Rates, Speed & Conversions
Speed-distance-time, currency, per-capita and work-rate problems.
This page is a sequence, not a document: try a question cold, learn the methods, drill until they’re automatic, then prove it under a clock. Skipping straight to reading is the least effective thing you can do here — that’s not opinion, it’s the testing effect.
Try one first
Before any teaching — a real question tells you honestly where you stand.
£1 = €1.25. How many euros for £550?
The methods
Every rates, speed & conversions pattern the tests use — the method in a few lines, then a worked example.
Per-capita rate
Per-person rate = total amount ÷ population. Watch the magnitude (thousands vs millions).
Worked example
A city council's annual budget is £18 million and the city's population is 240,000. What is the council's spend per person?
Answer: £75.00
Spend per person = budget ÷ population = 18,000,000 ÷ 240,000 = £75. The £7.50 and £750 foils slip the magnitude by a factor of ten; £13.33 inverts the rate (population ÷ budget); £60 uses the wrong population.
Currency conversion
Multiply by the exchange rate when converting from the base currency (£1 = €r → pounds × r). Divide to convert back.
Worked example
The exchange rate is £1 = €1.20. How many euros do you receive for £450?
Answer: €540
Multiply by the rate: 450 × 1.20 = €540. The €375 foil divides (inverts the rate); €90 gives only the 20% uplift; €450 forgets to convert.
Rate per unit
Rate = quantity ÷ the units named (per hour, per worker…). Check which divisor the question asks for.
Worked example
A factory produced 8,400 units in a 35-hour week. What was the production rate per hour?
Answer: 240
Rate per hour = units ÷ hours = 8,400 ÷ 35 = 240. The 210 foil uses 40 hours; 1,680 divides by days; 280 uses 30 hours; 24 slips the magnitude.
Chained currency
Convert leg by leg in order, multiplying or dividing by each rate as you cross it.
Worked example
You change £300 into US dollars at £1 = $1.25, then change it all back at $1 = £0.78. How much do you end up with?
Answer: £292.50
First leg: 300 × 1.25 = $375. Second leg: 375 × 0.78 = £292.50. The £375 foil stops at dollars; £300 assumes a round trip returns the original; £480.77 divides on the way back; £234 skips the first conversion.
Currency with commission
Apply the commission first (× (1 − rate)), then convert at the quoted rate.
Worked example
A bureau charges 2% commission, deducted from your pounds before conversion. You exchange £500 to euros at £1 = €1.18. How many euros do you receive?
Answer: €578.20
Deduct 2%: 500 × 0.98 = £490. Convert: 490 × 1.18 = €578.20. The €590 foil forgets the commission; €601.80 adds 2% instead of deducting; €566.40 deducts twice; €490 stops at the pound figure.
Time duration
Subtract times column by column, borrowing 60 minutes from the hours when needed.
Worked example
A train departs at 09:48 and arrives at 14:15. How long is the journey?
Answer: 4h 27min
From 09:48 to 14:15: the minutes need a borrow — 75 − 48 = 27 min, and 13 − 9 = 4 hours, so 4h 27min. The 4h 33min foil subtracts the minutes the wrong way (48−15); 5h 27min over-borrows an hour.
Speed, distance, time
Average speed = total distance ÷ total time. Never average the speeds themselves.
Worked example
A car travels 60 miles at 40 mph, then a further 60 miles at 60 mph. What is its average speed for the whole journey?
Answer: 48 mph
Average speed = total distance ÷ total time. Times: 60/40 = 1.5h and 60/60 = 1h, total 2.5h. 120 ÷ 2.5 = 48 mph. The 50 mph foil averages the two speeds — wrong, because more time is spent at the slower speed (the harmonic-mean trap).
Work rate
Add the rates (jobs per hour), not the times: combined time = 1 ÷ (1/t₁ + 1/t₂).
Worked example
Worker A can paint a fence in 6 hours and Worker B can paint it in 3 hours. Working together, how long do they take?
Answer: 2 hours
Add the rates: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 fence per hour, so it takes 2 hours. The 4.5 foil averages the times; 9 adds them; 3 just takes the faster worker.
Currency-adjusted comparison
Convert both amounts into the same currency first, then compare.
Worked example
Branch A earned $480,000 and Branch B earned €450,000. At $1 = £0.80 and €1 = £0.86, how much more (in £) did the higher-earning branch make?
Answer: £3,000
Convert both to pounds: A = 480,000×0.80 = £384,000; B = 450,000×0.86 = £387,000. B is higher by £3,000. The £30,000 foil compares the raw amounts before converting.
Drill it
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Prove it under a clock
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