Addition of Fractions
Fractions are categorized into two types: like fractions and unlike fractions. Two or more fractions that have the same denominators are called like fractions. For example, \(\frac{2}{3}and\frac{4}{3}\) are like fractions as they have the same denominator, which is 3. Two or more fractions which consist of different denominators are called unlike fractions. For example \(\frac{4}{11}and\frac{5}{9}\) are unlike fractions.
One aspect that we have to keep in mind is that the process of adding unlike fractions is slightly different from the process of adding like fractions.
To find the sum of two or more like fractions we can directly add the numerators. However, it is different in case of unlike fractions. Wondering how? The next section will take us through it.
How do we Add Two or More Unlike Fractions?
As we know when the denominators are different, then, these fractions are called unlike fractions. To add such fractions, we need to convert them into like fractions. This can be done by rewriting one or both addends as equivalent fractions such that both fractions have the same denominator.
Equivalent fractions are those fractions which, when simplified, give the same fraction. So, while adding unlike fractions we may come across two instances.
Case: 1
Let’s consider the first example.
\(\frac{7}{9}+\frac{5}{3}\)
Here the denominators are 9 and 3. 9 is a factor of 3. So we can express the second fraction as an equivalent fraction with a denominator equal to 9, that is, we write\(\frac{5}{3}as\frac{15}{9}\)by multiplying both the numerator and the denominator of\(\frac{5}{3}\)by 3.
So the original equation becomes:
\(=\frac{7}{9}+\frac{15}{9}\)
Now we can follow the usual method of adding fractions, that is adding numerators and simplifying the result to its simplest form.
\(=\frac{7+15}{9}\) [Add the numerators]
\(=\frac{22}{9}\)
Case: 2
Let’s consider another example.
Suppose we want to add the fractions \(\frac{9}{7}and\frac{11}{2}\)
Here the denominators are 7 and 2.
In such cases we will express both fractions as equivalent fractions with a common denominator.
The common denominator value is equal to the lowest common multiple of the two denominator values.
Here the denominators are 2 and 7 whose lowest common multiple is 14.
So,
\(\frac{9}{7}\)is multiplied by\(\frac{2}{2}and\frac{11}{2}\) is multiplied by\(\frac{7}{7}\)
\(=\frac{9}{7}\times\frac{2}{2}+\frac{11}{2}\times\frac{7}{7}\) [Equivalent fractions]
\(=\frac{18}{14}+\frac{77}{14}\) [Multiply]
\(=\frac{18+77}{14}\) [Add]
\(=\frac{95}{14}\) [Simplify]
So,\(=\frac{9}{7}+\frac{11}{2}=\frac{95}{14}\)
Here the denominators are 5 and 10, in which 10 is a factor of 5.
So express 35 as an equivalent fraction with the denominator equal to 10 by multiplying both the numerator and the denominator of\(=\frac{3}{5}\)
by 2, that is,
\(=\frac{3\times2}{5\times2}+\frac{7}{10}\)
\(=\frac{6}{10}+\frac{7}{10}\) [Rewrite \(=\frac{3}{5}\) as \(=\frac{6}{10}\) for equivalent fraction]
\(=\frac{6+7}{10}\) [Add the numerators]
\(=\frac{13}{10}\)
Thus, \(\frac{3}{5}+\frac{7}{10}\) is \(\frac{13}{10}\)
Rapid Recall:
Solved Examples
Example 1:
Find \(\frac{47}{15}+\frac{35}{3}\)?
Solution:
The equation provided: \(\frac{47}{15}+\frac{35}{3}\)
Since the fractions are unlike fractions, write the second fraction as an equivalent fraction based on the denominator of the first fraction.
\(=\frac{47}{15}+\frac{35}{3}\) [Write the fractions]
\(=\frac{47}{15}+\frac{35\times5}{3\times5}\) [Multiply \(=\frac{35}{3}\)by \(=\frac{5}{5}\) for equivalent fraction]
\(=\frac{47}{15}+\frac{175}{15}\) [Equivalent fraction]
\(=\frac{47+175}{15}\) [Add the numerators]
\(=\frac{222}{15}\) [Result]
As a result, \(=\frac{47}{15}+\frac{35}{3}=\frac{222}{15}\)
Example 2:
Find the value of \(=\frac{19}{12}+\frac{3}{5}\) .
Solution:
The expression stated in the question: \(=\frac{19}{12}+\frac{3}{5}\)
Here, the fractions are unlike fractions.
Let’s express each fraction with a common denominator, that is,
\(12\times5=60\).
\(=\frac{19}{12}+\frac{3}{5}\) [Write the fractions]
Rewriting \(\frac{19}{2}\) as \(\frac{19\times5}{12\times5}\) and \(\frac{3}{5}\) as \(\frac{3\times12}{5\times12}\)
\(=\frac{19\times5}{12\times5}+\frac{19\times5}{5\times12}\) [Rewriting addends as equivalent fractions]
\(=\frac{95}{60}+\frac{36}{60}\)
\(=\frac{95+36}{60}\) [Add the numerators]
\(=\frac{131}{60}\) [Result]
Thus, the sum of the fractions \(\frac{19}{12}\) and \(\frac{3}{5}\) is \(\frac{131}{60}\).
Example 3:
All of the participants in a math quiz are asked to find out the sum of \(\frac{1}{7}+\frac{2}{21}\). Answers from Ashely, Phoebe, and Justin are \(\frac{5}{2},\frac{1}{21}and\frac{4}{21}\) respectively. Find out who gave the correct answer?
Solution:
As mentioned in the question, Ashely, Phoebe, and Justin gave their answers as \(\frac{15}{21},\frac{1}{21}and\frac{4}{21}\) respectively.
To find out who gave the right answer, let’s calculate the sum \(\frac{1}{7}+\frac{2}{21}and\frac{1}{7}+\frac{2}{21}\)
\(=\frac{1\times3}{7\times3}+\frac{2}{21}\) [Multiply \(\frac{1}{7}\) by \(\frac{3}{3}\)for equivalent fraction]
\(=\frac{3}{21}+\frac{2}{21}\) [Simplify]
\(=\frac{3+2}{21}\) [Add the numerators]
\(=\frac{5}{21}\) [Add]
Hence, the answer is \(=\frac{5}{21}\).
So, Ashley gave the correct answer.
Equivalent fractions are fractions which, when simplified, give the same original fraction. For example, \(\)\frac{1}{2} and \frac{2}{4}\(\) are the equivalent as \(\)\frac{2}{4}\(\) when simplified gives us \(\)\frac{1}{2}\(\).
The fundamental rule for adding fractions is that the denominators of the fractions should be the same. We can add the numerators, if the fractions have the same denominator. However, we must convert the fractions into like fractions if the denominators are different.
Yes. By writing a whole number as an improper fraction with a denominator equal to 1, we can add it to fractions following the usual process.
We have three major types of fractions in math, which are, proper fractions, improper fractions and mixed numbers.