Question 10: Ratio of Boat Speed to Water Current Speed A boat going upstream takes 8 hours...
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Question 10: Ratio of Boat Speed to Water Current Speed
A boat going upstream takes 8 hours 24 minutes to cover a certain distance, while it takes 5 hours to cover 5/7 of the same distance running downstream. What is the ratio of the speed of the boat to the speed of the water current?
Show answer & explanation
Let boat speed = b, current speed = c, and distance = d. Upstream: d/(b-c) = 8.4 hours. Downstream: (5d/7)/(b+c) = 5 hours. Solving these two equations yields b/c = 11/5. The ratio of boat speed to current speed is 11:5.
Step-by-step Derivation:
Step 1: Convert 8 hours 24 minutes to decimal: 8 + 24/60 = 8.4 hours.
Step 2: Set up equations using distance = speed × time.
- Upstream: d = (b - c) × 8.4, so b - c = d/8.4
- Downstream: 5d/7 = (b + c) × 5, so b + c = 5d/(7×5) = d/7
Step 3: From upstream equation: b - c = d/8.4 = 10d/84 = 5d/42
Step 4: From downstream equation: b + c = d/7 = 6d/42
Step 5: Add the two equations:
(b - c) + (b + c) = 5d/42 + 6d/42
2b = 11d/42
b = 11d/84
Step 6: Subtract to find c:
(b + c) - (b - c) = 6d/42 - 5d/42
2c = d/42
c = d/84
Step 7: Calculate ratio:
b/c = (11d/84) / (d/84) = 11/5
Therefore, b:c = 11:5