Adding and Subtracting Rational Numbers

Rational numbers include integers, fractions, and decimals. When adding and subtracting rational numbers, you combine the rules you already know for fractions and decimals with the sign rules for integers.

The Rules of Signs

Whether you are working with fractions or decimals, the rules for positive and negative signs remain exactly the same as they are for integers:

• Same Signs: Add their absolute values and keep the original sign.
• Different Signs: Subtract the smaller absolute value from the larger one. Keep the sign of the number with the larger absolute value.
• Subtracting: Change the subtraction problem into an addition problem by adding the opposite: $a - b = a + (-b)$.

Working with Fractions

To add or subtract fractions, they must have a common denominator. Once the denominators are the same, apply the sign rules to the numerators.

Example 1: Evaluate $-\frac{3}{4} + \frac{1}{2}$

1. Find a common denominator, which is $4$. Convert $\frac{1}{2}$ to $\frac{2}{4}$.
2. Rewrite the problem: $-\frac{3}{4} + \frac{2}{4}$
3. Combine the numerators using integer rules (different signs, so subtract): $-3 + 2 = -1$.
4. The answer is $-\frac{1}{4}$.

Example 2: Evaluate $\frac{3}{5} + \left(-\frac{7}{10}\right)$

1. Find a common denominator, which is $10$. Convert $\frac{3}{5}$ to $\frac{6}{10}$.
2. Rewrite the problem: $\frac{6}{10} + \left(-\frac{7}{10}\right)$
3. Combine the numerators: $6 + (-7) = -1$.
4. The answer is $-\frac{1}{10}$.

Working with Decimals

For decimals, apply the sign rules first to determine what operation to perform, then align the decimal points to calculate.

Example 3: Evaluate $-2.5 - 1.3$

1. Change subtraction to adding the opposite: $-2.5 + (-1.3)$.
2. Because both numbers have the same sign (negative), you add their absolute values: $2.5 + 1.3 = 3.8$.
3. Keep the negative sign.
4. The answer is $-3.8$.