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Adding and subtracting rational numbers, with their properties
Mathematics Prep 1 Term 1, Unit 1, Lesson 5
GoalAdd and subtract rational numbers with unlike denominators and decimals, using the properties.
New words
common denominator · a number that both denominators divide intoadditive identity · 0, because a + 0 = aadditive inverse · −a is the inverse of a, because a + (−a) = 0associative · grouping does not change a sum
Explanation
To add or subtract fractions with different denominators, first change them to a common denominator: 2/3 + 1/4 = 8/12 + 3/12 = 11/12. Subtracting is adding the opposite: 1/2 − 3/4 = 1/2 + (−3/4) = 2/4 − 3/4 = −1/4. With decimals, line up the digits and use the integer rules: 3.6 − 5.1 = 3.6 + (−5.1) = −1.5. The properties of addition still hold for rational numbers: the sum is always rational (closure), order and grouping do not matter, a + 0 = a, and a + (−a) = 0. Use them to calculate cleverly: −2/5 + 3/7 + 2/5 = 3/7.
★Remember
Common denominator first. Subtracting is adding the opposite. a + (−a) = 0.
Worked examples
| Fraction | Denominator 12 |
|---|---|
| 2/3 | 8/12 |
| 1/4 | 3/12 |
| sum | 11/12 |
Find −3/4 + 1/2. The common denominator is 4: −3/4 + 2/4 = −1/4. Find 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2.
Calculate cleverly: −0.7 + 2.5 + 0.7. Group the opposites: (−0.7 + 0.7) + 2.5 = 2.5. A diver is at −12.5 m and rises 4.5 m: −12.5 + 4.5 = −8 m.
Class activity
Fraction strips. With a grown-up, fold three paper strips into halves, thirds and sixths. Use them to see why 1/2 + 1/3 = 5/6. Then check 1/2 − 1/3 with the strips.
Practice
1
Find 1/2 + 1/3.
3/6 + 2/6 = 5/6.
2
Find −5/6 + 1/3.
−5/6 + 2/6 = −3/6 = −1/2.
3
Find 3.6 − 5.1.
3.6 + (−5.1) = −1.5.
4
Calculate cleverly: 4/9 + (−7/5) + (−4/9).
(4/9 + (−4/9)) + (−7/5) = 0 + (−7/5) = −7/5.
5
Hana says 1/2 + 1/3 = 2/5 (she added the tops and the bottoms). What is wrong?
The denominators must be made the same first. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.