Problem 15 · 2023 AMC 8
Hard
Ratios, Rates & Proportions
distance-speed-timeproportion
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Answer: B — 4.2 mph.
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Hint 1 of 2
The deadline doesn't move — he still has the same 5 minutes left. What changed is only the distance: the detour swaps 1 block for 3, adding 2 extra blocks.
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Hint 2 of 2
So in the leftover 5 minutes he must now cover 7 blocks instead of 5. Turn both into mph-friendly units: each block is 0.05 mile, 5 minutes is 1/12 hour.
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Approach: convert blocks to miles, time to hours
- The anchor is that his arrival time is unchanged — so his time budget for the rest of the walk is fixed, and only his distance grew. That's what forces a faster speed.
- Usual walk: 10 blocks = 0.5 mile in 10 min, so each block is 0.05 mile and 5 blocks normally take 5 minutes. After the detour, the detour swaps 1 block for 3, so the remaining trip is 5 + 2 = 7 blocks = 0.35 mile.
- He still has only 5 minutes = 1/12 hour. Speed = 0.35 ÷ (1/12) = 0.35 × 12 = 4.2 mph. This transfers: when a deadline is fixed and only the distance changes, hold time constant and let speed = new distance ÷ that fixed time.
Another way — scale up his old speed by the extra distance:
- His normal pace covers those last 5 blocks in exactly the 5 minutes he has. The detour makes him cover 7 blocks in that same time — 7/5 as far.
- So he must go 7/5 times his usual 3 mph: 3 × 7/5 = 21/5 = 4.2 mph.
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