A number times a bracket
At each bus stop, 5 people enter the bus and 3 leave. How many more people are on it after 4 stops?
- One stop: 5 + (−3). Five positive tokens, three negative.
- Four stops: 4 × (5 + (−3)).
- Cut the rows in two: 4 × 5 = 20 and 4 × (−3) = −12.
- Add the two blocks: 20 + (−12) = 8.
- Or add the bracket first: 5 + (−3) = 2. Then 4 × 2 = 8.
After 4 stops, 8 more people are on the bus. The bracket is the part in ( ): here 5 + (−3).
Rule: Multiply the number by each part of the bracket, then add. The answer matches adding the bracket first. It still holds when a number is below zero. Distributive property — the number multiplies each part of the bracket.
| Expression | Bracket first | Each part first |
|---|---|---|
| 4 × (5 + (−3)) | 4 × 2 = 8 | 20 + (−12) = 8 |
| (−3) × (5 + (−2)) | (−3) × 3 = −9 | −15 + 6 = −9 |
| (−2) × (7 + (−9)) | (−2) × (−2) = 4 | −14 + 18 = 4 |
| (−1) × (6 + (−9)) | (−1) × (−3) = 3 | −6 + 9 = 3 |
The number outside the bracket is the multiplier. A multiplier of −1 flips the sign of every part. The last row shows it: 6 becomes −6, and −9 becomes 9.
Watch out!
The multiplier must reach every part. (−3) × 4 + (−5) is not (−3) × (4 + (−5)). The first gives −17. The second gives 3.
Pick the line that equals the bracket. The check shows both values.
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