Two consultants can type up a report in 12.5 hours and edit

Break down the problem into two pieces: typing and editing.

"Mary needs 30 hours to type the report" --> Mary's typing rate = 1/30 (rate reciprocal of time) (point 1 in theory below);
"Mary and Jim can type up a report in 12.5" and --> 1/30+1/x=1/12.5=2/25 (where x is the time needed for Jim to type the report alone) (point 2&3 in theory below)--> x=150/7;

"Jim needs 12 hours to edit the report" --> Jim's editing rate = 1/12;
"Mary and Jim can edit a report in 7.5" and --> 1/y+1/12=1/7.5=2/15 (where y is the time needed for Mary to edit the report alone) --> y=20;

"How many hours will it take if Jim types the report and Mary edits it immediately after he is done" --> x+y=150/7+20=~41.4

Answer: A.

THEORY
There are several important things you should know to solve work problems:

1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems.

\(time*speed=distance\) <--> \(time*rate=job \ done\). For example when we are told that a man can do a certain job in 3 hours we can write: \(3*rate=1\) --> \(rate=\frac{1}{3}\) job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then \(5*(2*rate)=1\) --> so rate of 1 printer is \(rate=\frac{1}{10}\) job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then \(3*(2*rate)=12\) --> so rate of 1 printer is \(rate=2\) pages per hour;

So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate).

2. We can sum the rates.

If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is \(rate_a=\frac{job}{time}=\frac{1}{2}\) job/hour and B's rate is \(rate_b=\frac{job}{time}=\frac{1}{3}\) job/hour. Combined rate of A and B working simultaneously would be \(rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\) job/hour, which means that they will complete \(\frac{5}{6}\) job in one hour working together.

3. For multiple entities: \(\frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T}\), where \(T\) is time needed for these entities to complete a given job working simultaneously.

For example if:
Time needed for A to complete the job is A hours;
Time needed for B to complete the job is B hours;
Time needed for C to complete the job is C hours;
...
Time needed for N to complete the job is N hours;

Then: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T}\), where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously.

For two and three entities (workers, pumps, ...):

General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:

Given that \(t_1\) and \(t_2\) are the respective individual times needed for \(A\) and \(B\) workers (pumps, ...) to complete the job, then time needed for \(A\) and \(B\) working simultaneously to complete the job equals to \(T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2}\) hours, which is reciprocal of the sum of their respective rates (\(\frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T}\)).

General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:

\(T_{(A&B&C)}=\frac{t_1*t_2*t_3}{t_1*t_2+t_1*t_3+t_2*t_3}\) hours.

Some work problems with solutions:
time-n-work-problem-82718.html?hilit=reciprocal%20rate
facing-problem-with-this-question-91187.html?highlight=rate+reciprocal
what-am-i-doing-wrong-to-bunuel-91124.html?highlight=rate+reciprocal
gmat-prep-ps-93365.html?hilit=reciprocal%20rate
questions-from-gmat-prep-practice-exam-please-help-93632.html?hilit=reciprocal%20rate
a-good-one-98479.html?hilit=rate
solution-required-100221.html?hilit=work%20rate%20done
work-problem-98599.html?hilit=work%20rate%20done
hours-to-type-pages-102407.html?hilit=answer%20choices%20or%20solve%20quadratic%20equation.%20R

Hope it helps.
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Yes it is a tricky one but you can use some logic to arrive at the answer quickly. The explanation I will write down will be long but when you do it in your head, it will take less than a minute, I promise.

Total time taken to type the report = 12.5 hrs
Time taken by Mary alone = 30 hrs
Time taken by Jim alone? Let's see. If Jim were to take 30 hrs alone too, together they would have taken 15 hrs. But together they took only 12.5 hrs. My guess is that Jim takes close to 20 hrs alone.
Let's see: 1/30 + 1/20 = 5/60
Time taken together = 60/5 = 12 hrs. Close! It means Jim takes a little more than 20 hrs and I would take it as 21 and move on.

Total time taken to edit the report = 7.5 hrs
Time taken by Jim alone = 12 hrs
Time taken by Mary alone? Now, if Mary were to take 12 hrs too, they could have edited it together in 6 hrs. But they took 7.5 hrs together. So Mary must take more than 12 hrs to edit it alone. I would guess 20 again.
Let's see: 1/12 + 1/20 = 8/60
Time taken together = 60/8 = 7.5 hrs - Exactly!
So Jim alone takes 21 hrs (apprx) to type it and Mary alone takes 20 hrs to edit it. Together they take 21+20 = 41 hrs (apprx)
Answer (A)
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Hi there. I'm happy to help with this.

The BIG idea to keep in mind: when two people are working together, what you add are the rates. You never add or subtract the times it takes to work. You add rates.

The question: Two consultants can type up a report in 12.5 hours and edit it in 7.5 hours. If Mary needs 30 hours to type the report and Jim needs 12 hours to edit it alone, how many hours will it take if Jim types the report and Mary edits it immediately after he is done?

I'm going to use the notation:
Rmt = the rate at which Mary types
Rme = the rate at which Mary edits
Rjt = the rate at which Jim types
Rje = the rate at which Jim edits
Rct = the combined typing rate
Rtt = the combined editing rate

The first two numbers tell us about combined rates.
If they type a report together in 12.5 = 25/2 hr, then their combined typing rate is Rtt = (1 report)/(25/2 hr) = 2/25.
If they edit a report together in 7.5 = 15/2 hr, then their combined editing rate is Rte = (1 report)/(15/2 hr) = 2/15.

Mary types one report in 30 hours, so Rmt = 1/30.

ADD RATES --> Rtt = Rmt + Rjt --> Rjt = Rtt - Rmt = (2/25) - (1/30) = (2/25)(6/6) - (1/30)(5/5) = 12/150 - 5/150 = 7/150

Jim's typing rate is 7/150, so he types one report in a time of 150/7 hr, approx 21 & change hours.

Jim edits one report in 12 hours, so Rje = 1/12

ADD RATES --> Rte = Rme + Rje --> Rme = Rte - Rje = 2/15 - 1/12 = (2/15)(4/4) - (1/12)(5/5) = 8/60 - 5/60 = 3/60 = 1/20

Mary's editing rate is 1/20, so she edits one report in a time of 20 hr.

So, the total time = (21 & change) hours for Jim to type + 20 hrs for Mary to edit = 41 and change hours

That's closest to answer choice A.

I'm sorry, but I disagree with what you posted as the OA. Is it possible that you miscopied?

Does my work here make sense? Please let me know if you have any questions on what I've said here.

Mike
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By the way, I have assumed that you will understand some things (e.g. if two people working alone take 30 hrs each, together they will take 15 hrs) since you got a Q42 in your last GMAT. If you need an explanation of these, let me know.
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Mary

Rate of type \(= \frac{1}{30}\)

Rate of edit \(= \frac{1}{b} (Consider)\)

Jim

Rate of type \(= \frac{1}{c} (Consider)\)

Rate of edit \(= \frac{1}{12}\)

Typing rate combined\(= \frac{1}{30} + \frac{1}{c} = \frac{1}{12.5}\)......... (1)

Editing rate combined \(= \frac{1}{b} + \frac{1}{12} = \frac{1}{7.5}\) ........... (2)

Solving above equations for values of b & c

Time taken for Jim typing \(= c = \frac{150}{7}\)

Time taken for Mary editing = b = 20

Total time taken for Jim typing & Mary editing

= b+c

= \(\frac{150}{7} + 20\)

\(= \frac{290}{7}\)

= 41.4 = Answer = A

Seriously, how to solve such a question in 2 minutes. It already takes 1 minute to put everything together or even more than one minute ...

We can let Jim’s typing time = j, and thus his rate = 1/j. Since Mary needs 30 hours to type the report alone, Mary’s typing rate is 1/30. Since together they can type the report in 12.5 hours, their combined rate is 1/12.5 and we can create the following equation to determine j:

1/j + 1/30 = 1/12.5

1/j + 1/30 = 2/25

Multiplying the equation by 150j, we have:

150 + 5j = 12j

150 = 7j

150/7 = j

j = 21.4

We can let Mary’s editing time = m, and thus her editing rate = 1/m. Since Jim needs 12 hours to edit the report, his editing rate is 1/12. Since together they can edit the report in 7.5 hours, their combined editing rate is 1/7.5 and we can create the following equation to determine m:

1/m + 1/12 = 1/7.5

1/m + 1/12 = 2/15

Multiplying the equation by 120m, we have:

120 + 10m = 16m

120 = 6m

20 = m

Thus, if Jim types the report and Mary edits the report, it will take 21.4 + 20 = 41.4 hours.

Answer: A
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The question stem should ask for the approximate time, as shown below:



Let the report = 300 words
Let the "two consultants" be Mary and Jim.

Since Mary and Jim together take 12.5 hours to type the 300-word report, their combined rate = work/time = 300/12.5 = 600/25 = 24 words per hour.
Since Mary on her own takes 30 hours to type the 300-word report, Mary's rate = work/time = 300/30 = 10 words per hour.
Thus, Jim's rate = (combined rate) - (Mary's rate) = 24-10 = 14 words per hour.
Since Jim types the 300-word report at a rate of 14 words per hour, Jim's time to type the report = work/rate = 300/14 = 150/7 ≈ 21.4 hours.

Since Mary and Jim together take 7.5 hours to edit the 300-word report, their combined rate = work/time = 300/7.5 = 600/15 = 40 words per hour.
Since Jim on his own takes 12 hours to edit the 300-word report, Jim's rate = work/time = 300/12 = 25 words per hour.
Thus, Mary's rate = (combined rate) - (Jim's rate) = 40-25 = 15 words per hour.
Since Mary edits the 300-word report at a rate of 15 words per hour, Mary's time to edit the report = work/rate = 300/15 = 20 hours.

Total time = 21.4 + 20 = 41.4 hours


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1/30+1/J = 2/25

J = 150/7

For editing
1/M = 2/15-1/12 = 1/20
M = 20

J + M = 150/7 + 20 = 41.4

A

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