A car tyre has 2 punctures.
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A car tyre has 2 punctures. The first puncture alone can make the tyre flat in 12 minutes and the second puncture alone can make the tyre flat in 15 minutes. If air leaks out at a constant rate from both punctures, how long will it take to make the tire flat?
Show answer & explanation
This is a classic combined work-rate problem. The first puncture deflates the tyre at a rate of 1/12 per minute, and the second at 1/15 per minute. Working together, their combined rate is 1/12 + 1/15 = 9/60 per minute. Therefore, time = 1 ÷ (9/60) = 60/9 ≈ 6.67 minutes.
Step-by-step Derivation:
Step 1: Define work rates as fractions of the tyre flattened per minute.
- Puncture 1 rate: 1/12 tyre/min
- Puncture 2 rate: 1/15 tyre/min
Step 2: Find the combined rate by adding the individual rates.
- Combined rate = 1/12 + 1/15
- Find common denominator (LCM of 12 and 15 is 60):
- 1/12 = 5/60
- 1/15 = 4/60
- Combined rate = 5/60 + 4/60 = 9/60 tyre/min
Step 3: Calculate time to flatten one complete tyre.
- Time = Work ÷ Rate = 1 ÷ (9/60) = 60/9 = 6.666... minutes
- Rounded: 6.7 minutes
Verification: In 6.67 min, puncture 1 deflates 6.67/12 ≈ 0.556 of tyre, puncture 2 deflates 6.67/15 ≈ 0.444 of tyre. Total ≈ 1.0 ✓