A train travels at the rate of 10 miles/hr for the first hou

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.

I think the easiest way to solve this problem would be simply to count the number of miles it travels per hour (and in total)


Hour miles/hour Total miles
1 10 10
2 20 30
3 30 60
4 40 100
5 50 150
6 60 210
7 70 280
8 80 360
9 90 450

It takes a total of nine hours to cover the 450 mile distance.

ANSWER: C. 9

Also remember that the sum of the first 5 integers is 15 and the sum of the first 10 integers is 55. Therefore, he will need only 9 hours to get to 450.

total miles/hours (h)=5(h+1)
450/h=5(h+1)
h^2+H-90=0
h=9 hours

The total rate of the trip must be a multiple of 10, since the different rates are increasing constantly by 2.
Total Time= Total Distance/Total Rate. Plugging in the total time using the answer choices and resolving for the rate, we get only one time value that generates a rate multiple of 10 and that is T=9. (450/9 =50)

in first hour it travels 10 miles
in second hour it travels 20 miles
like wise adding up
10 +20 +30.........10n = 450
n/2(10 +10n) = 450
or n = 9
therefore it will cover the distance in 9 hrs

Let us assume the train completes the journey after n hours. Then,

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

n(n + 1)/2 = 45

n(n + 1) = 90

n = 9 or n = -10

Since n cannot be negative, then n = 9.

Answer: C

First check the answer choices (ALWAYS check the answer choice before deciding on an approach to a question)

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

Hour - Distance
1st - 10
2nd - 20
3rd - 30
4th - 40
5th - 50
6th - 60
7th - 70
8th - 80
9th - 90

9 hrs = 450 miles

(C)

Formula:

1) Speed = \(\frac{Distance}{Time}\)

Distance = Speed*Time

2) Sum of first n natural numbers = \(\frac{n(n+1)}{2}\)

Let the hours taken by train to complete a 450-mile journey be "n"

Distance covered by train in the 1st hour = 10*1 = 10 miles

Distance covered by train in the 2nd hour = 20*1 = 20 miles

Distance covered by train in the nth hour = 10n*1 = 10n miles

Total distance covered by train in n hours = \(10 + 20 + 30 + ...... 10n = 450\)

\(10 ( 1 + 2 + ..... n) = 450\)

10 \(\frac{(n)(n+1)}{2}\) = 450

\(n(n+1) = 90\)

\(n^2 + n - 90 = 0\)

\(n^2 + 10n - 9n - 90 = 0\)

\(n(n+10) - 9(n+10) = 0\)

\((n-9)(n+10) = 0\)

\(n = 9 OR -10\)

Since n is the number of hours, it cannot be negative. Therefore n = 9 hours

The correct answer is C

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