Problem 12 · 2018 AMC 8
Medium
Ratios, Rates & Proportions
proportionratio
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?
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Answer: B — 6:00.
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Hint 1 of 2
The first 30-minute shopping trip is a free calibration: it tells you that whenever the car clock ticks 35, only 30 real minutes have passed. The clock runs fast, so the true time is always less than what the car shows.
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Hint 2 of 2
The technique is a conversion factor: real time = car time × (30/35). Build the fraction so the fast clock's bigger number is on the bottom, guaranteeing you shrink it.
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Approach: convert car-time to actual via the rate
- Calibrate from the shopping trip: 30 real minutes per 35 car minutes, so real = (30/35) × car = (6/7) × car.
- The car shows 7:00, i.e. 420 minutes past noon. Real elapsed = 420 × 6/7 = 360 minutes = 6 hours.
- Six real hours after noon is 6:00. Sanity check: the clock gains, so real time must be earlier than 7:00 — 6:00 is, while traps like 8:10 are not.
- You'll see it again: any "broken clock running at a steady wrong rate" is a unit-conversion problem — find the real-per-fake ratio once, then scale.
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