Rates, ratios & proportions · Ages 13–14
Rates
A rate is one quantity ÷ another, with its units.
How to do it
Method
A rate is one quantity ÷ another, with its units
36 ÷ 4 = 9 ($/kg), 9 × 6 = 54
Another way
The multiplier
6 ÷ 4 = 1.5, 36 × 1.5 = 54
In full
A rate compares two different kinds of quantity, so it has units: a speed (km/h), a unit price ($/kg), a typing speed (words/minute). Comparing two amounts of the same kind is a ratio, which has none. To find a rate, divide one quantity by the other: $36 for 4 kg of rice is 36 ÷ 4 = $9kg. One way on from there, the unit rate times the amount: $9 a kg, so 6 kg cost 9 × 6 = $54. Another, the multiplier: 6 kg is 1.5 times 4 kg, so the price is 36 × 1.5 = $54. Changing units changes both of them: 72 km/h = 72 × 1000 m ÷ 3600 s = 20 m/s. To find the better buy, compare the price of one unit.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
4 ballpoint pens cost $68: $ each, and 6 of them cost $.
Fill in each blank
Show the answer and working
Answer: 17, 102
The unit rate first: 68 ÷ 4 = 17; then 17 × 6 = 102.
Example 2
54 km/h = m/s
Show the answer and working
Answer: 15
1 km = 1000 m and 1 hour = 3600 seconds: 54 × 1000 ÷ 3600 = 15. Both units change.
Example 3
How many m/s is 126 km/h?
- A. 126000 m/s
- B. 453.6 m/s
- C. 35 m/s
- D. 2100 m/s
Choose the right answer
Show the answer and working
Answer: C. 35 m/s
126 × 1000 ÷ 3600 = 35: 1000 ÷ 3600 is 1 ÷ 3.6.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A printer prints 39 pages in 3 minutes. What is its printing rate?
- A. 117 pages a minute
- B. 36 pages a minute
- C. 13 pages a minute
- D. 42 pages a minute
Choose the right answer
Answer:
2.A car travels 100 km in 2 hours: its average speed is 50 km/h.
True or false?
Answer:
3.The same cereal: a small box of 500 g costs $75, a big box of 1000 g costs $170. Which is the better buy?
- A. The big box
- B. The small box
- C. They are as good as each other
- D. They can't be compared
Choose the right answer
Answer:
4.90 km/h = m/s
Answer:
5.How many m/s is 90 km/h?
- A. 90000 m/s
- B. 324 m/s
- C. 1500 m/s
- D. 25 m/s
Choose the right answer
Answer:
All the answers
- 1. C. 13 pages a minute
- 2. True
- 3. B. The small box
- 4. 25
- 5. D. 25 m/s
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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