Rates, ratios & proportions · Ages 13–14
Average speed and mixtures
Total over total, not the average of the rates.
How to do it
Method
Total over total, not the average of the rates
(40 × 2 + 60 × 3) ÷ 5 = 52
Another way
Weight by the shares
40 × 25 + 60 × 35 = 16 + 36 = 52
In full
Average speed = total distance ÷ total time, not the two speeds added and halved. 120 km at 60 km/h takes 2 hours, and back the same way at 40 km/h takes 3, so the average speed is 240 ÷ 5 = 48 km/h, not 50 km/h: the slower half lasts longer. A mixture's average price works alike: total price ÷ total weight. 2 kg of tea at $40 a kg mixed with 3 kg at $60 a kg: one way, (40 × 2 + 60 × 3) ÷ 5 = 260 ÷ 5 = $52kg. Another, weight each price by its share: the weights are 2 : 3, so 40 × 25 + 60 × 35 = 16 + 36 = 52. The other way round, in what ratio do they mix to make $52kg? 40 is 12 below 52 and 60 is 8 above, so the weights are 8 : 12 = 2 : 3: the closer a price is to the average, the more of it there is.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A car drives 2 hours at 30 km/h, then 2 hours at 60 km/h. Its average speed for the whole trip = km/h
Show the answer and working
Answer: 45
Total distance 30 × 2 + 60 × 2 = 180 km over 4 hours: 180 ÷ 4 = 45 km/h.
Example 2
A car goes from P to Q at 50 km/h and comes back the same way at 75 km/h. What is its average speed for the whole trip?
- A. 125 km/h
- B. 62.5 km/h
- C. 25 km/h
- D. 60 km/h
Choose the right answer
Show the answer and working
Answer: D. 60 km/h
Say each way is d km: the time is d50 + d75, so the average speed = 2d ÷ (d50 + d75) = 2 × 50 × 75 ÷ (50 + 75) = 60 km/h.
Example 3
There are 5 kg of sweets at $24 a kg. How many kg of sweets at $38 a kg must be added to make the mixture $31 a kg?
Show the answer and working
Answer: 5 kg
Add x kg: 24 × 5 + 38x = 31(5 + x), so x = 5 × (31 − 24) ÷ (38 − 31) = 5.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A car goes from P to Q at 75 km/h and comes back the same way at 15 km/h. What is its average speed for the whole trip?
- A. 25 km/h
- B. 60 km/h
- C. 90 km/h
- D. 45 km/h
Choose the right answer
Answer:
2.5 kg of rice at $17 a kg is mixed with 3 kg at $57 a kg. The total price is $, an average of $ a kg.
Fill in each blank
Answer:
3.Tea at $46 a kg is mixed with tea at $88 a kg to make a blend at $64 a kg. The weights (the $46 tea : the $88 tea) are :
Fill in each blank
Answer:
4.There are 2 kg of sweets at $44 a kg. How many kg of sweets at $74 a kg must be added to make the mixture $62 a kg?
Answer:
5.Chi-keung cycles 12 km in 2 hours, rests 45 minutes, then cycles 18 km in 1 hours. Counting the rest, what is the average speed for the whole trip?
- A. 12 km/h
- B. 8 km/h
- C. 3.75 km/h
- D. 10 km/h
Choose the right answer
Answer:
All the answers
- 1. A. 25 km/h
- 2. 256, 32
- 3. 4, 3
- 4. 3 kg
- 5. B. 8 km/h
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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