What are Mixed Numbers?
A mixed number is a fraction that consists of a whole number and a fraction.
For instance, \(5\frac{8}{11}\) is a mixed number in which 5 is a whole number and \(\frac{8}{11}\) is a fraction.
How to Add Two Mixed Numbers?
There are two different methods for adding mixed numbers.
Method 1:
Step 1: First, add all the whole number parts of the given mixed numbers. This provides the sum of the whole numbers.
Step 2: Second, add the fractional parts of the mixed numbers. Here, the fractions may be of two types, fractions with like denominators and fractions with unlike denominators.
Step 3: If the fractions have like denominators, we can directly add the numerators and then simplify the result to arrive at the sum of the fractional parts.
Step 4: If the fractions have unlike denominators, use the equivalent fractions to make the denominators the same. In this way, we can convert unlike fractions into like fractions. Once we get like fractions, we follow the previous step to obtain the sum of the fractional parts of the mixed numbers.
Step 5: Now, merge both the results that we obtained after adding the whole number parts and the fractional parts. This will be the final result.
For example: Add \(2\frac{3}{7}+1\frac{2}{7}\)
\(=\left( 2+1 \right)+\left( \frac{3}{7}+\frac{2}{7} \right)\)
\(=3 + \frac{5}{7}\)
\(=3 \text{ }\frac{5}{7}\)
Method 2:
Write each mixed number as an equivalent fraction and add them.
For example: \(4\frac{2}{8}\text{ }+\text{ }2\frac{7}{8}\)
Let us consider the first mixed number, that is \(4\frac{2}{8}\)
\(= \frac{32}{8} + \frac{2}{8}\)
\(=\frac{34}{8}\)
And, the second mixed number is \(2\frac{7}{8}=\frac{16}{8}+\frac{7}{8}\)
\(=\frac{23}{8}\)
Now, add both the numbers.
\(\frac{34}{8}+\frac{23}{8}=\frac{57}{8}\)
Rapid Recall:
Solved Examples:
Example 1:
Find the value of \(7\frac{4}{11}\text{ }+\text{ }8\frac{9}{11}\).
Solution:
\(7\frac{4}{11}\text{ }+\text{ }8\frac{9}{11}=\text{ }?\)
Add the whole numbers in the equation, that is, 7 + 8 = 15
Since both the mixed numbers have like denominators, we can directly add the numerators and simplify.
Then,
\(\frac{4}{11}+\frac{9}{11}=\frac{13}{11}=1\frac{2}{11}\)
Now, the final sum will be \(15+1\frac{2}{11}=16\frac{2}{11}.\)
Therefore, \(7\frac{4}{11}+8\frac{9}{11}=16\frac{2}{11}.\)
Example 2:
What will be the value of \(9\frac{5}{18}\text{ }+\text{ }24\frac{10}{3}\)?
Solution:
\(9\frac{5}{18}\text{ }+\text{ }24\frac{10}{3}=\) ?
First, we will add the whole numbers = \(=9 + 24\)
\(=33\)
Here, the fractions have unlike denominators. Let us convert them into like denominators.
That is,
\(=\frac{5}{18} + \frac{10}{3}\)
\(=\frac{5}{18} + \frac{10 \times 6}{3 \times 6}\) [Multiply 6 with both the numerator and the denominator]
\(=\frac{5}{18} + \frac{60}{18}\)
\(=\frac{65}{18}\)
\(=3\frac{11}{18}\)
Merging both the values, \(33\text{ }+\text{ }3\frac{11}{18}\text{ }=\text{ }36\frac{11}{18}\)
Therefore, \(9\frac{5}{18}\text{ }+\text{ }24\frac{10}{3}\text{ }=\text{ }36\frac{11}{18}\)
Example 3:
Janice bought two strings of fairy lights to decorate her room. One is \(7\frac{5}{16}\text{ }m\) long and the other is \(18\frac{9}{8}\text{ }m\) long. Will both the strings of lights be sufficient to decorate her room, if the required length of the strings is \(25\text{ } m\) for the complete room?
Solution:
It is mentioned that the length of one string is \(7\frac{6}{16}\text{ }m\) and the other one is \(18\frac{9}{8}\text{ }m.\)
Add the length of both the strings to check whether they are enough to cover a room.
Then,
Adding both whole numbers \(=\text{ }1 + 18\)
\(=25\)
Let us convert them into like fractions, as both the fractions have unlike denominators.
That is,
\(\Rightarrow \frac{9}{16} + \frac{9}{8}\)
\(\Rightarrow \frac{9}{16} + \frac{9 \times 2}{8 \times 2}\) [Multiply 2 with both the numerator and the denominator]
\(\Rightarrow \frac{9}{16} + \frac{18}{16}\)
\(\Rightarrow \frac{24}{16}\)
\(\Rightarrow \frac{3}{2}\) [Divide both the numerator and the denominator by 8]
\(\Rightarrow 1.5\) [Convert into decimal]
Now, merge both the values to get the total sum, 25 + 1.5 = 26.5
The length of both the strings of the lights = 26.5 m
The total required length is 25 m. Hence, both the strings are sufficient to decorate Janice’s room.
There are two ways to add two or more mixed numbers. One way is to add the whole number parts and fractional parts individually. The alternative is to convert the mixed numbers into equivalent fractions and then add them.
Proper fractions are those in which a numerator is less than a denominator. For example, 4/9 is a proper fraction.
If a fraction has a numerator that is greater than the denominator, then it is called an improper fraction.
When two or more fractions have the same denominator, such fractions are called like fractions. If the fractions have different denominators, then they are known as unlike fractions.