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Equivalent fractions are the fractions that represent the same value, even though they look different. For example, if you have a cake, cut it into two equal pieces, and eat one of them, you will have eaten half the cake. If you cut a cake into eight equal pieces and eat four of them, you will still have eaten half the cake. These are equivalent fractions.
Let’s understand what is an equivalent fraction.
What are Equivalent Fractions?
Equivalent fractions are the fractions that have different numerators and denominators but still, the values of the two fractions are equal.
For example, $\frac{1}{2}$ and $\frac {3}{6}$ are equivalent fractions. Also, $\frac{6}{12}$ is equivalent to both $\frac{1}{2}$ and $\frac {3}{6}$. Both of these fractions $\frac {3}{6}$, $\frac{6}{12}$ can be reduced to the lowest form of $\frac{1}{2}$.
Visual Representation of Equivalent Fractions
Let’s visualize fractions $\frac{1}{2}$, $\frac{2}{4}$ and $\frac{3}{6}$ by dividing a circle into equal parts (sectors).
All these fractions have different numerators and denominators. But they all represent half part of a circle. Hence, the fractions $\frac{1}{2}$, $\frac{2}{4}$ and $\frac{3}{6}$ are equivalent fractions and they all reduce to the lowest/simplest form $\frac{1}{2}$.
Let’s consider one more example.
All these fractions also have different numerators and denominators. But they all represent three-fourths of a square. Hence, the fractions $\frac{3}{4}$, $\frac{12}{16}$ and $\frac{48}{64}$ are equivalent fractions and they all reduce to the lowest/simplest form $\frac{3}{4}$.
How to Find Equivalent Fractions?
You can find equivalent fractions of any given fraction by
- multiplying both the numerator and the denominator by a number
- dividing both the numerator and the denominator by a number
Note:
- The method of multiplying is generally used when the numerator and denominator of the given fraction are smaller
- The method of dividing is generally used when the numerator and denominator of the given fraction are larger
Finding Equivalent Fractions by Multiplying
In order to find the equivalent fractions of any fraction multiply both the numerator and the denominator by the same number.
Let’s understand it by the following example.
You want to find equivalent fractions of $\frac {5}{7}$. Let’s multiply both the numerator and the denominator by numbers $3$, $5$, $6$, $7$, $8$ and $10$.
Note: You can choose any number.
Multiplying the numerator and the denominator of $\frac {5}{7}$ by $3$: $\frac {5 \times 3}{7 \times 3}$ = $\frac {15}{21}$
Multiplying the numerator and the denominator of $\frac {5}{7}$ by $5$: $\frac {5 \times 5}{7 \times 5}$ = $\frac {25}{35}$
Multiplying the numerator and the denominator of $\frac {5}{7}$ by $6$: $\frac {5 \times 6}{7 \times 6}$ = $\frac {30}{42}$
Multiplying the numerator and the denominator of $\frac {5}{7}$ by $7$: $\frac {5 \times 7}{7 \times 7}$ = $\frac {35}{49}$
Multiplying the numerator and the denominator of $\frac {5}{7}$ by $8$: $\frac {5 \times 8}{7 \times 8}$ = $\frac {40}{56}$
Multiplying the numerator and the denominator of $\frac {5}{7}$ by $10$: $\frac {5 \times 10}{7 \times 10}$ = $\frac {50}{70}$
$5$ equivalent fractions of $\frac {5}{7}$ are $\frac {15}{21}$, $\frac {25}{35}$, $\frac {30}{42}$, $\frac {35}{49}$, and $\frac {40}{56}$.
Note:
- For any fraction, there are countless equivalent fractions.
- You can find any number of equivalent fractions by multiplying the numerator and the denominator by the same number.
Finding Equivalent Fractions by Dividing
In order to find the equivalent fractions of any fraction divide both the numerator and the denominator by the same number.
Let’s understand it by the following example.
You want to find equivalent fractions of $\frac {42}{126}$.
Factors of $42$ are $1$, $2$, $3$, $6$, $7$, $14$, $21$, and $42$.
Factors of $126$ are $1$, $2$, $3$, $6$, $9$, $14$, $18$, $21$, $42$, $63$, and $126$.
Common factors of $42$ and $126$ are $1$, $2$, $3$, $6$, $14$, $21$, and $42$
You can divide numerator and denominator by any of these numbers – $2$, $3$, $6$, $14$, $21$, and $42$.
Note: $1$ is not considered as dividing by $1$ results in the same number.
Dividing the numerator and the denominator of $\frac {42}{126}$ by $2$: $\frac {42 \div 2}{126 \div 2}$ = $\frac {21}{63}$
Dividing the numerator and the denominator of $\frac {42}{126}$ by $3$: $\frac {42 \div 3}{126 \div 3}$ = $\frac {14}{42}$
Dividing the numerator and the denominator of $\frac {42}{126}$ by $6$: $\frac {42 \div 6}{126 \div 6}$ = $\frac {7}{21}$
Dividing the numerator and the denominator of $\frac {42}{126}$ by $14$: $\frac {42 \div 14}{126 \div 14}$ = $\frac {3}{9}$
Dividing the numerator and the denominator of $\frac {42}{126}$ by $21$: $\frac {42 \div 21}{126 \div 21}$ = $\frac {2}{6}$
Dividing the numerator and the denominator of $\frac {42}{126}$ by $42$: $\frac {42 \div 42}{126 \div 42}$ = $\frac {1}{3}$
$6$ equivalent fractions of $\frac {42}{126}$ are $\frac {21}{63}$, $\frac {14}{42}$, $\frac {7}{21}$, $\frac {3}{9}$, $\frac {2}{6}$ and $\frac {1}{3}$.
How Do You Know Fractions Are Equivalent?
You can check whether two or more fractions are equivalent or not by simplifying the fractions. The idea behind the process of simplification is that every equivalent fraction reduces to the same fraction when written in the simplest form.
The methods that you can use to check whether given fractions are equivalent or not are
- Making Denominators Same
- Converting Fractions to Decimals
- Cross Multiplication Method
Let’s use these methods one by one.
Making Denominators Same
The denominators of two or more equivalent fractions can be made the same by finding the L.C.M. of the denominators of the fractions.
Let’s consider the following examples to understand the process.
Ex 1: $ \frac {12}{20}$ and $ \frac {18}{30}$
The two denominators are $20$ and $30$.
L.C.M. of $20$ and $30$ is $60$.
For first fraction $60 \div 20 = 3$
For first fraction $60 \div 30 = 2$
Now, multiplying the numerators and the denominators of the first fraction by $3$ and the second by $2$.
$ \frac {12 \times 3}{20 \times 3} = \frac {36}{60}$
$ \frac {18 \times 2}{30 \times 2} = \frac {36}{60}$
Since, both the fractions $ \frac {12}{20}$ and $ \frac {18}{30}$ can be written as \frac {36}{60}$, therefore these fractions are equivalent fractions.
Ex 2: $ \frac {18}{36}$ and $ \frac {5}{9}$
L.C.M. of $36$ and $9$ is $36$.
For first fraction $36 \div 36 = 1$
For first fraction $36 \div 9 = 4$
Note: The quotient obtained for $ \frac {18}{36}$ is $1$, therefore, there no need to multiply, as multiplying any number by $1$ gives the same number.
Multiply the numerator and the denominator of the second fraction by $4$.
$ \frac {5 \times 4}{9 \times 4} = \frac {20}{36}$
As the fractions $ \frac {18}{36}$ and $ \frac {20}{36}$ are not the same, therefore, the fractions $ \frac {18}{36}$ and $ \frac {5}{9}$ are not equivalent.
Converting Fractions to Decimals
Two or more fractions will be equivalent fractions if they convert to the same decimal numbers.
Ex 1: $\frac {12}{18}$ and $\frac {4}{6}$
$\frac {12}{18} = 0.666666…$
$\frac {4}{6} = 0.666666…$
Here, the decimal represent of both the fractions is $0.666666…$, therefore, these fractions are equivalent fractions.
Ex 2: $\frac {5}{12}$ and $\frac {35}{48}$
$\frac {5}{12} = 0.666666…$
$\frac {35}{48} = 0.7291666…$
The fractions $\frac {5}{12}$ and $\frac {35}{48}$ are not equivalent fractions.
Cross Multiplication Method
In the cross multiplication method, multiply the numerator of the first fraction by the denominator of the second fraction and then multiply the denominator of the first fraction by the numerator of the second fraction. If the two products are equal, the fractions are equivalent otherwise not.
Ex 1: $\frac {3}{7}$ and $\frac {18}{42}$
$3 \times 42 = 126$ and $7 \times 18 = 126$
$\frac {3}{7}$ and $\frac {18}{42}$ and are equivalent fractions.
Ex 2: $\frac {35}{55}$ and $\frac {60}{75}$
$35 \times 75 = 2625$ and $55 \times 60 = 3300$
$\frac {35}{55}$ and $\frac {60}{75}$ are not equivalent fractions.
Conclusion
The two or more fractions are called equivalent fractions if they have different numerators and denominators but represent the same value. There exist countless equivalent fractions for any fraction and these equivalent fractions can be checked by using either of these methods – by making denominators the same, converting fractions to decimals, or cross multiplication method.
Practice Problems
- Find five equivalent fractions for each of the following fractions:
- $\frac {1}{2}$
- $\frac {4}{5}$
- $\frac {18}{33}$
- $\frac {65}{75}$
- $\frac {42}{77}$
- Check whether given fractions are equivalent or not
- $\frac {5}{6}$ and $\frac {45}{54}$
- $\frac {3}{9}$ and $\frac {18}{54}$
- $\frac {7}{13}$ and $\frac {77}{169}$
- $\frac {56}{91}$ and $\frac {49}{84}$
- $\frac {60}{135}$ and $\frac {64}{144}$
