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work-rate question

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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work-rate question

by sukh » Sun Aug 07, 2011 8:50 am
Three secretaries working together can type 36 pages of text in half an hour. If the typing speeds of the secretaries are in a 4:3:2 ratio, how long will it take the slowest-typing secretary working alone to type 52 pages of text? (A) 1 hour and 34 minutes (B) 1 hour and 45 minutes (C) 2 hours and 24 minutes (D) 3 hours and 15 minutes (E) 3 hours and 25 minutes


Lionel left his house and walked towards Walt's house, 48 miles away. Two hours later, Walt left his house and ran towards Lionel's house. If Lionel's speed was 4 miles per hour and Walt's 6 miles per hour, how many miles had Lionel walked when he met Walt? (A) 12 (B) 16 (C) 20 (D) 24 (E) 28


Andrew and Stephen drive on the highway in the same direction at respective rates of 72 and 80 kmh. If Stephen is 4 km behind Andrew, by how much does he have to increase his speed to catch up with Andrew in 20 minutes? (A) 1 kmh (B) 2 kmh (C) 3 kmh (D) 4 kmh (E) 5 kmh
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by VivianKerr » Sat Aug 13, 2011 8:27 pm
Here's the first one:

4x + 3x + 2x = 36

9x = 36

x = 4

So the secretaries type at 16, 12, and 8 pages per 1/2 hr respectively. The slowest-typing is 8 pages per 1/2 hour, or 16 pages per hour. To type 52 pages it will take her 52/16 hours, or 3.25.

The answer is D.
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by sumgb » Sun Aug 14, 2011 8:53 am
2nd Que -
Lionel left his house and walked towards Walt's house, 48 miles away. Two hours later, Walt left his house and ran towards Lionel's house. If Lionel's speed was 4 miles per hour and Walt's 6 miles per hour, how many miles had Lionel walked when he met Walt? (A) 12 (B) 16 (C) 20 (D) 24 (E) 28
in 2 hours L has already walked (2 * 4) = 8 miles. so when W starts running the distance between the two is 40 miles.
since direction of L and W is opposite we can their relative speed = 4 + 6 = 10 mph.
so it would take 4 hours to cover distance of 40 miles

In those 4 hours, L would have covered 4 * 4 = 16 miles

so total distance covered by L = 8 + 16 = 24 miles. Ans D.

Hope this helps...

QUe 3-
Andrew and Stephen drive on the highway in the same direction at respective rates of 72 and 80 kmh. If Stephen is 4 km behind Andrew, by how much does he have to increase his speed to catch up with Andrew in 20 minutes? (A) 1 kmh (B) 2 kmh (C) 3 kmh (D) 4 kmh (E) 5 kmh
the relative speed of A and S = 80 - 72 = 8 kmph

so with this relative speed S would take 30 mins to cover the distance of 4 km
however S needs to cover the distance of 4 km in 20 mins that is 1/3 hour.

so since speed = dist / time
speed = 4 / (1/3) = 12 kmph
But the relative speed is 8 kmph so S would need to increase his speed by 12 - 8 = 4 kmph

hope this helps...
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by pinchharmonic » Sun Aug 14, 2011 8:16 pm
Sumgb's answer is the best, and simplifies the question considerably using relative speed. My approach was different, because I wasn't able to see the relative speed, but I believe this is easier to understand.

S needs to catch up to A, and A is already 4km ahead. Not only that, but whatever time you need to gain that 4km, A is going to continue moving ahead

D=RT, and the D is going to be the distance we want S to cover.

4km + distance A would move = Speed of S x T, let T be the time.

4km + 72*T = 80*T.

if you solved for T here, you get the amount of time it takes for S to catch up to A going at their existing speeds as question stated. But we don't want to solve for T. We know T = 20 minutes, so another item must vary, and that is 80 because that is speed of S and we're asked to vary the speed of S.

20 minutes = 1/3 hour

4km + 72/3 = (80+V)*1/3 where V is the difference in velocity S needs. Notice all the 20 minutes are now plugged in as 1/3.

4 + 24 = 80/3 + V/3
28 = 80/3 + V/3
84 = 80 + V
4 = V
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