Addition of Rational Number with Different Denominator

We will learn the addition of rational number with different denominator. To find the sum of two rational numbers which do not have the same denominator, we follow the following steps:

Step I: Let us obtain the rational numbers and see whether their denominators are positive or not. If the denominator of one (or both) of the numerators is negative, re-arrange it so that the denominators become positive.

Step II: Obtain the denominators of the rational numbers in step I.

Step III: Find the lowest common multiple of the denominators of the two given rational numbers.

Step IV: Express both the rational numbers in step I so that the lowest common multiple of the denominators becomes their common denominator.

Step V: Write a rational number whose numerator is equal to the sum of the numerators of rational numbers obtained in step IV and denominators is the lowest common multiple obtained in step III.

Step VI: The rational number obtained in step V is the required sum (simplify if required).

Following examples will illustrate the above procedure.

1. Add $\frac{4}{7}$ and 5

Solution:

We have, 4 = $\frac{4}{1}$

Clearly, denominators of the two rational numbers are positive. We now re-write them so that they have a common denominator equal to the LCM of the denominators.

In this case the denominators are 7 and 1.

The LCM of 7 and 1 is 7.

We have, 5 = $\frac{5}{1}$ = $\frac{5 × 7}{1 × 7}$ = $\frac{35}{7}$

Therefore, $\frac{4}{7}$ + 5

= $\frac{4}{7}$ + $\frac{5}{1}$

= $\frac{4}{7}$ + $\frac{35}{7}$

= $\frac{4 + 35}{7}$

= $\frac{39}{7}$

2. Find the sum: $\frac{-5}{6}$ + $\frac{4}{9}$

Solution:

The denominators of the given rational numbers are 6 and 9 respectively.

LCM of 6 and 9 = (3 × 2 × 3) = 18.

Now, $\frac{-5}{6}$ = $\frac{(-5) × 3}{6 × 3}$ = $\frac{-15}{18}$

and $\frac{4}{9}$ = $\frac{4 × 2}{9 × 2}$ = $\frac{8}{18}$

Therefore, $\frac{-5}{6}$ + $\frac{4}{9}$

            = $\frac{-15}{18}$ + $\frac{8}{18}$

            = $\frac{-15 + 8}{18}$

            = $\frac{-7}{18}$

3. Simplify: $\frac{7}{-12}$ + $\frac{5}{-4}$

Solution:

First we write each of the given numbers with positive denominator.

$\frac{7}{-12}$ = $\frac{7 × (-1)}{(-12) × (-1)}$ = $\frac{-7}{12}$, [Multiplying the numerator and denominator by -1]

⇒ $\frac{7}{-12}$ = $\frac{-7}{12}$

$\frac{5}{-4}$ = $\frac{5 × (-1)}{(-4) × (-1)}$ = $\frac{-5}{4}$, [Multiplying the numerator and denominator by -1]

⇒ $\frac{5}{-4}$ = $\frac{-5}{4}$

Therefore, $\frac{7}{-12}$ + $\frac{5}{-4}$ = $\frac{-7}{12}$ + $\frac{-5}{4}$

Now, we find the LCM of 12 and 4.

The LCM of 12 and 4 = 12

Rewriting $\frac{-5}{4}$ in the form in which it has denominator 12, we get

$\frac{-5}{4}$ = $\frac{(-5) × 3}{4 × 3}$ = $\frac{-15}{12}$

Therefore, $\frac{7}{-12}$ + $\frac{5}{-4}$

= $\frac{-7}{12}$ + $\frac{-5}{4}$

= $\frac{-7}{12}$ + $\frac{-15}{12}$

= ($\frac{(-7) + (-15)}{12}$

= $\frac{-22}{12}$

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

Thus, $\frac{7}{-12}$ + $\frac{5}{-4}$ = $\frac{-11}{6}$

4. Simplify: 5/-22 + 13/33

Solution:

First we write each one of the given rational numbers with positive denominator.

Clearly, denominator of 13/33 is positive.

The denominator of 5/-22 is negative.

The rational number 5/-22 with positive denominator is -5/22.

Therefore, 5/-22 + 13/33 = -5/22 + 13/33

The LCM of 22 and 33 is 66.

Rewriting -5/22 and 13/33 in forms having the same denominator 66, we get

-5/22 = (-5) × 3/22 × 3, [Multiplying the numerator and denominator by 3]

⇒ -5/22 = -15/66

13/33 = 13 × 2/33 × 2, [Multiplying the numerator and denominator by 2]

⇒ 13/33 = 26/66

Therefore, 5/-22 + 13/33

= 22/-5 + 13/33

= -15/66 + 26/66

= -15 + 26/66

= 11/66

= 1/6

Therefore, 5/-22 + 13/33 = 1/6

If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers such that b and d do not have a common factor other than 1, i.e., HCF of b and d is 1, then

$\frac{a}{b}$ + $\frac{c}{d}$ = $\frac{a × d + c × b}{b × d}$

For example, $\frac{5}{18}$ + $\frac{3}{13}$ = $\frac{5 × 13 + 3 × 18}{18 × 13}$ = $\frac{65 + 54}{234}$ = $\frac{119}{234}$

And $\frac{-2}{11}$ + $\frac{3}{14}$ = $\frac{(-2) × 14 + 3 × 11}{11 × 14}$ = $\frac{-28 + 33}{154}$ = $\frac{5}{154}$

● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number