A boat traveled upstream 90 miles at an average speed of

can someone explain what is wrong with my explanation below and why I am not getting the right answer?

Upstream
D= 90 miles
S= v-3 mph
T=H

Downstream
d= 90 miles
s= v+3
t=h

H=h+1/2

distance=speed x time
Upstream

90=(v-3) (h+1/2).....(i)

Downstream

90=(v+3)h ....(ii)

with equation i and ii

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

After this, I am stuck. Is the above logic right at all?

pra1785, in a different kind of question, the logic would work more efficiently. Your logic is not wrong here. It just makes things a lot harder, I think.

Distances are equal, so it does seem like a natural step to set each leg's (r * t) equal, but . . .

The problem is that you have defined one unknown variable in terms of another. You have two unknown variables and one equation. The answer choices are values, not variable expressions (in which case your equation, without more, might work, see below).

You will have to use a known value at some point. IMO, it's easier to do so at the beginning.

Your equation yields \(\frac{1}{2}v - 6h = \frac{3}{2}\)

"Solve" for time, which you have called h:

h = \(\frac{(v - 3)}{12}\)

Now you must "go back" and use the known value of 90.

h\(_2\) = \(\frac{(v-3)}{12}\)

h\(_1\) = \(\frac{90}{(v+3)}\)

\(\frac{(v-3)}{12}\) = \(\frac{90}{(v+3)}\)

\(v^2 - 9 = 1080\)
\(v^2 = 1089\)
\(v = 33\)

Finally (whew!), if h = \(\frac{(v - 3)}{12}\), then

\(\frac{(33 - 3)}{12}\) = h

h = \(\frac{30}{12}\) = 2.5 hrs

If answer choices WERE variable expressions, depending on the prompt, the variable-variable solution ("h = ...") to your equation as-is (i.e., not taken further as I did, with D = 90) would be among them.

But the answer choices have actual values. So you need one variable and one known in your equation. Known D / unknown rates = known time.

When distances are equal, it often works efficiently to write \(r_1*t_1 = r_2*t_2\). I don't think it's efficient here. But if you are very quick and very accurate with algebra, it's probably fine.

Does that help?
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Hi All,

This is a layered story-problem and takes a lot of effort to solve using a traditional "math approach". Here's how you can solve it with a bit of logic and TESTing THE ANSWERS:

From the prompt, we can create 2 equations:

D = R x T

90 = (V-3)(T + 1/2)
90 = (V+3)(T)

We're asked for the value of T.

From the prompt, I find it interesting that the distance is a nice, round number (90)…. because when looking at the answer choices, most of them are NOT nice decimals. When multiplying two values together (as we do in BOTH equations), if you end up with a round number, chances are that either….

1) both numbers are round numbers
2) one of the numbers includess a nice fraction (e.g. 1/2) which can be multiplied and the end result will be a round number.

This gets me thinking that 2.5 is probably the answer, but I still have to prove it….I'm going to plug in THAT value for T and see what happens to the 2 equations….

90 = (V-3)(3)
90 = (V+3)(2.5)

30 = (V-3)
36 = (V+3)

33 = V
33 = V

Notice how both values of V are THE SAME? That means that we have the solution. V=33 and T=2.5

Final Answer:

GMAT assassins aren't born, they're made,
Rich
_________________

Let's try this way

B = Speed of Boat
S = Speed of Stream

Upstream Speed = Net speed against the flow of current = B - S
Downstream Speed = Net speed with the flow of current = B + S

Time upstream = (1/2)Hour + Downstream Time

\(\frac{90}{v-3} = (\frac{1}{2}) + \frac{90}{v+3}\)

90/30 = 3
90/36 = 2.5
Diffference = 1/2

i.e. v-3 = 30 and v+3 = 36 i.e. v = 33

Downstream time = 90/36 = 2.5

Answer: Option A
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how am i getting this wrong?
time = t for downstream and 1.5t for upstream
Distance upstream = distance downstream (90miles = 90miles)
therefore,
(v-3) 1.5t = (v+3)t
1.5vt-4.5t = vt + 3t
therefore, v = 15mph
Downstream speed = v+3 = 15+3 = 18mph
therefore, time t= d/s = 90/18 = 5hrs

The question says "the trip upstream took a half hour longer than the trip downstream"
So, if t = the travel time downstream (in hours), then t + 0.5 = the travel time upstream (in hours)

I have a feeling that you accidentally wrote: t + 0.5t = the travel time upstream (which simplifies to be 1.5t)
_________________

We can PLUG IN THE ANSWERS, which represent the number of hours to travel downstream.
The time upstream is 0.5 hour longer than the time downstream.
The correct answer choice is almost CERTAIN to be A -- the only option that implies two times that divide evenly into 90 (2.5 hours and 3 hours).
When the correct answer is plugged in, the difference between the rate downstream (v+3) and the rate downstream (v-3) will be 6 mph:
(v+3) - (v-3) = 6

Answer choice A: 2.5 hours to travel downstream, implying 3 hours to travel upstream
Rate downstream \(= \frac{distance}{time} = \frac{90}{2.5} = 36\) mph
Rate upstream \(= \frac{distance}{time}= \frac{90}{3} = 30\) mph
Rate difference \(= 36-30 = 6\) mph
Success!


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—-

Hi Bunuel,

Thanks for explaining this so well. Could u pls post a list of similar boat stream Qs

Posted from my mobile device

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Dear experts, can you help me in this approach. I am getting it wrong.

(i) Upstream
\(v-3 = \frac{90}{t+\frac{1}{2}}\) .... eqn 1

(i) Downstream
\(v+3 = \frac{90}{t}\)

\(v = \frac{90}{t} - 3\) , substituting this in eqn 1

\(\frac{90-6t}{t} = \frac{180}{2t+1}\)

(90-6t)(2t+1)= 180t

90=18t

t=5.

where have i gone wrong?

Hi GRS,

This equation does NOT 'simplify' in the way that you wrote.

(90-6t)(2t+1)= 180t

FOIL it out, combine like terms and then divide by 3... You will end up with 2 solutions (because it's a Quadratic) and the one positive solution is the answer to the original prompt.

GMAT assassins aren't born, they're made,
Rich
_________________

90/(v+3) - 90/(v-3)=0.5
v=33
So, t=90/36=2.5

Upstream speed = V - 3
Downstream Speed = V + 3

\(\frac{90}{v-3} - \frac{90}{v+3} = \frac{1}{2}\)

\(\frac{90(v+3) - 90(v-3)}{ (v+3)(v-3) }= \frac{1}{2}\)

\(\frac{90(v+3-v+3)}{(v^2 -9)} = \frac{1}{2}\)

\(90*6 *2 = v^2 -9\)
\(v^2 = 1080 + 9 = 1089\)
v= 33

So the time taken by boat to travel downstream = 90/(v+3) = 90/36 = 2.5 hrs

Option A is the correct answer.

Another approach I would suggest is to try logical elimination of the options.
Most of the cases, V will be an integer.
so the Downstream speed = V + 3 will also be an integer.
\(\frac{90}{Downstream speed }= time \)
=> 90/time taken = Downstream Speed
That means, when you divide 90 by the options choices ,you should get an integer as well.

Option A will give an integer .You can also crosscheck by finding Upstream time and see if it is 3 hrs i.e. 2.5 + .5 = 3 hrs
Opt C, D,E can be eliminated as it give a recurring decimal.

Thanks,
Clifin J Francis
GMAT SME
_________________

Normally with this type of question the answer choices will be easy to work backwards from, so it's surprising to see decimals here.
Here are links to two other official problems of the same type:
https://gmatclub.com/forum/a-store-currently-charges-the-same-price-for-each-towel-that-144780.html
https://gmatclub.com/forum/at-his-regular-hourly-rate-don-had-estimated-the-labor-cost-136642.html

In this case, we're told that the 90 miles take half an hour longer if the speed is 6mph slower. So I would just try answer choice A, because adding half an hour to it gets us an integer.

(A) 90 miles in 3 hours means 30mph, and I must check whether 2.5 * 36mph is also 90 miles. Yes, it is (I check quickly by "borrowing a factor of 4 from the 36 and "giving" it to the 2.5, thereby getting 10*9). So the answer is A.

*it is inappropriate, in my opinion, to solve this problem, or the other two that I linked to, using algebra.
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Question: 90/(v+3)?

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

=> 180/(v-3) - 180/(v+3) = 1

=> 180/(33-3) - 180/(33+3) = 1 [because we need a number that after reduction will give us the difference of one - since GMAT numbers fit well; numbers around 10,20,40 wouldn't work]

=> So, 90/(v+3)
=> 90/(33+3)
=> 90/(36)
=> 2.5

My GMAT mock scores are 590. I am not a genius so don't take my method seriously.
Anyways My approach is a little different.

speed x time = distance
downstream time = y (assume)
1)y*(v+3) = 90
2)(y+0.5)*(v-3) = 90
FACTOR OF 90 = 3*3*5*2
B,C,D and E are out because unnecessary prime will be introduced So answer is A.
Reason for elimination of C and D when calculating eq 1 then they will have extra prime 11 and 23 which are not factors of 90 and do not have anything that can cancel them out.
Reason for elimination of B and E 0.5 hour extra is there when travelling uptsream so when added to downstream time B and E will become prime which are not factor of 90 and do not have anything that can cancel them out.
Still want to check then check for 2.5 because remember you need to add 0.5hr in upstream time so
2.5= 5/2
so now we need two 2 and two 3
(5/2)*2*2*3*3=90 i.e v=33
check in eq 2
(2.5+0.5)(33-3)=90
voila. ans is option A

Bunuel KarishmaB thanks for your explanations in the posts above. Can you please suggest some similar questions to practice doing the math faster? Thanks!

Have you checked this post?
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UserMaple5

For further concept discussion and practice questions, you can also check out my TSD module which has a section on effective speed (boats and streams, elevators etc).
Here is the link: https://anglesandarguments.com/study-module
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