## SUBTRACTING FRACTIONS

• Same denominators
• Different denominators
• Mixed numbers

Type 1: Subtracting fractions with same denominators

For subtracting fractions having same denominators, subtract numerators and just write the common denominator in the final difference.

Example:

Subtract 2/5 from 3/5 Or Find 3/5 – 2/5

Solution:

Since denominator 5, is same in both 2/5 and 3/5, therefore:

In the final answer, i.e the difference of the two fractions, write 5 in denominator and 3 – 2 = 1 in numerator.

So, 3/5 – 2/5 = (3 – 2)/5

Note:

To subtract 3/5 from 2/5, i.e. to find 2/5 – 3/5

So, 2/5 – 3/5 = (2 – 3)/5 = -1/5

Type 2: Subtracting fractions having different denominators

For subtracting fractions having different denominators, follow the below steps

Step 1: Find LCM of the various denominators in the fractions.

Step 2: Write equivalent fractions of the original fractions with LCM in denominator. (to make denominators same)

Step 3: Add the equivalent fractions.

Example:

Subtract 3/4 from 5/6

Solution:

Step 1:

First, what is the LCM of the denominators 4 and 6?

4 = 2 × 2, and 6 = 2 × 3

So, LCM of 4 and 6 = 2 × 2 × 3 = 12

Step 2:

Now, write equivalent fractions for 3/4 and 5/6 having LCM 12 as the common denominator.

Equivalent fraction for 3/4

Now, 4 × ? = 12, it is

12 ÷ 4 = 3

So, multiply 3 to 1 and 4 in ¼ to get 12 in denominator.

3/4 = (3 × 3)/ (3 × 4) = 9/12

Equivalent fraction for 5/6

6 × ? = 12, it’s

12 ÷ 6 = 2

So, multiply 2 to both 5 and 6 in 5/6 to get 12 in denominator

5/6 = (5 × 2)/ (6 × 2) = 10/12

Step 3:

Finally,

To find the difference: 10/12 – 9/12

Subtract the equivalent fraction 9/12 from 10/12

10/12 – 9/12 = 1/12

Therefore,

(5/6) – (3/4) = 1/12

Short-cut for subtracting fractions having different denominators

To find (a/b) – (c/d)

First, find (a × d – b × c)/(b × d)

Next, reduce the fraction so obtained into simplest term.

Example:

Find (5/6) – (3/4)

• First, (5/6) – (3/4)

= (5 × 4 – 3 × 6)/6 × 4

= (20 – 18)/24

= 2/24

2. Next, reduce 2/24, by cancelling the common factor 2, as below:

2/24 = (2 × 1)/ (2 × 12) = 1/12

Type 3: Subtracting fractions which are mixed numbers

Subtract mixed numbers in two different methods. They are:

1st method:

• First, subtract the whole numbers and the fractions separately.
• Next, add the differences of the whole numbers and the fractions obtained in the first step

2nd method:

• First, convert the mixed fractions into improper fractions
• Next, subtract the two improper fractions.

Example:

Find the difference: 32/3 – 23/4

Below, let us apply both of the above methods for subtracting fractions.

Method 1:

Step 1:

Subtract the whole numbers: 3 – 2 = 1, and

Subtract the fractions: 2/3 – 3/4

Now, lcm of 3 and 4 is 3 × 4 = 12

{If two numbers do not have any common factor, then their LCM is their product}

2/3 = (2 × 4)/ (3 × 4) = 8/12, and

3/4 = (3 × 3)/ (4 × 3) = 9/12

So, 2/3 – 3/4 = (8/12) – (9/12) = -1/12

Step 2:

Next, add the fractions’ difference and the whole numbers’ difference

1 + (-1/12) = 1 – 1/12 = (12/12) – (1/12) = 11/12

Method 2 is

Subtracting fractions by converting them into improper fractions.

The mixed fraction 32/3 = (3 × 3 + 2)/3 = 11/3, and

The mixed fraction 23/4 = (4 × 2 + 3)/4 = 11/4.

Step 2:

Next, subtract the above two improper fractions, for subtracting fractions

32/3 and 23/4

i.e. 32/3 – 23/4 = (11/3) – (11/4)

= (11× 4)/(3 × 4) – (11 × 3)/(4 × 3)

{converting into equivalent fractions}

= (44/12) – (33/12) = 11/12

32/3 – 23/4

Or, using the above short-cut for subtracting fractions, we have

(11/3) – (11/4)

= (11× 4 – 11× 3)/ ( 3 × 4) =

= (44 – 33)/12 = 11/12

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