Learning Objectives
Choose suitable representations for signed numbers, fractions, and percentages.
Solve a short multi-step problem while keeping units and the percentage base clear.
Check whether a numerical answer is reasonable before accepting it.
Prerequisites: the three number-foundation lessons in this section. Keep a written line for each important operation. Clear working makes it easier to spot a sign error or a misplaced denominator.
1. Choose a Helpful Representation
Situation | Useful representation |
|---|---|
Compare negative values | Positions on a number line |
Add fractions | Equivalent fractions with a common denominator |
Find a percentage of an amount | A decimal multiplier |
Compare differently sized packages | A unit rate |
Compare a part with the whole | A fraction using the total as denominator |
Different representations can describe the same value. For example, 3/4, 0.75, and 75% are equal. Choose the version that makes the calculation clear rather than converting automatically in every problem.
2. A Multi-Step Example
Worked Example: Mixing a Drink
A recipe uses concentrate and water in the ratio 1:4. You need 2.5 litres of the mixture. Find the concentrate and water amounts.
There are 1 + 4 = 5 total parts. Each part represents 2.5 ÷ 5 = 0.5 litre.
Concentrate uses one part, 0.5 litre. Water uses four parts, 2 litres.
Check the total: 0.5 + 2 = 2.5 litres. Check the ratio: 0.5:2 simplifies to 1:4. Concentrate makes up 1/5 = 20% of the drink.
3. Estimate and Reverse the Operation
A rough estimate can catch an unreasonable answer. If you calculate 10% of 48, the result should be close to 5, not 50. To check a division, multiply the quotient by the divisor. To check a subtraction, add the subtracted amount back.
Common Mistakes
A result can be arithmetically correct but answer the wrong question. Label whether it is a total, a fraction, a rate, or a change.
When two steps use percentages, identify the base separately for each step.
Do not remove a negative sign just because an answer looks unfamiliar. Check it on a number line or by reversing the operation.
4. Mixed Practice
1. Evaluate −4 + 3(2 − 5).
2. Calculate 7/8 − 1/4.
3. You have 1 1/2 metres of ribbon. Each piece is 3/8 metre long. How many complete pieces can you cut, assuming no cutting loss?
4. In a collection, the ratio of circles to squares is 3:5. There are 32 shapes in total. How many are circles?
5. A $40 item has a 20% discount. Find its sale price.
6. A value rises from 100 to 120, then falls by 20% of its new value. Find the final value and explain why it is not 100.
7. Which has the lower unit price: 4 notebooks for $6.80 or 6 notebooks for $9.90?
Worked Solutions
1. 2 − 5 = −3, so −4 + 3(−3) = −4 − 9 = −13.
2. Rewrite 1/4 as 2/8. The difference is 5/8.
3. Convert 1 1/2 to 3/2. Then (3/2) ÷ (3/8) = (3/2)(8/3) = 4 pieces. Four lengths of 3/8 total 12/8 = 1 1/2 metres.
4. There are 8 total parts, so each part is 32/8 = 4 shapes. Circles occupy 3 parts, giving 12 circles.
5. The discount is 0.20 × 40 = $8. The sale price is $32.
6. The decrease is 0.20 × 120 = 24, so the final value is 96. The increase used 100 as its base, while the decrease used 120.
7. The unit prices are $1.70 and $1.65 per notebook. The six-pack has the lower price per notebook, although its total price is higher.
5. Summary
- Choose a representation that fits the calculation.
- Keep signs, units, and the whole or base amount visible.
- Use an estimate or the reverse operation to check your answer.
- Explain what the result means in the original question.