Equivalent Fractions Using Number Lines & Area Models
Find Equivalent Fractions by using Area model and Number Line
There are three friends John, Robert, and David. Each of them took the pizza slices as shown.
John cut the pizza into two equal parts and took 1 of them.
Robert cut the pizza into 6 equal parts and took 3 of them.
David cut the pizza into 8 equal pieces and took 4 of them.
Now, who has taken more pizza? Observing each friend’s pizza slice, we can see that everyone has taken half the pizza. They have taken an equivalent amount of pizza.
But in the fraction notation, we can see,
John has taken \( \frac{1}{2} \) of pizza, Robert has taken \( \frac{3}{6} \) of pizza and David has taken \( \frac{4}{8} \) of pizza.
This indicates that the three fractions \( \frac{1}{2},~\frac{3}{6} \) and \( \frac{4}{8} \) are equal fractions but appear as different fractions. Although these fractions appear different because of different numerators and denominators, but they have the same value.
What are Equivalent Fractions?
We know that two or more numbers that have the same value are called equivalent. For example 2 = 2 , 3 = 3 and so on. Similarly, two or more fractions that represent the same parts of a whole are called equivalent fractions. Equivalent fractions always represent the same point on the number line.
For example,
\( \frac{2}{3} \) is equivalent to \( \frac{4}{6} \) and \( \frac{6}{10} \) because they represent the same part of a whole.
So, \( \frac{2}{3}=\frac{4}{6} \) and \( \frac{2}{3}=\frac{6}{10} \)
There are two ways to represent equivalent fractions
1. Fraction bars
2. Number line
Fraction Bars or Area Model
We can draw an area model that shows a whole divided into different parts. In this model we draw an undivided rectangle to represent a whole. To represent any fraction in the model, we divide the whole (rectangle) into as many equal parts as the denominator. Then color the required parts which would come from the numerator of the fraction.
Here are some fraction bars of the whole (1), one half \( (\frac{1}{2}) \), one third \( (\frac{1}{3}) \), one fourth \( (\frac{1}{4}) \), one sixth \( (\frac{1}{6}) \) and so on.
Similarly, we can represent \( \frac{3}{4} \) using the fraction bars. We divide the whole into 4 parts as this is equal to the denominator. Then shade 3 parts as equal to the numerator. The bar model is obtained as,
Now, to represent equivalent fractions on a bar model we first draw a bar model of a fraction. Then, we subdivide each part into equal parts and count total parts and shaded parts. The number of shaded parts and total number of parts written in numerator and denominator respectively represent the equivalent fraction.
For example,
We can write the equivalent fraction of \( \frac{1}{2} \) as,
We first draw a bar which represent the whole. The divide the bar into 2 parts as equal to the denominator. Shade the one part to represent fraction \( \frac{1}{2} \).
Then subdivide each part into halves. We get a total number of parts 4 and 2 shaded parts. So, the equivalent fraction is \( \frac{2}{4} \).
Again, by further subdividing into parts we can get, \( \frac{3}{6},~\frac{4}{8},~\frac{5}{10}, \) and so on are the equivalent fractions.
Therefore, \( \frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\ldots \)
Number Line
The number line method to represent equivalent fractions is the same as that of the area model. In this method, to represent any number, we draw a number line with whole numbers. Then, we divide the space between each whole into equal parts, and the number of equal parts needs to match the number in the denominator. Then mark the point as equal to the numerator.
For example,
We can represent fraction \( \frac{2}{6} \) in the following manner.
First, draw a number line through 0 and 1. This represents a whole. Then divide the whole into 6 equal parts as this number is equal to the denominator. Then look into the numerator and mark the second point as equal to the numerator. So, \( \frac{2}{6} \) on the number line can be represented as,
To represent equivalent fractions on the number line, we subdivide each part into equal parts. Then count the total number of parts and the position of the marked point. At last, put the position of marked point at numerator and total number of parts in the denominator.
For example,
We can find equivalent fraction of \( \frac{1}{2} \) as,
First draw a number line through 0 and 1. Then, divide the whole into two parts as equal to the denominator. Then mark \( \frac{1}{2} \) on the line. Then subdivide each part into two equal parts. Then we can see for the same position the new fraction is \( \frac{2}{4} \) because there are 4 equal parts and the marking is after two equal parts.
So, \( \frac{1}{2} \) and\( \frac{2}{4} \) are equivalent fractions. Again by subdividing into equal parts we can show, \( \frac{1}{2},~\frac{2}{4},~\frac{3}{6} \) and so on are equivalent fractions.
Therefore, \( \frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\ldots \)
Solved Equivalent Fractions Using Number Lines Examples
Example 1: Find an equivalent fraction of \( \frac{4}{5} \) using the area model.
Solution:
To find the equivalent fraction of \( \frac{4}{5} \) using the area model, we first draw a rectangular box that represents a whole.
Then divide the whole into 5 equal parts as equal to the denominator of the fraction.
Then shade 4 equal parts as equal to the numerator of the fraction to represent \( \frac{4}{5} \).
Then divide each part into halves.
Count the total equal boxes and shaded boxes. The number of shades boxes at numerator and total number of boxes at denominator of a fraction. This obtained fraction is \( \frac{8}{10} \).
Therefore \( \frac{4}{5} \) is an equivalent fraction of \( \frac{8}{10} \).
So, \( \frac{4}{5}=\frac{8}{10} \).
Example 2:
Use a number line to find an equivalent fraction of \( \frac{1}{4} \).
Solution:
First represent the fraction \( \frac{1}{4} \) on the number line.
Make a number line through 0 and 1 which represents a whole. Then divide the whole into 4 equal parts and mark \( \frac{1}{4} \).
Then subdivide each part into halves. Count the new position of the same point on the number line.
Count the total number of parts. We can see that the new fraction for the same position is , \( \frac{2}{8} \).
Therefore, \( \frac{1}{4} \) and \( \frac{2}{8} \) are equivalent fractions.
So, \( \frac{1}{4}=\frac{2}{8} \).
Example 3:
Your notebook is \( \frac{1}{6} \) feet long. Your friend’s notebook is \( \frac{3}{12} \) feet long. Are the notebooks of the same length?
Solution:
To solve the problem, we check whether the fractions are equivalent or not. If the fractions are equivalent then the length of the notebooks will be equal. Otherwise the notebooks are of different sizes.
We first draw a rectangular box to represent a whole. In the fraction \( \frac{1}{6} \) the numerator is 1 and the denominator is 6. Then, divide the whole into 6 equal parts and shade 1 of the part.
Then subdivide each part into two equal parts. And count the shaded and the total number of parts.
There are 2 shaded part among a total of 12 parts. So, the equivalent fraction of \( \frac{1}{6} \) is \( \frac{2}{12} \)
The length of your notebook is \( \frac{2}{12} \) foot.
But, the friend’s notebook is \( \frac{3}{12} \) foot long.
Therefore, the length of two notebooks are not equal.
Two or more fractions that have the same values but appears different are called equivalent fractions.
Or,
Two or more fractions that name the same part of a whole are called equivalent fractions. Equivalent fractions always represent the same point on the numbers line.
For example, \( \frac{1}{2} \) and \( \frac{2}{4},~\frac{2}{3},~\frac{4}{6} \) and \( \frac{6}{10} \) etc.
\( \frac{2}{3} \) is equivalent to \( \frac{4}{6} \) and \( \frac{6}{10} \) because they represent the same part of a whole.
So, \( \frac{2}{3}=\frac{4}{6} \) and \( \frac{2}{3}=\frac{6}{10} \).
To find if two fractions are equivalent or not, we draw their area models or line diagrams. Then observe their positions to see whether they are at the same position or not. If they represent the same part of a whole, or the same point on the number line, then we say them equivalent fractions.
Otherwise, we can make the denominator of the two fractions same and compare the numerator. If the numerator of the fractions after multiplication are same then the fractions are said to be equivalent.
For example,
We can find that \( \frac{2}{5} \) and \( \frac{4}{10} \) are equivalent or not as,
Multiply by 2 on both numerator and denominator of the fraction \( \frac{2}{5} \)
\( \frac{2}{5}=\frac{2\times 2}{5\times 2}=\frac{4}{10} \)
The second fraction is \( \frac{4}{10} \). So, both the fractions are equal.
Therefore, \( \frac{2}{5} \) and \( \frac{4}{10} \) are equivalent fractions.