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16 Sep 2014, 00:23
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A
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Difficulty:
55%
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Question Stats:
70% (02:39) correct
30%
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wrong
based on 721
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A soccer coach takes \(x\) hours to cycle from his house to his office. When riding at 24 km/h, he arrives 5 minutes late, but at 30 km/h, he arrives 4 minutes early. What is the distance between his house and the office?
A. 18 km
B. 24 km
C. 36 km
D. 40 km
E. 72 km
A. 18 km
B. 24 km
C. 36 km
D. 40 km
E. 72 km
16 Sep 2014, 00:23
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Official Solution:
A soccer coach takes \(x\) hours to cycle from his house to his office. When riding at 24 km/h, he arrives 5 minutes late, but at 30 km/h, he arrives 4 minutes early. What is the distance between his house and the office?
A. 18 km
B. 24 km
C. 36 km
D. 40 km
E. 72 km
Denote the distance as \(d\).
Keep in mind that the rates are given in kilometers per hour, while the time is in minutes. Convert the minutes to hours: 5 minutes is \(\frac{1}{12}\) hours, and 4 minutes is \(\frac{1}{15}\) hours.
The equation for the first scenario is \(\frac{d}{24} = x + \frac{1}{12}\). Multiplying by 24, we get: \(d = 24x + 2\).
The equation for the second scenario is \(\frac{d}{30} = x - \frac{1}{15}\). Multiplying by 30, we get: \(d = 30x - 2\).
Subtract the second equation from the first one to obtain: \(6x = 4\).
Solving for \(x\), we find that \(x = \frac{2}{3}\).
Now, substitute this value of \(x\) into either equation to find the distance \(d\). For example, using the first equation: \(d = 24*\frac{2}{3} + 2 = 18\).
Therefore, the distance between the coach's house and his office is 18 kilometers.
Answer: A
A soccer coach takes \(x\) hours to cycle from his house to his office. When riding at 24 km/h, he arrives 5 minutes late, but at 30 km/h, he arrives 4 minutes early. What is the distance between his house and the office?
A. 18 km
B. 24 km
C. 36 km
D. 40 km
E. 72 km
Denote the distance as \(d\).
Keep in mind that the rates are given in kilometers per hour, while the time is in minutes. Convert the minutes to hours: 5 minutes is \(\frac{1}{12}\) hours, and 4 minutes is \(\frac{1}{15}\) hours.
The equation for the first scenario is \(\frac{d}{24} = x + \frac{1}{12}\). Multiplying by 24, we get: \(d = 24x + 2\).
The equation for the second scenario is \(\frac{d}{30} = x - \frac{1}{15}\). Multiplying by 30, we get: \(d = 30x - 2\).
Subtract the second equation from the first one to obtain: \(6x = 4\).
Solving for \(x\), we find that \(x = \frac{2}{3}\).
Now, substitute this value of \(x\) into either equation to find the distance \(d\). For example, using the first equation: \(d = 24*\frac{2}{3} + 2 = 18\).
Therefore, the distance between the coach's house and his office is 18 kilometers.
Answer: A
20 Jul 2015, 07:25
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I solved this problem in a very prevaricate way. I got the right answer but I'm just concerned I'm not solving these the fastest way. It took me almost three minutes and involved some calculation.
The overall approach was to create two equations, set them equal to each other and after finding t, and plugging t back into the original equation to find d.
Let d equal distance to office.
Let t equal scheduled time to office.
\(d=rt\)
\(d=24 km/h*(t + 5)\)
\(d=30 km/h*(t - 4)\)
\(30(t - 4) = 24(t+5)\)
\(30t - 120 = 24t + 120\)
\(6t=240\)
\(t=40\)
Plugging t into the first equation:
\(d=24(40+5) = 1080 km/h\)
But that's per hour. We need to find the distance, so we must divide that by 60.
\(\frac{1080}{60} = 18 km\)
Answer choice A.
Any tips on how I can see different ways to solve this? I'm getting closer to solving these harder problems, but I'm not quite there yet to doing it flawlessly.
The overall approach was to create two equations, set them equal to each other and after finding t, and plugging t back into the original equation to find d.
Let d equal distance to office.
Let t equal scheduled time to office.
\(d=rt\)
\(d=24 km/h*(t + 5)\)
\(d=30 km/h*(t - 4)\)
\(30(t - 4) = 24(t+5)\)
\(30t - 120 = 24t + 120\)
\(6t=240\)
\(t=40\)
Plugging t into the first equation:
\(d=24(40+5) = 1080 km/h\)
But that's per hour. We need to find the distance, so we must divide that by 60.
\(\frac{1080}{60} = 18 km\)
Answer choice A.
Any tips on how I can see different ways to solve this? I'm getting closer to solving these harder problems, but I'm not quite there yet to doing it flawlessly.
