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Two candles of same height are lighted at the same time. The first is consumed in 3 hours and second in 2 hours. Assuming that each candles burns at a constant rate, in how many hours after being lighted, the ratio between the first and second candles becomes 2:1?
- a)2 hour
- b)2.5 hour
- c)4 hour
- d)4.5 hour
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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Problem: Two candles of the same height are lighted at the same time. The first is consumed in 3 hours and the second in 2 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted, the ratio between the first and second candles becomes 2:1?
Solution:
Let the height of each candle be h units and let the rate of burning of the first candle be x units per hour and that of the second candle be y units per hour. We have to find the time t such that the ratio of the heights of the candles is 2:1.
Step 1: Finding the rate of burning of each candle
The first candle is consumed in 3 hours, so the total amount of wax in the candle is consumed in 3x hours. Therefore, we have
h = 3x ............(1)
Similarly, the second candle is consumed in 2 hours, so the total amount of wax in the candle is consumed in 2y hours. Therefore, we have
h = 2y ............(2)
Step 2: Finding the time t such that the ratio of the heights of the candles is 2:1
Let the ratio of the heights of the candles after t hours be 2:1. Then, we have
h - xt = 2/3 h ............(3)
h - yt = 1/3 h ............(4)
Substituting the values of h from equations (1) and (2) into equations (3) and (4), we get
3x - xt = 2(3x)/3
2x = xt
t = 2 hours
and
2y = yt
t = 2 hours
Therefore, the ratio of the heights of the candles becomes 2:1 after 2 hours. Hence, the correct answer is option (d).
Solution:
Let the height of each candle be h units and let the rate of burning of the first candle be x units per hour and that of the second candle be y units per hour. We have to find the time t such that the ratio of the heights of the candles is 2:1.
Step 1: Finding the rate of burning of each candle
The first candle is consumed in 3 hours, so the total amount of wax in the candle is consumed in 3x hours. Therefore, we have
h = 3x ............(1)
Similarly, the second candle is consumed in 2 hours, so the total amount of wax in the candle is consumed in 2y hours. Therefore, we have
h = 2y ............(2)
Step 2: Finding the time t such that the ratio of the heights of the candles is 2:1
Let the ratio of the heights of the candles after t hours be 2:1. Then, we have
h - xt = 2/3 h ............(3)
h - yt = 1/3 h ............(4)
Substituting the values of h from equations (1) and (2) into equations (3) and (4), we get
3x - xt = 2(3x)/3
2x = xt
t = 2 hours
and
2y = yt
t = 2 hours
Therefore, the ratio of the heights of the candles becomes 2:1 after 2 hours. Hence, the correct answer is option (d).
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