Hi,

I have a few (2) math questions from the GMAT Prep Exam, mainly figuring out faster/efficient ways to solve them:

1. A boat traveled upstream a distance of 90 miles at an average speed of (v-3) miles per hour and then traveled the same distance downstream at an average speed of (v+3) miles per hour. If the trip upstream took half an hour longer than the trip downstream, how many hours did it take the boat to travel downstream?

A: 2.5
B: 2.4
C: 2.3
D: 2.2
E: 2.1

Q: I have seen an explanation to the problem that would take more than 2 minutes to complete. My question is what is an alternative way to figure out the answer within a reasonable amount of time or a method that could immediately narrow down the answer choices (I think it's hard since the answer choices do not deviate much from one another).
The answer choice I viewed:


2. The number 75 can be written as the sum of the squares of 3 different positive integers. What is the sum of these 3 integers?
A: 17
B: 16
C: 15
D: 14
E: 13

Q: Is there a quick and easy way to solve this problem? My first intuition was to break 75 into its primes, but realized that did not help. After reviewing the question I wrote out all numbers from 1-8 and their squares and eye balled to see what squared values would sum to 75.

For the first one i think thats the only method i know off

For the second one 1^2 + 5^2 + 7^2 = 75 Hence 13

As a general rule, please split different problems into separate threads. I'll go ahead and answer both of these now, but please do this in future. Thanks!

I'll give the intuitive way first and the algebraic way second. Totally up to you which you prefer.

#1
The distance is the same for both trips. The difference in speed for the two trips is 6mph (that is, the boat goes 6mph faster downstream than upstream). So, for every hour that the boat travels downstream, it travels 6 miles more in that time than it does during an hour spent traveling upstream. Imagine two boats doing the downstream and upstream portions simultaneously. At the one hour mark, the downstream boat is 6 miles ahead. At the two hour mark, the downstream boat is 12 miles ahead. And so on.

A 6mph difference over 90 miles translates to a half hour difference in travel time (given by the problem). So the upstream boat needs half an hour extra to make up the distance it loses by going 6mph slower.

Your choices are all between 2.1 and 2.5 hours, so let's start at 2 hours to make this a little easier. Assume that the downstream boat takes 2 hours, which means it is 12 miles ahead of the upstream boat. Also, calculate the speed: to go 90 miles in 2 hours, the boat must be traveling 45mph. If the upstream boat is 6mph slower, it must be traveling 39mph. In half an hour, that boat would travel 19.5 miles. So that upstream boat must be going significantly slower, because it only has 12 miles to make up in that half hour, not 19.5. 2 hours is obviously not the right answer, because it isn't there, but neither are the numbers very close to 2, because 12 and 19.5 are pretty far apart.

So swing to the other end of the spectrum: try 2.5. If the downstream boat takes 2.5 hours, it will be 15 miles ahead of the upstream boat and its speed will be 90/2.5 = 180/5 = 36 mph. The upstream boat, then, must be traveling 30mph and, in half an hour, would travel 15 miles. Hey! That's exactly the distance we said it would be behind! That's the right answer.

Or, if you want to do something more like the above algebraic solution, there are a lot of things you can do to make it easier. Start by replacing (v-3) and (v+3) with r = rate for upstream trip and r+6 = rate for downstream trip. That gives you:
90 = r(t+0.5)
90 = (r+6)t
and these are much nicer to work with. Make sure to set the plain variable t for the downstream portion of the trip, since that's what the problem asks you to find.

90 = rt + 0.5r upstream
90 = rt + 6t downstream
subtract
0 = 0.5r - 6t

next, solve for r to substitute, not t, since we want to solve for t eventually
6t = 0.5r
12t = r

substitute into 2nd equation (downstream) b/c that one's easier
90 = 12t(t) + 6t
90 = 12t^2 + 6t
0 = 12t^2 + 6t - 90 simplify: divide by 6
0 = 2t^2 + t - 15
= (2t-5)(t+3)
t = either 2.5 or -3. And it can't equal a negative, so it must be 2.5.

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

Very surprised to see an "upstream/downstream" question in GMAT. Such questions are common in India's b school entrance exams.

Anyway, the most "viable" method would be the obvious one:

90/(v-3) - 90/(v+3) = 1/2

Now comes the time taking part:

Solve for v.

V comes to 33.

And since downstream's time has been asked,

Distance/downstream's speed

90/(33+3)=2.5

But yes, this is a time consuming problem. Perhaps, one could compensate by spending lesser time on a following DS question.

Thanks all!

_________________
Jamie Nelson
ManhattanGMAT Instructor