Addition and Subtraction of Fractions

So far in our study we have learnt about natural numbers, whole numbers and then integers. In the present chapter, we are learning about fractions, a different type of numbers.

Whenever we come across new type of numbers, we want to know how to operate with them.

Can we combine and add them? If so, how? Can we take away some number from another i.e., can we subtract one from the other and so on. Which of the properties learnt earlier about the numbers hold now?

Which are the new properties? We also see how these help us deal with our daily life situations.

1. My mother divided an apple into 4 pieces and gave me two pieces while my brother got one piece. How much of the total apple did my mother give away to us?

2. Mother asked Neelu and her brother to pick stones from the wheat. Neelu picked one fourth of the total stones in it and her brother also picked up one fourth of the stones. What fraction of the stones did both pick up together?

3. Sohan was putting covers on his note books.He put one fourth of the covers on Monday. He put another one fourth on Tuesday and the remaining on Wednesday. What fraction of the covers did he put on Wednesday?

Look at the following examples: A tea stall owner consumes in her shop 2 x 12 litres of milk in the morning and 1 x 12 litres of milk in the evening in preparing tea. What is the total amount of milk she uses in the stall?

Or Shekhar ate 2 chapatis for lunch and 1 x 12 chapatis for dinner. What is the total number of chapatis he ate?

Clearly, both the situations require the fractions to be added. Some of these additions can be done orally and the sum can be found quite easily.

All fractions cannot be added orally. We need to know how they can be added in different situations and learn the procedure for it. We begin by looking at addition of like fractions.

Take a 7 × 4 grid sheet as shown (the dark tealed grid). The sheet has seven boxes in each row and four boxes in each column.

How many boxes are there in total?

Colour five of its boxes in green.

What fraction of the whole is the green region? 528523

Now colour another four of its boxes in yellow.

What fraction of the whole is this yellow region? 428424

What fraction of the whole is coloured altogether? 928919

since, we can see: 528 + 428 = 928919

In Fig we have 2 quarter parts of the figure shaded. This means we have 2 parts out of 4 shaded or 12 of the figure shaded.

That is, 14 + 14 = 1+14 = 24 = 1214

Similarly, 19 + 19 + 19 = 1+1+19 = 39 = 1319

Sharmila had 56 of a cake. She gave 26 out of that to her younger brother. How much cake is left with her?

Thus, we can say that the difference of two like fractions can be obtained as follows:

Step 1 : Subtract the smaller numerator from the bigger numerator.

Step 2 : Retain the (common) denominator.

Step 3 : Write the fraction as : Result of Step 1Result of Step 2

We have learnt to add and subtract like fractions. It is also not very difficult to add fractions that do not have the same denominator. When we have to add or subtract fractions we first find equivalent fractions with the same denominator and then proceed.

What added to 15 gives 12? This means subtract 15 from 12 to get the required number.

Since 15 and 12 are unlikelike fractions, in order to subtract them, we first find their equivalent fractions with the same denominator.

These are 210 and 510 respectively.

This is because 12 = 1×52×5 = 510105 and 12 = 1×25×2 = 2101012

Therefore, 12 - 15 = 510 - 210 = 5−210 = 310103

Note that: is the least common multiple (LCM) of 2 and 5.

Subtract 34 from 56.

Instruction

Solution :

Add 25 and 13.

Solution :

The LCM of 5 and 3 is

Therefore, 25 + 13 = 2×35×33×34×2 + 1×53×51×34×2 = + = 11151511

Simplify 35 - 720.

Solution :

The LCM of 5 and 20 is .

Therefore, 35 - 720 = 3×45×44×45×4 - 720

= - 720

= (Simplified form)

1. Add 25 and 37.

The LCM of 5 and 7 is

Therefore, 25 + 37 = 2×75×74×35×2 + 3×57×52×53×5 = + =

2. Subtract 25 and 57.

We need to find equivalent fractions of 25 and 57, which have the same denominatore.

This denominator is given by the LCM of 5 and 7 that is

Therefore, 25 - 57 = 2×75×75×56×5 - 5×57×55×57×7 = - =

Mixed fractions can be written either as a whole part plus a proper fraction or entirely as an improper fraction.

One way to add (or subtract) mixed fractions is to do the operation seperately for the whole parts and the other way is to write the mixed fractions as improperproper fractions and then directly add (or subtract) them.

Add 2 45 and 356.

2 45 + 3 56 = + 45 + 56

Since LCM of 5 and 6 = ,

45 + 56 = 4×65×66×65×5 + 5×56×56×55×6

= + =

= 30+193030+1530 = + 19301530

Thus, 5 + 45 + 56 = 5 + 1 + 1930 = + = 6 19305 1930

Therefore, 2 45 + 3 56 = 6 1930

Find 4 25 - 2 15.

Instruction

Simplify: 814 - 256.

Instruction