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

Addition or Subtraction of Fractions With Different Denominators

Definition

When the denominators of any fractions are unequal or are different those fractions are called unlike fractions.

Operations like addition and subtraction cannot be done directly on unlike fractions.

These unlike fractions are first converted into like fractions by finding the least common denominator of these fractions and rewriting the fractions into equivalent fractions with same denominators (LCD)

Adding unlike fractions − Formula

When fractions with different or unlike fractions are to be added, first the least common denominator of the fractions is found. The equivalent fractions of given fractions are found with LCD as the common denominator. The numerators are now added and the result is put over the LCD to get the sum of fractions.

• We find the least common denominator of all the fractions.
• We rewrite the fractions to have the denominators equal to the LCD obtained in first step .
• We add the numerators of all the fractions keeping the denominator value equal to the LCD obtained in first step.
• We then express the fraction in lowest terms.

Subtracting unlike fractions − Formula

When fractions with different or unlike fractions are to be subtracted, first the least common denominator of the fractions is found. The equivalent fractions of given fractions are found with LCD as the common denominator. The numerators are now subtracted and the result is put over the LCD to get the difference of the given fractions.

• We find the least common denominator of all the fractions.
• We rewrite the fractions to have the denominators equal to the LCD obtained in step 1.
• We subtract the numerators of all the fractions keeping the denominator value equal to the LCD obtained in step 1.
• We express the fraction in lowest terms.

Problem 1:

Add 15$\frac{1}{5}$ + 27$\frac{2}{7}$

Solution

Step 1:

Add 15$\frac{1}{5}$ + 27$\frac{2}{7}$

Here the denominators are different. As 5 and 7 are prime the LCD is their product 35.

Step 2:

Rewriting

15$\frac{1}{5}$ + 27$\frac{2}{7}$ = (1×7)(5×7)$\frac{(1\times 7)}{(5\times 7)}$ + (2×5)(7×5)$\frac{(2\times 5)}{(7\times 5)}$ = 735$\frac{7}{35}$ + 1035$\frac{10}{35}$

STEP 3:

As the denominators have become equal

735$\frac{7}{35}$ + 1035$\frac{10}{35}$ = (7+10)35$\frac{(7+10)}{35}$ = 1735$\frac{17}{35}$

STEP 4:

So, 15$\frac{1}{5}$ + 27$\frac{2}{7}$ = 1735$\frac{17}{35}$

Problem 2:

Subtract 215$\frac{2}{15}$ − 110$\frac{1}{10}$

Solution

Step 1:

Subtract 215$\frac{2}{15}$ − 110$\frac{1}{10}$

Here the denominators are different. The LCM of 10 and 15 is 30.

Step 2:

Rewriting

215$\frac{2}{15}$ − 110$\frac{1}{10}$ = (2×2)(15×2)$\frac{(2\times 2)}{(15\times 2)}$ − (1×3)(10×3)$\frac{(1\times 3)}{(10\times 3)}$ = 430$\frac{4}{30}$ − 330$\frac{3}{30}$

STEP 3:

As the denominators have become equal

430$\frac{4}{30}$ − 330$\frac{3}{30}$ = (4−3)30$\frac{(4-3)}{30}$ = 130$\frac{1}{30}$

STEP 4:

So, 215$\frac{2}{15}$ − 110$\frac{1}{10}$ = 130