Add or Subtract Fractions With the Same Denominator

Definition

Fractions that have exact same denominators are called like fractions.

Fractions such as 15$\frac{1}{5}$ and 45$\frac{4}{5}$ are like fractions because they have a common denominator 5.

In other words, fractions with like denominators are categorized as like fractions. Performing any mathematical operations on like fractions is comparatively easier as we can make use of the common denominator for fraction operations like addition and subtraction.

Adding like fractions − Formula

If fractions with same denominators are to be added, we need to add the numerators only and keep the same denominator.

• We add the numerators.
• We keep the common denominator.
• Then the Sum of the fractions = (Sum−of−the−Numerators)(Common−Denominator)$\frac{(Sum-of-the-Numerators)}{(Common-Denominator)}$
• Sum of the fractions = ac$\frac{a}{c}$ + bc$\frac{b}{c}$ = (a+b)c$\frac{(a+b)}{c}$, where a, b and c are any three real numbers.

Subtracting like fractions − Formula

If fractions with same denominators are to be subtracted, we need to subtract the numerators only and keep the same denominator.

• We subtract the numerators.
• We keep the common denominator.
• Then the Difference of the fractions = (Difference−of−the−Numerators)(Common−Denominator)$\frac{(Difference-of-the-Numerators)}{(Common-Denominator)}$
• Difference of the fractions = ac$\frac{a}{c}$ − bc$\frac{b}{c}$ = (a−b)c$\frac{(a-b)}{c}$, where a, b and c are any three real numbers

Problem 1:

Add 37$\frac{3}{7}$ + 27$\frac{2}{7}$

Solution

Step 1:

Here, the denominators are the same 7. We add the numerators 3 + 2 = 5 and put the result 5 over the common denominator 7 to get the answer.

37$\frac{3}{7}$ + 27$\frac{2}{7}$ = (3+2)7$\frac{(3+2)}{7}$ = 57$\frac{5}{7}$

Step 2:

So, 37$\frac{3}{7}$ + 27$\frac{2}{7}$ = 57$\frac{5}{7}$

Problem 2:

Subtract 56$\frac{5}{6}$ − 46$\frac{4}{6}$

Solution

Step 1:

Here, the denominators are the same 6. We subtract the numerators; 5 − 4 = 1 and put the result 1 over the common denominator to get the answer.

56$\frac{5}{6}$ − 46$\frac{4}{6}$ = (5−4)6$\frac{(5-4)}{6}$ = 16$\frac{1}{6}$

Step 2:

So, 56$\frac{5}{6}$ − 46$\frac{4}{6}$ = 16

Add or Subtract Fractions With the Same Denominator and Simplification

Adding like fractions and simplification − Formula

If fractions with same denominators are to be added, we add the numerators only and keep the same denominator. If necessary, we simplify the resulting fraction to lowest terms.

• Sum of the fractions = ac$\frac{a}{c}$ + bc$\frac{b}{c}$ = (a+b)c$\frac{(a+b)}{c}$, where a, b and c are any three real numbers.

Subtracting like fractions and simplification − Formula

If fractions with same denominators are to be subtracted, we subtract the numerators only and keep the same denominator. If necessary, we simplify the resulting fraction to lowest terms.

• Difference of the fractions = ac$\frac{a}{c}$ − bc$\frac{b}{c}$ = (a−b)c$\frac{(a-b)}{c}$, where a, b and c are any three real numbers.

Problem 1:

Add 38$\frac{3}{8}$ + 18$\frac{1}{8}$

Solution

Step 1:

Add 38$\frac{3}{8}$ + 18$\frac{1}{8}$

Here, the denominators are the same 8. Since this is an addition operation,

We add the numerators 3 + 1 = 4 and put the result 4 over the common denominator to get the answer.

So 38$\frac{3}{8}$ + 18$\frac{1}{8}$ = (3+1)8$\frac{(3+1)}{8}$ = 48$\frac{4}{8}$

Step 2:

Reducing the fraction to lowest terms

48$\frac{4}{8}$ = 12$\frac{1}{2}$

So, 38$\frac{3}{8}$ + 18$\frac{1}{8}$ = 12$\frac{1}{2}$

Problem 2:

Subtract 56$\frac{5}{6}$ − 16$\frac{1}{6}$

Solution

Step 1:

Subtract 56$\frac{5}{6}$ − 16$\frac{1}{6}$

Here, the denominators are same 6. Since this is a subtraction operation, we subtract the numerators, 5 − 1 = 4 and put the result 4 over the common denominator 6.

So 56$\frac{5}{6}$ − 16$\frac{1}{6}$ = (5−1)6$\frac{(5-1)}{6}$ = 46$\frac{4}{6}$

Step 2:

Simplifying to the lowest terms,

46$\frac{4}{6}$ = 23$\frac{2}{3}$

So, 56$\frac{5}{6}$ − 16$\frac{1}{6}$ = 23

Finding the LCD of Two Fractions

Definition

When we add or subtract fractions, their denominators need to be same or common. If they are different, we need to find the LCD (least common denominator) of the fractions before we add or subtract.

To find the LCD of the fractions, we find the least common multiple (LCM) of their denominators. LCD can be found by two methods. In the first method, LCD of two or more fractions is found as the smallest of all the possible common denominators.In second method, we find the prime factors of the denominators. Then we look for the most occurrence of each of those prime factors and then take their product. This gives the LCD of the fractions.

Formula 1:

Here is how to find out LCD of any two fractions; for example 1/3 and 1/6:

Their denominators are 3 and 6 and the multiples of 3 and 6 are

List the multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...

List the multiples of 6: 6, 12, 18, 24, ...

The common multiples are 6, 12, 18...The least among these common multiples is 6. So, 6 is the Least Common Denominator of 1/3 and 1/6.

Formula 2:

Here is how to find out LCD of any two fractions; for example 1/8 and 7/12:

The denominators of the fractions are 8 and 12

Their prime factorizations are

8 = 2 × 2 × 2

12 = 2 × 2 × 3

The most occurrences of the primes 2 and 3 are 2 × 2 × 2 (in 8) and 3 (in 12).

Their product is 2 × 2 × 2 × 3 = 24

So, 24 is the LCD of these two fractions.

Problem 1:

Find the LCD of 38$\frac{3}{8}$, 512$\frac{5}{12}$

Solution

Step 1:

Since the denominators of the fractions are different, we need to find the LCD of the fractions.

The denominators of the fractions are 8 and 12.

Step 2:

To find their LCD, we find their multiples

8: 8, 16, 24, 32, 40, 48...

12: 12, 24, 36, 48,....

Step 3:

The common multiples of 8 and 12 are 24, 48....

Step 4:

The least of the common multiples is 24. So, 24 is the LCD of these two fractions.

Problem 2:

Find the LCD of 34$\frac{3}{4}$, 79$\frac{7}{9}$

Solution

Step 1:

Since the denominators of the fractions are different, we need to find the LCD of the fractions.

The denominators of the fractions are 4 and 9.

Step 2:

To find their LCD, we find their prime factorization.

4 = 2 × 2

9 = 3 × 3

Step 3:

The most occurrences of the primes 2 and 3 are 2 × 2 (in 4) and 3 × 3 (in 9). Their product is 2 × 2 × 3 × 3 = 36

Step 4:

So 36 is the LCD of these two fractions.

Addition or Subtraction of Unit Fractions

Definition

A unit fraction is a fraction where the numerator is always one and the denominator is a positive integer. Addition or subtraction of unit fractions can be of two types; one, where the denominators are same; two, where the denominators are different.

Rules for Addition of unit fractions

• When the unit fractions have like denominators, we add the numerators and put the result over the common denominator to get the answer.
• When the unit fractions have unlike or different denominators, we first find the LCD of the fractions. Then we rewrite all unit fractions to equivalent fractions using the LCD as the denominator. Now that all denominators are alike, we add the numerators and put the result over the common denominator to get the answer.

Rules for Subtraction of unit fractions

• When the unit fractions have like denominators, we subtract the numerators and put the result over the common denominator to get the answer.
• When the unit fractions have unlike or different denominators, we first find the LCD of the fractions. Then we rewrite all unit fractions to equivalent fractions using the LCD as the denominator. Now that all denominators are alike, we subtract the numerators and put the result over the common denominator to get the answer.

Problem 1:

Add 13$\frac{1}{3}$ + 19$\frac{1}{9}$

Solution

Step 1:

Add 13$\frac{1}{3}$ + 19$\frac{1}{9}$

Here the denominators are different. As 9 is a multiple of 3, the LCD is 9 itself.

Step 2:

Rewriting

13$\frac{1}{3}$ + 19$\frac{1}{9}$ = (1×3)(3×3)$\frac{(1\times 3)}{(3\times 3)}$ + 19$\frac{1}{9}$ = 39$\frac{3}{9}$ + 19$\frac{1}{9}$

Step 3:

As the denominators have become equal

39$\frac{3}{9}$ + 19$\frac{1}{9}$ = (3+1)9$\frac{(3+1)}{9}$ = 49$\frac{4}{9}$

Step 4:

So, 13$\frac{1}{3}$ + 19$\frac{1}{9}$ = 49$\frac{4}{9}$

Problem 2:

Subtract 19$\frac{1}{9}$ − 112$\frac{1}{12}$

Solution

Step 1:

Subtract 19$\frac{1}{9}$ − 112$\frac{1}{12}$

Here the denominators are different. The LCD of the fractions is 36.

Step 2:

Rewriting

19$\frac{1}{9}$ − 112$\frac{1}{12}$ = (1×4)(9×4)$\frac{(1\times 4)}{(9\times 4)}$ − (1×3)(12×3)$\frac{(1\times 3)}{(12\times 3)}$ = 436$\frac{4}{36}$ − 336$\frac{3}{36}$

Step 3:

As the denominators have become equal

436$\frac{4}{36}$ − 336$\frac{3}{36}$ = (4−3)36$\frac{(4-3)}{36}$ = 136$\frac{1}{36}$

Step 4:

So, 19$\frac{1}{9}$ − 112$\frac{1}{12}$ = 136