Mathematics Year 7: Adding and subtracting rational numbers
Mathematics, Year 7 (Portugal): Rational numbers add and subtract with the rules of integers, as fractions or decimals; subtracting is adding the opposite, and the properties of addition make mental calculation shorter.
Same rules, new kinds of numbers
You already add and subtract integers with signs. With rational numbers the rules do not change, only the numbers do: they can be fractions or decimals. Subtracting a number is the same as adding its opposite: a − b = a + (−b). If the fractions have different denominators, first write them with a common denominator and then add the numerators. For example −1/2 + 1/3 = −3/6 + 2/6 = −1/6, and 0.7 − 1.2 = 0.7 + (−1.2) = −0.5.
Properties that let you regroup
Addition of rational numbers is commutative (a + b = b + a) and associative ((a + b) + c = a + (b + c)); adding 0 changes nothing, and a number plus its opposite is 0. So you can regroup the terms that are easy to add: equal denominators, decimals that make a whole, or opposites. Subtraction is neither commutative nor associative: 1/2 − 3/4 = −1/4 but 3/4 − 1/2 = 1/4. So first turn subtractions into additions of opposites, then regroup freely.
One algorithm, one shortcut
A: −5/6 + 3/4. Step 1: the lowest common multiple of 6 and 4 is 12. Step 2: −5/6 = −10/12 and 3/4 = 9/12. Step 3: −10/12 + 9/12 = −1/12. B: 4.8 − 7.3 + 1.2 in your head. Write 4.8 + (−7.3) + 1.2 and swap the order: (4.8 + 1.2) + (−7.3) = 6 − 7.3 = −1.3. Check: 4.8 − 7.3 = −2.5 and −2.5 + 1.2 = −1.3.
Minus a negative number
Calculate 3/4 − (−1/2). Many people write 3/4 − 1/2 = 1/4 and lose the second minus sign. But subtracting −1/2 is adding its opposite, +1/2: 3/4 − (−1/2) = 3/4 + 1/2 = 3/4 + 2/4 = 5/4. The result grows, just as taking away a debt makes you richer. Quick check: a minus in front of a negative number in brackets becomes a plus.
Balances, depths, temperatures
An account with a balance of −18.40 € receives 25 € and pays 9.75 €: −18.40 + 25 − 9.75 = −3.15 €, so it is still overdrawn. A diver at −7.5 m who rises 2.25 m is at −5.25 m. In every case you add the changes to the starting value, and the sign of the result says on which side of zero you end. Spreadsheets and calculators do exactly this when they add a column of positive and negative decimals.
Test what you learned
Subtracting a number is the same as adding its opposite. Which addition is equal to 2/3 − 5/6?
- a) 2/3 + 5/6
- b) −2/3 + 5/6
- c) −2/3 − 5/6
- d) 2/3 + (−5/6)
Correct answer: d) 2/3 + (−5/6) — Yes: only the number being subtracted is replaced by its opposite. 2/3 + (−5/6) = 4/6 − 5/6 = −1/6.
At dawn a thermometer shows −3.5 °C. In the morning it rises 6.25 °C and in the afternoon it falls 4.5 °C. What does it show at the end?
- a) −5.25 °C
- b) 7.25 °C
- c) −1.75 °C
- d) 1.75 °C
Correct answer: c) −1.75 °C — Yes: −3.5 + 6.25 = 2.75, and 2.75 − 4.5 = −1.75. It ends below zero.
What is 1/3 − (−1/6)?
- a) 1/6
- b) 1/2
- c) −1/2
- d) 2/9
Correct answer: b) 1/2 — Yes: 1/3 − (−1/6) = 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2.
Keep exploring
Other languages
Loading MyLeoNes™…