Math Detours · Grade 9 · Connections · 6

The miles-per-gallon illusion

Swapping a 10 mpg car for a 20 mpg one saves far more fuel than swapping 30 for 60. Inverse variation fools almost everyone.

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Tonight's idea

A car's fuel economy is often given in miles per gallon (mpg): how far it goes on one gallon. Which swap saves more fuel: 10 mpg to 20 mpg, or 30 mpg to 60 mpg? Both double the mpg, and the second gains 30 mpg against 10. Most people pick the second, or call it a tie.

But what you pay for is gallons, and

\(\text{gallons used} = \dfrac{\text{miles driven}}{\text{mpg}}\).

For a fixed distance this is inverse variation: gallons × mpg stays constant, so doubling the mpg halves the fuel. Halving a big number saves a lot. Halving a small number saves little.

102030405060miles per gallon 0200400600 gallons used over 6,000 miles 10 → 20 saves 300 30 → 60 saves 100
Equal steps along the mpg axis are not equal steps in fuel. The curve is steep on the left and nearly flat on the right.

The honest measure turns the fraction over: gallons per 100 miles. Many countries quote litres per 100 km, which works the same way. On that scale, equal differences are equal savings.

Watch one: 6,000 miles in each car

At 10 mpg: \(6000 \div 10 = 600\) gallons. At 20 mpg: 300 gallons. The swap saves 300.

At 30 mpg: 200 gallons. At 60 mpg: 100 gallons. The swap saves 100.

The first swap saves three times as much fuel. In gallons per 100 miles the four cars use 10, 5, 3.33 and 1.67. The drops are 5 and 1.67, and the answer is in plain view.

Your turn

  1. Four cars each drive 12,000 miles. They get 15, 20, 30 and 40 mpg. (a) How many gallons does each use? (b) Which saves more fuel: going from 15 to 20 mpg, or from 30 to 40 mpg?
  2. (a) Convert to gallons per 100 miles: 25 mpg, 40 mpg, 50 mpg. (b) A truck uses 8 gallons per 100 miles. What is its mpg? (c) Which is the bigger saving, 25 → 40 mpg or 40 → 50 mpg?
  3. An engineer improves a 20 mpg car to 25 mpg. A second car gets 40 mpg. What must its mpg rise to, to save the same fuel over the same distance? Work in gallons per 100 miles.
  4. A family drives a 12 mpg truck and a 36 mpg car 9,000 miles a year each. Plan A swaps the truck for an 18 mpg one. Plan B swaps the car for a 72 mpg one. (a) How many gallons a year does each plan save? (b) Could any replacement for the car save more than Plan A? Explain.

Stretch: the average that is not 40

You drive 60 miles in town at 20 mpg, then 60 miles on the open road at 60 mpg. What is your mpg for the whole 120 miles? It is not 40. Then find how many open-road miles would bring the overall figure up to 40.

Answers

  1. (a) 800, 600, 400 and 300 gallons. (b) 15 → 20 saves 200 gallons. 30 → 40 saves 100. The smaller-looking step saves twice as much.
  2. (a) \(100/25 = 4\), \(100/40 = 2.5\), \(100/50 = 2\). (b) \(100/8 = 12.5\) mpg. (c) 25 → 40 saves 1.5 gallons per 100 miles; 40 → 50 saves 0.5. The first is three times the saving.
  3. 20 mpg is 5 gallons per 100 miles and 25 mpg is 4, a saving of 1. The second car uses 2.5, so it must come down to 1.5: \(100/1.5 \approx 66.7\) mpg. A gain of 5 mpg on the thirsty car matches a gain of nearly 27 mpg on the efficient one.
  4. (a) Truck: \(9000/12 = 750\) gallons, falling to \(9000/18 = 500\). Plan A saves 250. Car: \(9000/36 = 250\) gallons, falling to 125. Plan B saves 125. (b) No. The car burns only 250 gallons a year in total, so even a car that used no fuel at all would save 250, which only ties Plan A.

Stretch. Town: \(60/20 = 3\) gallons. Road: \(60/60 = 1\) gallon. Overall \(120 \div 4 = 30\) mpg. The slow-mpg miles use most of the fuel, so they pull the figure down. For 40 mpg with \(d\) road miles: \(60 + d = 40\left(3 + \dfrac{d}{60}\right)\), so \(60 + d = 120 + \tfrac23 d\) and \(d = 180\) miles. Check: 240 miles on \(3 + 3 = 6\) gallons is 40 mpg.

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