plot the fractions on the number line.
All the 3 fractions have the same denominator, which is 8.
\(\frac{1}{8}\) is farthest to the left. \(\frac{7}{8}\) is farthest to the right, and \(\frac{5}{8}\) is between the other two fractions.
So, the order from least to greatest is \(\frac{1}{8}\), \(\frac{5}{8}\), \(\frac{7}{8}\).
Hence we can state this as, \(\frac{1}{8}\) < \(\frac{5}{8}\)<\(\frac{7}{8}\)
Example 2: Compare fractions on the number line \(\frac{3}{4}\) and \(\frac{6}{8}\).
Solution:
First draw the fractions on the number line.
Draw a number line through 0 and 1. Then divide that into 8 equal parts. Mark the fractions on the number line as shown,
We can see that both the numbers are at the same point or position. Hence, we can conclude that the given fractions are equivalent.
Or, \(\frac{3}{4}\) = \(\frac{6}{8}\)
Example 3: Compare fractions on the number line \(\frac{4}{6}\) and \(\frac{4}{4}\).
Solution:
Draw a number line through 0 and 1. Then divide that into 4 equal parts. Mark the fractions on the number line as shown.
Clearly \(\frac{4}{4}\) = 1 and \(\frac{4}{6}\) can be seen below
We can see that \(\frac{4}{6}\) is on the left of \(\frac{4}{4}\).
Therefore \(\frac{4}{6}\) is smaller than \(\frac{4}{4}\).
Or, \(\frac{4}{6}\)<\(\frac{4}{4}\)
Example 4: Find three equivalent fractions of \(\frac{5}{6}\).
Solution:
The equivalent fractions can be obtained by multiplying an integer to the numerator and the denominator as shown:
\(\frac{5}{6}=\frac{5\times 2}{6\times 2}=\frac{10}{12}\)
\(\frac{5}{6}=\frac{5\times 3}{6\times 3}=\frac{15}{18}\)
\(\frac{5}{6}=\frac{5\times 4}{6\times 4}=\frac{20}{24}\)
Therefore, three equivalent fractions of \(\frac{5}{6}\) are \(\frac{10}{12}\) ,\(\frac{15}{18}\), \(\frac{20}{24}\)
Example 5:
Use a number line to find an equivalent fraction of \(\frac{3}{4}\).
Solution:
Step 1: Plot \(\frac{3}{4}\) on a number line.
Step 2: Divide the number line into eighths. Label the marks to show eighths.
The fractions that falls on the same point as \(\frac{3}{4}\) is \(\frac{6}{8}\).
Hence one of the equivalent fractions of \(\frac{3}{4}\) is \(\frac{6}{8}\).
The like fractions have the same positions on the number line. These are some examples of some fractions of number line \(\frac{1}{2}\) and \(\frac{2}{4}\) ,\(\frac{4}{5}\) and \(\frac{8}{10}\).
If the denominator of two or more fractions are equal then the fractions are called “like” fractions. For example, \(\frac{4}{5}\), \(\frac{3}{5}\), \(\frac{6}{5}\) and so on.
If the denominator of two or more fractions are not equal, then the fractions are called unlike fractions. For example, \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{4}{7}\) and so on.
There are two types of fractions, proper fractions and improper fractions. If the numerator of a fraction is less than that of the denominator, then the fraction is called a proper fraction.
For example: \(\frac{2}{3}\), \(\frac{3}{23}\), \(\frac{9}{17}\) proper fractions.
If the numerator is greater than the denominator, then the fraction is called an improper fraction.
For example: \(\frac{5}{4}\), \(\frac{3}{2}\), \(\frac{9}{5}\) are improper fractions.
Improper fractions can also be converted to mixed numbers or mixed fractions. Examples of mixed numbers are :
\(1\frac{2}{3}\), \(3\frac{5}{6}\) are mixed numbers or mixed fractions.