6th Grade - Adding and Subtracting Rational Numbers

Introduction

• Rational Number =$\frac{\mathrm{numerator}}{\mathrm{denominator}}=\frac{x}{y}, where y \ne 0$
• A rational number is the ratio of two integers, meaning both the numerator and the denominator should be integers.
• Zero is a rational number.

Addition and Subtraction of Rational Numbers

•  To add/subtract rational numbers with the same (like) denominators:

1. Follow the rules of adding and subtracting integers.

Click here to learn the rules of addition and subtraction of integers.

2. Add/subtract the numerator and keep the common denominator unchanged.

• To add or subtract rational numbers with unlike (different) denominators:

1. Find the Least Common Denominator (LCD).  

To find LCD:

• • • List the multiples of the denominators until the least common multiple(LCM) is found. The LCM is the LCD.

OR

• • • Multiply the denominators of both fractions; the product will be the LCD.

2. Create an equivalent ratio by multiplying the denominator and the numerator of a rational number with the desired number that can yield the same denominators.  

3. Add/Subtract the numerators (follow the rules of adding and subtracting integers), keeping the same common denominator.

E.g.  $\frac{1}{2} + \frac{2}{3}$

Multiples of 2: 2, 4, 6 (Stop at this number as it is the least (smallest) common multiple.)

Multiples of 3: 3, 6  (Stop at this number because it is the least (smallest) common multiple.)

OR multiply the denominators (3 x 2) = 6.

Since the least common denominator is 6,  the denominators should be multiplied by a number that can yield 6. The numerator should also be multiplied with the same number as the denominator to create an equivalent fraction.

Cheat sheet

Follow the rules of addition and subtraction of integers while adding or subtracting the rational numbers.

Solved Examples

Example 1: Add the given rational numbers: $\frac{7}{2} + \left(-\frac{2}{3}\right)$

Solution: Follow the rule of adding the integers with different signs.

$\frac{7}{2} - \frac{2}{3}$

Find the LCD by multiplying the denominators. Create equivalent fractions by multiplying both the numerator and the denominator by the same number.

$= \left(\frac{7 \times 3}{2 \times 3}\right) - \left(\frac{2 \times 2}{3\times 2}\right)$

$= \frac{21}{6} - \frac{4}{6}$

$= \frac{17}{6}$

 

Example 2:  Subtract the given rational numbers: $\frac{-3}{7} - \frac{-5}{4}$

Solution: Follow the rule of subtracting integers.

$= -\frac{3}{7} + \frac{5}{4}$

Find the LCD by multiplying the denominators. Create equivalent fractions by multiplying both the numerator and the denominator by the same number.

$= \frac{-3 \times 4}{7 \times 4} + \frac{5 \times 7}{4 \times 7}$

$= \frac{-12}{28} + \frac{35}{28}$

$= \frac{23}{28}$