A train travels at the rate of 10 miles/hr for the first hour of a trip, at 20 miles/hr for the second hour, at 30 miles/hr for the third hour and so on. How many hours will it take the train to complete a 450-mile journey? Assume that the train makes no intermediate stops.
A. 8
B. 8.5
C. 9
D. 9.5
E. 10
The OA is C.
Is there a fast way (a formula or something) that can help to solve this PS question? I would appreciate any help.
A train travels at the rate of 10 miles/hr for the first
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First check the answer choices (ALWAYS check the answer choice before deciding on an approach to a question)M7MBA wrote:A train travels at the rate of 10 miles/hr for the first hour of a trip, at 20 miles/hr for the second hour, at 30 miles/hr for the third hour and so on. How many hours will it take the train to complete a 450-mile journey? Assume that the train makes no intermediate stops.
A. 8
B. 8.5
C. 9
D. 9.5
E. 10
We see that all answer choices are between 8 hours and 10 hours.
So, let's test the answer choices, starting with 8 hours, and then go from there (if need be).
A) 8 hours
Distance = (rate)(time)
So, the train travels 10 miles in the 1st hour, 20 miles in the 2nd hour, 30 miles in the 3rd hour, . . . . . 80 miles in the 8th hour
So, TOTAL distance traveled = 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 = 360 miles
We want the train to travel 450 miles, so we need to keep going
ASIDE: We know that the train travels 90 miles in the 9th hour.
So, AFTER 9 hours the distance the train has traveled = 360 miles + 90 miles = 450 VOILA!!
Answer: C
Cheers,
Brent
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Let us assume the train completes the journey in n hours. Then,M7MBA wrote:A train travels at the rate of 10 miles/hr for the first hour of a trip, at 20 miles/hr for the second hour, at 30 miles/hr for the third hour and so on. How many hours will it take the train to complete a 450-mile journey? Assume that the train makes no intermediate stops.
A. 8
B. 8.5
C. 9
D. 9.5
E. 10
10 + 20 + 30 + ... + 10n = 450
Let's factor 10 from the left hand side:
10(1 + 2 + 3 + ... + n) = 450
1 + 2 + 3 + ... + n = 45
The formula for the sum of n consecutive integers is n(n + 1)/2, so we have:
n(n + 1)/2 = 45
n(n + 1) = 90
n^2 + n - 90 = 0
(n - 9)(n + 10) = 0
n = 9 or n = -10
Since n cannot be negative, then n = 9.
Answer: C
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