General Form of a Fraction
A fraction is essentially what we get when we divide a whole number by another whole number, other than zero. So, we can also use fractions to represent division operations.
The general form of a fraction is \( \frac{a}{b}\). Here, ‘a’ is known as the numerator and ‘b’ is known as the denominator. If \( b~> ~a\) in a fraction, the fraction is known as a proper fraction. On the other hand, if \( a~\geq ~b\) in a fraction, the fraction is known as an improper fraction.
What are ‘Like’ Fractions and ‘Unlike’ Fractions?
Like fractions are fractions that have the same denominator. So, the value of ‘b’ in like fractions will be the same. On the contrary, unlike fractions have different numbers as their denominators. So, the value of ‘b’ in unlike fractions will be different.
Like fractions and unlike fractions are also known as similar fractions and dissimilar fractions, respectively.
Like Fractions
The fractions that have the same denominator are known as like fractions or similar fractions. In the case of like fractions, the numerator can be any whole number. Consider an example of a pizza divided into 5 parts.
Here, the fractions \( \frac{1}{5},~\frac{2}{5},~\frac{3}{5},~\frac{4}{5},\) and \( \frac{5}{5}\) are like fractions.
Unlike Fractions
If two fractions have different denominators, they are known as unlike fractions or dissimilar fractions. The fractions used to represent the parts of the two pizzas divided into different numbers of slices are examples of unlike fractions.
In this case, the first pizza is divided into 4 parts. So each slice can be represented using the fraction \( \frac{1}{4}\). The second pizza is divided into 5 parts, and each slice of this pizza can be represented using the fraction \( \frac{1}{5}\). Since the denominators of these fractions are different, they are unlike fractions.
Addition of Like and Unlike Fractions
When we add two like fractions, the result will have the same denominator. The numerator of the result will be the sum of the numerators of the two like fractions.
For example, \( \frac{1}{8}~+~\frac{4}{8}~=~\frac{5}{8}\). Here, the denominator of the result remains 8, and the numerator of the result is the sum of the numerators of the two addends.
We cannot perform direct addition or subtraction on two unlike fractions. To perform these operations, we need to convert these unlike fractions into like fractions using simple multiplication and division operations.
To convert unlike fractions into like fractions, we can either try to simplify them, or we should make the denominators the same. It is always better to check whether they are in the simplest form. Two equivalent, unlike fractions can be simplified to get like fractions.
For example, \( \frac{1}{4}\) and \( \frac{2}{8}\) are equivalent and unlike fractions. But the second fraction can be simplified by dividing both numerator and denominator by 2 to get \( \frac{1}{4}\). Now, they are like fractions, and they can be added directly.
If the fractions don’t have the same denominator even after simplification, we should go ahead and perform some mathematical operations to make the denominators the same. In order to do this, we need to find the lowest common multiple (LCM) of the denominators. Now, we need to multiply the numerator and denominator of each fraction with a number so that the denominators become the LCM of the original fractions.
For example, to add \( \frac{1}{3}\) and \( \frac{2}{5}\), we need to find the LCM of 3 and 5.
Multiples of 3: 3, 6, 9, 12, 15, 18, 21…
Multiples of 5: 5, 10, 15, 20, 25, 30, 35…
So, the LCM of 3 and 5 is 15. Now, we need to multiply the numerator and denominator of the two fractions such that the denominators become 15.
\( \frac{1}{3}~=~\frac{1~\times~5}{3~\times~5}~=~\frac{5}{15}\)
Similarly, \( \frac{2}{5}~=~\frac{2~\times~3}{5~\times~3}~=~\frac{6}{15}\)
We know that \( \frac{5}{15}\) and \( \frac{6}{15}\) are similar fractions. We can add them directly to get \( \frac{11}{15}\) as the sum.
Solved Like & Unlike Fractions Examples
Example 1: Determine whether \( \frac{12}{25}\) and \( \frac{11}{25}\) are like fractions. If yes, find the sum of the two fractions.
Solution: Like fractions are fractions that have the same denominator. That means \( \frac{12}{25}\) and \( \frac{11}{25}\) are like fractions as they have the same denominator, 25.
So, the sum of the two fractions is \( \frac{12}{25}~+~\frac{11}{25}~=~\frac{23}{25}\).
Example 2: Sam gave 2 slices of a pizza and 3 slices of pie to his friend Marie. If the pizza and the pie had the same number of slices, determine whether the fractions that represent the slices of pizza and pie given to Marie are like fractions or not.
Solution: Sam gave 2 slices of a pizza and 3 slices of a pie to Marie, so the minimum number of slices that the pie or pizza can be split into is 3. But the number of slices is unknown.
Let the number of slices be \( x\). Since the number of slices of pie and pizza are the same, \( x\) represents a whole number greater than or equal to 3.
That means Marie gets \( \frac{2}{x}\) of the pizza and \( \frac{3}{x}\) of the pie. Since both fractions have the same denominator \( x\), they are like fractions.
Example 3: Pick out the like fractions from the following: \( \frac{3}{8}\), \( \frac{5}{10}\), \( \frac{11}{15}\), \( \frac{8}{8}\), \( \frac{9}{15}\), \( \frac{13}{14}\)
Solution: There are two pairs of like fractions among the provided fractions. They are \( \frac{5}{8}\) and \( \frac{8}{8}\) and \( \frac{11}{15}\) and \( \frac{9}{15}\).
Like fractions are the fractions that have the same denominators. For example, 1/55 and 32/55 are like fractions.
Like fractions and equivalent fractions are not the same. Like fractions have the same denominator. On the other hand, equivalent fractions are fractions that represent the same value; they can be simplified to represent the same fraction.
Since 1/15 and 12/15 have the same denominator, they are like fractions.