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16 Sep 2014, 00:35
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A cyclist traveled for two days. On the second day, she rode 4 hours longer and at an average (arithmetic mean) speed 10 miles per hour slower than on the first day. If over these two days, she covered a total of 280 miles in 12 hours, what was her average speed on the second day?
A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph
A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph
16 Sep 2014, 00:35
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Official Solution:
A cyclist traveled for two days. On the second day, she rode 4 hours longer and at an average (arithmetic mean) speed 10 miles per hour slower than on the first day. If over these two days, she covered a total of 280 miles in 12 hours, what was her average speed on the second day?
A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph
Approach 1 - Algebra:
Given that on the second day the cyclist traveled for 4 hours more than on the first day and the total travel time over two days was 12 hours, we can set up the equation \(t + (t + 4) = 12\). Solving for \(t\), we get \(t = 4\). This means she traveled for 4 hours on the first day and 8 hours on the second day.
If we let the rate on the second day be \(r\) miles per hour, then the equation for the total distance covered on both days is: \(4(r + 10) + 8r = 280\). Solving for \(r\), we find \(r = 20\) miles per hour.
Approach 2 - Logic:
The average rate of the cyclist is \(\frac{\text{total distance}}{\text{total time}}=\frac{280}{12}=23\frac{1}{3}\). Now, since the weighted average of 2 individual averages (\(r\) and \(r+10\)) must lie between these individual averages, we have \(r \lt 23\frac{1}{3} \lt r+10\). Only answer choice C fits, as the rate from the correct answer choice must be less than \(23\frac{1}{3}\), and that rate plus 10 must be more than \(23\frac{1}{3}\).
Answer: C
A cyclist traveled for two days. On the second day, she rode 4 hours longer and at an average (arithmetic mean) speed 10 miles per hour slower than on the first day. If over these two days, she covered a total of 280 miles in 12 hours, what was her average speed on the second day?
A. 5 mph
B. 10 mph
C. 20 mph
D. 30 mph
E. 40 mph
Approach 1 - Algebra:
Given that on the second day the cyclist traveled for 4 hours more than on the first day and the total travel time over two days was 12 hours, we can set up the equation \(t + (t + 4) = 12\). Solving for \(t\), we get \(t = 4\). This means she traveled for 4 hours on the first day and 8 hours on the second day.
If we let the rate on the second day be \(r\) miles per hour, then the equation for the total distance covered on both days is: \(4(r + 10) + 8r = 280\). Solving for \(r\), we find \(r = 20\) miles per hour.
Approach 2 - Logic:
The average rate of the cyclist is \(\frac{\text{total distance}}{\text{total time}}=\frac{280}{12}=23\frac{1}{3}\). Now, since the weighted average of 2 individual averages (\(r\) and \(r+10\)) must lie between these individual averages, we have \(r \lt 23\frac{1}{3} \lt r+10\). Only answer choice C fits, as the rate from the correct answer choice must be less than \(23\frac{1}{3}\), and that rate plus 10 must be more than \(23\frac{1}{3}\).
Answer: C
24 Nov 2019, 15:08
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Bunuel
Another approach: You can also solve this one with another sort of logic and number sense. Focus first on the time. Namely, 12 hours across 2 days can be broken down readily enough to 6 hours per day on average. Since we know there was a 4-hour difference between days one and two, we can take the 4 and divide it by 2 and skew our 6-hour split by going 2 hours in either direction:
6 - 2 hours = 4 hours (Day 1)
6 + 2 hours = 8 hours (Day 2)
--------------------------
12 hours altogether over two days, with the correct split
If the question asks about the second day, pick a number in the middle, say, (B), 10 mph:
10 mph * 8 hours = 80 miles (Day 2)
20 mph * 4 hours = 80 miles (Day 1)
-------------------------------
12 hours altogether over two days, but only 160 total miles traveled--TOO LOW. Out with (A) and (B), onto (D). Why (D)? Because regardless of whether it ends up being the answer, we will know definitively whether it is too high or too low.
30 mph * 8 hours = 240 miles (Day 2)
40 mph * 4 hours = 160 miles (Day 1)
--------------------------------
12 hours altogether over two days, but with 400 miles traveled--TOO HIGH. Thus, (D) and (E) are out, and (C), 20 mph, must be the answer.
This mental math is simple enough and takes about a minute to work through, maybe a minute and a half in a Day 2-type situation.
Cheers,
Andrew
