Addition and Subtraction of Mixed Numbers with like Denominator (Definition, Types and Examples) - BYJUS

# Addition and Subtraction of Mixed Numbers with like Denominator

Mixed numbers are numbers that have a whole number part and a fractional part. We can perform mathematical operations on mixed fractions just like we do with whole numbers and fractions. Learn the steps involved in the addition and subtraction of mixed numbers that have the same denominator....Read MoreRead Less

## Mixed Number

A mixed number is a number that comprises a whole number and a proper fraction. Numbers that are more than a whole number but lesser than the next whole number are expressed in the form of mixed numbers, also called mixed fractions

Improper fractions can be used to create mixed numbers. Improper fractions represent those numbers that can’t be divided evenly.

Let’s understand this with an example:

You own a collection of baseball trading cards of your favorite teams. There are 18 cards in total. One card is a fraction of the entire set and it can be represented fractionally as ‘$$\frac{1}{18}$$’. The total number of cards in the collection is represented as ‘$$\frac{18}{18}$$’.

What if a friend gifts you a few extra trading cards? You count the fresh cards. You now have two sets, but the second set has only 5 cards. This means that we can represent the total number of cards (including set 1 and set 2) as a mixed number.

So, how many sets of baseball cards do you have?

Original set = $$\frac{18}{18}$$ cards which is one whole set

The new set of cards added = $$\frac{5}{18}$$

So, one can say that you have 1 and  $$\frac{5}{18}$$ (one and five eighteenth) sets of baseball trading cards.

Like fractions and unlike fractions are two different types of fractions. Like fractions are those fractions which have the same denominator (lower digit).

For example, 13, 23, 53, 73 are like fractions.

‘Unlike’ fractions have different denominators.

For example, 17, 29, 54, 72 are unlike fractions.

It’s easy to find the sum and difference of like fractions. Also if two or more fractions or mixed numbers have the same denominator it becomes very easy to compare their values.

## Addition and Subtraction of Mixed Numbers

There are two ways to add or subtract mixed numbers with the same denominator. One method is to solve the fractional and whole part of the mixed numbers separately and then add the two to find the final sum or difference.

The second method involves writing each mixed fraction as a fraction and then adding or subtracting them.

Method 1:

Let us take the help of an example to understand this.

Suppose, you bought two pies from place A and two pies from place B. Each of these pies were divided into four equal pieces. You ate three slices from the second pie bought from place A. You are left with a total of five slices or in other words one whole pie and $$\frac{1}{4}^{th}$$ of a whole pie.

From place B you bought two pies and ate two slices from the second pie. You decided to bring the rest back home. You are left with six slices of pie or in other words, one pie along with $$\frac{2}{4}^{th}$$ of the other pie.

Now, to find the total number of pies, we need to add $$1\frac{1}{4}$$ and $$1\frac{2}{4}$$

$$1\frac{1}{4}+1\frac{2}{4}$$

$$(1+1)=\left(1\frac{1}{4}+1\frac{2}{4}\right)$$      [Separate the whole and fractional parts]

$$2\left(\frac{3}{4}\right)$$                               [Add]

After returning home, you remember that a total of $$2\left(\frac{3}{4}\right)$$ slices of pie were taken in total from both places. However, you only remember that you took $$1\left(\frac{1}{4}\right)$$ slices of pie from place-A. Now, you wish to find out how many slices were taken home from place-B. For this, you need to subtract $$2\left(\frac{3}{4}\right)$$ from $$2\left(\frac{3}{4}\right)$$.

Method 2:

The second way to go about addition and subtraction of mixed numbers is to express them as fractions and then solve as shown below by performing the correct operations. The way to simplify mixed fractions is shown below.

Let us take the same example as taken in method 1.

$$1\frac{1}{4}+1\frac{2}{4}$$

$$\frac{5}{4}+\frac{6}{4}=\frac{11}{4}$$

Subtraction:

$$2\frac{3}{4}-1\frac{2}{4}$$

$$\frac{11}{4}-\frac{6}{4}=\frac{5}{4}$$

## Solved Examples

Example 1:

Simplify $$7\frac{7}{8}-5\frac{3}{8}$$

Solution:

Method 1:

$$7\frac{7}{8}-5\frac{3}{8}$$

= (7 – 5) +$$\left(\frac{7}{8}-\frac{3}{8}\right)$$                          [Separate the whole and fractional parts]

= (2) + $$\left(\frac{4}{8}\right)=2\frac{4}{8}=2\frac{1}{2}$$                   [Add]

Method 2:

$$7\frac{7}{8}-5\frac{3}{8}$$

= $$\frac{63}{8}-\frac{43}{8}$$                                                  [Simplify]

= $$\frac{20}{8}$$                                                         [Subtract]

= $$\frac{5}{2}$$                                                             [Simplify]

Hence the difference is $$2\frac{1}{2}$$ or $$\frac{5}{2}$$.

Example 2:

Simplify $$8\frac{9}{7}+2\frac{2}{7}$$

Solution:

Method 1:

$$8\frac{9}{7}+2\frac{2}{7}$$

= (8+2) $$+\frac{9}{7}+\frac{2}{7}$$                                            [Separate the whole and fractional parts]

= (10) $$+\left(\frac{9+2}{7}\right)=10\frac{11}{7}$$                     [Add]

Method 2:

$$8\frac{9}{7}+2\frac{2}{7}$$

= $$\frac{56+9}{7}+\frac{14+2}{7}$$                                          [Simplify]

= $$\frac{56+9+14+2}{7}$$                                  [Add]

= $$\frac{81}{7}$$

Hence the sum is $$10\frac{11}{7}$$ or $$\frac{81}{7}$$.

Example 3:

Jeremy added $$3\frac{1}{2}$$ cups of flour to bake a cake. He later realised that he needed more cake as a few more friends were joining him. He added $$1\frac{1}{2}$$ more cups of flour. How many cups of flour did Jeremy use in total?

Solution:

The total amount of flour that has been used is the sum of the two amounts. That is;

$$3\frac{1}{2}+1\frac{1}{2}$$

= (3 + 1) $$+\left(\frac{1}{2}+\frac{1}{2}\right)$$                              [Separate the whole and fractional parts]

= 4 $$+\frac{2}{2}$$                                                                      [Add]

= 4 $$+\frac{1}{1}$$                                                                       [Simplify]

= 4 + 1                                                                                                       [Simplify]

Hence, Jeremy uses 5 cups of flour in total.

Example 4:

There was a total of $$4\frac{7}{2}$$ litres of soft drink in a jug. Three friends divided the drink among themselves. One of them drank $$1\frac{3}{4}$$ litres and the other drank $$1\frac{1}{4}$$ litres of the soft drink. Calculate how much soft drink is left for the third friend.

To find the amount of soft drink that remains, calculate the sum of the quantity of soft drink the first two friends drank and deduct the same from the total.

The total amount of soft drink the first two friends drank = $$1\frac{3}{4}l+1\frac{3}{4}l$$

(1 + 1) + $$\left(\frac{3}{4}+\frac{3}{4}\right)$$                                          [Separate the whole and fractional parts]

$$2+\frac{3+3}{4}=2\frac{6}{4}=2\frac{3}{2}$$                               [Add and Simplify]

The amount of soft drink left for the third friend is as follows.

$$4\frac{7}{2}-2\frac{3}{2}$$

= (4 – 2) $$+\left(\frac{7}{2}-\frac{3}{2}\right)$$                            [Separate the whole and fractional parts]

= 2 + $$\left(\frac{4}{2}\right)$$                                                   [Subtract]

= 2 + 2                                                                                                     [Simplify]