Algebraic Rate Problem

[Ides_of_marzo]

To see a firework display on the 4th of July, Sydney drives from his home to a local park. He drives at a rate of m miles per hour to the park. On his return trip he drives at a rate of n miles per hour. How far away from his home is the park if he spends a total of z hours in the car, making no stops along the way?

A. (n + z)/m - (z/n)
B. (m + n + z)/(m*n)
C. (m*n*z)/(m+n)
D. (m + z)/(m*n)
E. (m*z)/n

Highlight "C" for answer


Here's what I did that's wrong.

1. Find combined rate m + n = r
2. Plug into rate for formula d = r*t = (m+r)*z
3. divide by 2 to find half the total distance (m+r)*z*0.5

[GMATGuruNY]

To see a firework display on the 4th of July, Sydney drives from his home to a local park. He drives at a rate of m miles per hour to the park. On his return trip he drives at a rate of n miles per hour. How far away from his home is the park if he spends a total of z hours in the car, making no stops along the way?

A. (n + z)/m - (z/n)
B. (m + n + z)/(m*n)
C. (m*n*z)/(m+n)
D. (m + z)/(m*n)
E. (m*z)/n

Let the distance between home and the park = 10 miles.
Let m = 5 miles per hour and n = 2 miles per hour.

At a rate of 5 miles per hour, the time to travel the 10 miles to the park = d/r = 10/5 = 2 hours.
At a rate of 2 miles per hour, the time to travel the 10 miles home = 10/2 = 5 hours.
z = the total travel time = 2 + 5 = 7 hours.

Since the question stem asks for the distance between home and the park -- 10 miles -- the correct answer choice must yield a value of 10 when m=5, n=2 and z=7.
Only C works:
(mnz)/(m+n) = (5*2*7)/(5+2) = 10.

The correct answer is C.

[Ides_of_marzo]

Thank you!
Quick follow up question - How did you choose your strategy? ie why did you select plugging in numbers rather than solving algebraically? What indicators did you rely on to make your decision?

[OptimusPrep]

To see a firework display on the 4th of July, Sydney drives from his home to a local park. He drives at a rate of m miles per hour to the park. On his return trip he drives at a rate of n miles per hour. How far away from his home is the park if he spends a total of z hours in the car, making no stops along the way?

A. (n + z)/m - (z/n)
B. (m + n + z)/(m*n)
C. (m*n*z)/(m+n)
D. (m + z)/(m*n)
E. (m*z)/n

Highlight "C" for answer


Here's what I did that's wrong.

1. Find combined rate m + n = r
2. Plug into rate for formula d = r*t = (m+r)*z
3. divide by 2 to find half the total distance (m+r)*z*0.5

Assume the distance from home to park = d miles
Total time spent = z hours

Total time = time from home to park + time from park to home
z = d/m + d/n
z*m*n = d (m + n)
d = z*m*n/(m + n)

Option C

You cannot add the two speeds to get the resultant speed.
For example: Rickon drives at a speed of 50 miles/hr for x miles
And comes at a speed of 60 miles/hour between A and B.

This does not means that he drives at a speed of 110 miles /hour.

The average speed would be = Total distance/Total time = 2x/(x/50 + x/60)

Does this help?

[ceilidh.erickson]

Thank you!
Quick follow up question - How did you choose your strategy? ie why did you select plugging in numbers rather than solving algebraically? What indicators did you rely on to make your decision?

This tends to be a personal preference / case-by-case question. Personally, I almost always pick numbers rather than doing algebra when there are variables in the answer choices, unless the algebra seems easy or obvious. I find that rate questions are usually easier to do by picking #s rather than by algebra.

The only way you'll know which strategy you prefer is to try each question both ways. Practice using both strategies, and you'll quickly develop a sense for which works better for you, and in which circumstances.

[ceilidh.erickson]

By the way, what is the source of this question? The structure & answer choices have been pretty much directly plagiarized from this OG question:
https://www.beatthegmat.com/og-13-24-t288366.html#765096

Hopefully this should go without saying, but you shouldn't study from sources that plagiarize other materials.

[Matt@VeritasPrep]

Thank you!
Quick follow up question - How did you choose your strategy? ie why did you select plugging in numbers rather than solving algebraically? What indicators did you rely on to make your decision?

Can't speak for Guru, but my advice would be:

1:: If you can think of an algebraic approach that makes immediate sense, use it.

2:: If you can't, then this problem is an EXCELLENT candidate for a plug in, since it's a word problem with variables in the answers. That means that the correct equation must work for ANY values of m and d, so you can pick whichever ones you like to try to find the answer.

[Matt@VeritasPrep]

Also, let me propose an algebraic solution. This problem is pretty quick if we use the harmonic mean, an average that represents the average of two rates.

Taking that equation, we know that the average of the two rates must be

2mn/(m+n)

From there, we know that

D = RT

Our average rate = 2mn/(m+n)
Our time TO the park = half our total time = z/2

D = (z/2) * 2mn/(m+n) = zmn/(m+n)