- What is a common denominator?
- What are proper and improper fractions?
- What are equivalent fractions?
- How do we use equivalent fractions to write fractions with common denominators?
- Adding fractions with unlike denominators
- Subtracting fractions with unlike denominators
- What are mixed numbers?
- Adding mixed fractions with like denominators
- Adding and subtracting mixed fractions with unlike denominators
- Solved Examples
- Frequently Asked Questions
What is a common denominator?
When the denominators of two or more fractions are the same, they are known as common denominators.
For example, the fractions \(\frac{3}{5}\) and \(\frac{7}{5}\) have 5 as the common denominator.
What are proper and improper fractions?
A proper fraction is defined as a fraction in which the numerator is less than the denominator. For example,
An improper fraction is defined as a fraction in which the denominator is less than the numerator. For example,
What are equivalent fractions?
When two or more fractions are written in their simplest form and their value is the same, they are called equivalent fractions. For instance, the equivalent fractions of\(\frac{1}{5}\) are \(\frac{5}{25}\) , \(\frac{6}{30}\) , and \(\frac{4}{20}\) , all of which can be simplified to the same fraction, \(\frac{1}{5}\) .
How do we use equivalent fractions to write fractions with common denominators?
Step 1: Determine which denominator of each fraction is larger.
For example, \(\frac{7}{18}\) and \(\frac{5}{24}\) :
The larger of the two denominators, in this case, is 24.
Step 2: Examine whether the smaller denominator divides evenly into the larger one. Since 18 does not divide evenly into 24, you need to proceed to the next step.
You can skip step 3 if the smaller denominator divides the larger denominator evenly. For example, if the denominators are 3 and 9, you’ll find that 3 divides 9 evenly. Hence, 9 becomes the common denominator.
Step 3: Multiples of the larger denominator should be checked.
Examine multiples of the larger denominator until you find a number that the smaller denominator can divide evenly into as well.
2 × 24 = 48, but 18 does not evenly divide 48.
3 × 24 = 72, and 18 evenly divides 72. In this case, the common denominator is 72.
Step 4: Write the first fraction as an equivalent fraction with the common denominator. 18 divides into 72 four times, so the fraction \(\frac{7}{18}\) is multiplied by 4:
\(\frac{7}{18}\) x \(\frac{4}{4}\) = \(\frac{28}{72}\)
Step 5: With the common denominator, write the second fraction as an equivalent fraction.
Since 24 divides 72 three times, the fraction \(\frac{5}{24}\) is multiplied by 3:
\(\frac{5}{24}\) x \(\frac{3}{3}\) = \(\frac{15}{72}\)
Now both \(\frac{28}{72}\) and \(\frac{15}{72}\) have 72 as the common denominator.
Adding fractions with unlike denominators
Follow the steps to find the addition of fractions with unlike denominators:
Step 1: Write fractions with the same denominator, using equivalent fractions.
Step 2: Calculate the sum.
Example: Find the sum of fractions \(\frac{5}{8}\) + \(\frac{1}{6}\).
Step 1: To write fractions with a common denominator, use equivalent fractions.
8 is not a multiple of 6, so we rewrite each fraction with an 8 × 6 = 48, 48 as the denominator.
Rewrite, \(\frac{5}{8}\) as \(\frac{5\times 6}{8\times 6}\) = \(\frac{30}{48}\) and \(\frac{1}{6}\) as \(\frac{1\times 8}{6\times 6}\) = \(\frac{8}{48}\)
Step 2: Calculate the sum, as shown,
\(\frac{30}{48}\) + \(\frac{8}{48}\) = \(\frac{38}{48}\) = \(\frac{19}{24}\)
Subtracting fractions with unlike denominators
Follow the steps to perform the subtraction of fractions with unlike denominators:
Step 1: Write the fractions with the same denominator, using equivalent fractions.
Step 2: Calculate the difference.
Example: Calculate the difference \(\frac{2}{5}~-~\frac{1}{4}\) .
Step-1: To write fractions with a common denominator, use equivalent fractions.
5 is not a multiple of 4, so rewrite each fraction with the denominator 5 × 4 = 20, 20 in this case, is the common denominator.
Rewrite, \(\frac{2}{5}\) as \(\frac{2\times 4}{5\times 4}\) = \(\frac{8}{20}\) and \(\frac{1}{4}\) as \(\frac{1\times 5}{4\times 5}\) = \(\frac{5}{20}\)
Step 3: Calculate the difference.
\(\frac{8}{20}\) – \(\frac{5}{120}\) = \(\frac{8~-~5}{20}\) = \(\frac{3}{20}\)
What are mixed numbers?
A mixed number, also known as a mixed fraction, is a number made up of one integer and a proper fraction. It is a mixture of whole numbers and fractions.
Real-life examples include leftover pizza (you may have two and a quarter pizzas left), or unfinished drinks (you may have three and a half cans of cola left).
Adding mixed fractions with like denominators
Mixed fractions with like denominators are those mixed fractions that have the same denominator.
For example, \(3\frac{1}{2}\) and \(6\frac{1}{2}\) .
Let us take an example to understand the steps for the addition of mixed fractions with like denominators.
\(3\frac{1}{2}\) + \(6\frac{1}{2}\)
Step 1: Add the whole numbers of both fractions, that is, 3 + 6 = 9.
Step 2: Add the fractional parts of both numbers, that is, \(\left(\frac{1}{2}\right)\) + \(\left(\frac{1}{2}\right)\) = \(\frac{2}{2}\) .
Step 3: Convert the result from step 2 to its simplest form, \(\frac{2}{2}\) = 1.
Step 4: Add the results of steps 1 and 3, which equals 9 + 1 = 10.
Adding and subtracting mixed fractions with unlike denominators
We must convert two or more mixed fractions into improper fractions first. Then we must determine whether the denominators of the given fractions are equal. We can add or subtract them directly if they are equal, but if they are not, we must convert them to like denominators. Then, we can add or subtract the numerators while keeping the denominator the same.
Example:
Find the sum of the fractions \(2\frac{3}{5}\) + \(1\frac{3}{10}\) .
Solution:
Step 1: Convert mixed fractions to improper fractions.
\(2\frac{3}{5}\) + \(1\frac{2}{10}\)
\(2\frac{3}{5}\) = 2 + \(\frac{3}{5}\) = \(\frac{10}{5}\) + \(\frac{3}{5}\) = \(\frac{13}{5}\)
\(1\frac{3}{10}\) = 1 + \(\frac{3}{10}\) = \(\frac{10}{10}\) + \(\frac{3}{10}\) = \(\frac{13}{10}\)
\(2\frac{3}{5}\) + \(1\frac{3}{10}\) = \(\frac{13}{5}\) + \(\frac{13}{10}\)
Step 2: Change the fractions into fractions with a common denominator.
= \(\frac{13\times 2}{5\times 2}\) + \(\frac{13}{10}\)
= \(\frac{26}{10}\) + \(\frac{13}{10}\)
Step 3: Combine the like fractions and simplify the sum.
= \(\frac{26~~+~~13}{10}\)
= \(\frac{39}{10}\) = \(\frac{30~~+~~9}{10}\) = \(\frac{30}{10}\) + \(\frac{9}{10}\) = 3 + \(\frac{9}{10}\) = \(3\frac{9}{10}\) .
Solved Examples
Example 1:
Find the sum of the fractions \(\frac{3}{8}\) and \(\frac{7}{6}\).
Solution:
Step 1: Write the fractions with a common denominator using equivalent fractions.
8 is not a multiple of 6, so rewrite each fraction with the denominator 8 × 6 = 48 as the denominator.
Rewrite, \(\frac{3}{8}\) as \(\frac{3\times 6}{8\times 6}\) = \(\frac{18}{48}\) and \(\frac{7}{6}\) as \(\frac{7\times 8}{6\times 8}\) = \(\frac{56}{48}\)
Step 2: Calculate the sum.
\(\frac{18}{48}\) + \(\frac{56}{48}\) = \(\frac{74}{48}\) = \(\frac{37}{24}\)
Example 2:
Calculate the difference: \(\frac{2}{7}\) – \(\frac{1}{4}\)
.
Solution:
Step 1: Write the fractions with a common denominator using equivalent fractions.
7 is not a multiple of 4, so rewrite each fraction with the denominator 7 × 4 = 28.
Rewrite, \(\frac{2}{7}\) as \(\frac{2\times 4}{7\times 4}\) = \(\frac{8}{28}\) and \(\frac{1}{4}\) as \(\frac{1\times 7}{4\times 7}\) = \(\frac{7}{28}\)
Step 2: Calculate the difference.
\(\frac{8}{28}~~-\) \(\frac{7}{28}\) = \(\frac{8~-~7}{28}\) = \(\frac{1}{28}\)
Example 3:
Find the difference between the fractions \(5\frac{2}{3}\) – \(2\frac{1}{3}\).
Solution:
Step 1: Subtract the whole numbers of both fractions, that is, 5 – 2 = 3.
Step 2: Subtract the fractional parts of both numbers, that is, \(\left(\frac{2}{3}\right)\) – \(\left(\frac{1}{3}\right)\) = \(\frac{1}{3}\).
Step 3: Now add the results of steps 1 and 2, which equals \(3~+~\frac{1}{3}\) = \(3\frac{1}{3}\)
Example 4:
Find the sum of the fractions \(3\frac{1}{5}~+~2\frac{3}{10}\).
Solution:
Step 1: Convert the mixed fractions into improper fractions.
\(3\frac{1}{5}\) + \(2\frac{3}{10}\)
\(3\frac{1}{5}\) = 3 + \(\frac{1}{5}\) = \(\frac{15}{5}\) + \(\frac{1}{5}\) = \(\frac{16}{5}\)
\(2\frac{3}{10}\) = 2 + \(\frac{3}{10}\) = \(\frac{20}{10}\) + \(\frac{3}{10}\) = \(\frac{23}{10}\)
= \(\frac{16}{5}\) + \(\frac{23}{10}\)
Step 2: Change the fractions into fractions with a common denominator.
= \(\frac{16\times 2}{5\times 2}\) + \(\frac{23}{10}\)
= \(\frac{32}{10}\) + \(\frac{23}{10}\)
Step 3: We combine the like fractions and simplify the sum.
= \(\frac{32~+~23}{10}\)
\(\frac{55}{10}\) = \(\frac{11}{2}\) = \(\frac{10~+~1}{2}\) + \(\frac{10}{2}\) + \(\frac{1}{2}\) = 5 + \(\frac{1}{2} \) = 5\(\frac{1}{2}\) .
A fraction represents a portion of something larger. As a result, if we have to read a fraction, such as \(\frac{3}{4}\) , we should read it as three-fourths of a whole.
Let us understand a mixed fraction through an example. When \(2\frac{1}{3}\) is converted to an improper fraction, we get \(\frac{7}{3}\). Now, if we perform the division 7 ÷ 3 , we get 2 as the quotient, 1 as the remainder, and 3 as the divisor, which are the parts of the mixed fraction \(2\frac{1}{3}\).
Hence, the whole number of the mixed fraction denotes the quotient, the numerator of the fraction denotes the remainder, and the denominator of the fraction denotes the divisor.