Fluent Operations with Rational Numbers
Students build fluency with addition, subtraction, multiplication, and division of positive and negative rational numbers and apply the operations in context.
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Adding Positive and Negative Rational Numbers
Rational numbers include integers, fractions, and terminating or repeating decimals. When adding numbers with the same sign, add their absolute values and keep the common sign. For example, −4.5 + (−2.3) = −6.8. When the signs differ, subtract the smaller absolute value from the larger and use the sign of the number with the greater absolute value. For example, 7.2 + (−9.5) = −2.3 because 9.5 − 7.2 = 2.3 and −9.5 has the greater absolute value. A number line helps explain this operation. Starting at 7.2 and adding −9.5 means moving 9.5 units left, ending at −2.3. Always estimate first so the sign and size of the answer make sense.
Subtracting Rational Numbers
Subtraction can be rewritten as addition of the opposite. The expression a − b is equivalent to a + (−b). This rule works with integers, fractions, and decimals. For example, −6 − 4 becomes −6 + (−4), which equals −10. In 3.5 − (−2.1), subtracting a negative becomes adding a positive, so 3.5 + 2.1 = 5.6. For fractions, first rewrite the subtraction and then find a common denominator if necessary. For example, −1/2 − 3/4 becomes −1/2 + (−3/4). Rewriting −1/2 as −2/4 gives −2/4 + (−3/4) = −5/4, or −1 1/4. Parentheses are useful because they make each number’s sign clear.
Multiplying and Dividing Rational Numbers
The sign rules for multiplication and division are the same. Two numbers with the same sign produce a positive result, while two numbers with different signs produce a negative result. For example, (−8)(−3) = 24 because both factors are negative. In contrast, 42 ÷ (−6) = −7 because the signs differ. Apply the sign rule first, and then calculate using absolute values. Fractions may be multiplied by multiplying numerators and denominators. For example, (−3/5)(10/9) has a negative result. Simplifying before multiplying gives (−1/1)(2/3) = −2/3. Division by a fraction requires multiplying by its reciprocal. Thus, −3/4 ÷ 1/2 becomes −3/4 × 2/1 = −3/2.
Choosing Operations in Real-World Problems
Rational-number operations model changes in money, temperature, elevation, and other quantities. Identify what each sign means before calculating. Suppose a bank account has a balance of $18.50 and three charges of $7.25 each occur. The change is 3(−7.25) = −21.75, so the new balance is 18.50 + (−21.75) = −$3.25. The negative balance indicates that the account is overdrawn. In another example, a temperature falls 2.5 degrees per hour for four hours. The total change is 4(−2.5) = −10 degrees. If the starting temperature was 6 degrees, the final temperature is 6 + (−10) = −4 degrees. Labeling units and interpreting the sign are as important as performing the calculation correctly.