In a toy factory, when working together, Mary and Jim can assemble

Official Solution:

In a toy factory, when working together, Mary and Jim can assemble a model car in 12 hours and paint it in 4 hours. If it takes Mary 30 hours to assemble the car by herself, and it takes Jim 12 hours to paint the car by himself, then how many hours will it take if Jim assembles the car and Mary immediately paints it after he is finished?

A. 26
B. 24
C. 22
D. 20
E. 18


Let's divide the problem into two components: the assembly and the painting of the model car.

When it comes to assembling the car, Mary can complete the task alone in 30 hours, giving her an assembly rate of \(\frac{1}{30}\) job/hour.

When Mary and Jim work together, they can assemble the car in 12 hours. This gives \(\frac{1}{30} + \frac{1}{x}= \frac{1}{12}\), where \(x\) is the time it would take for Jim to assemble the car by himself. Solving this equation, we find that \(x=20\) hours.

Now, let's consider painting the car. Jim can complete this task alone in 12 hours, which means his painting rate is \(\frac{1}{12}\) job/hour.

When Mary and Jim work together, they can paint the car in 4 hours. This give the equation \(\frac{1}{y}+\frac{1}{12}=\frac{1}{4}\), where \(y\) is the time it would take for Mary to paint the car by herself. Solving this equation gives us \(y=6\) hours.

Finally, to find out the total time it would take if Jim assembles the car and Mary immediately paints it once he finishes, we simply add the time it takes each of them to complete their tasks individually. Therefore, \(x + y=26\) hours.


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:
https://gmatclub.com/forum/time-n-work- ... cal%20rate
https://gmatclub.com/forum/facing-probl ... reciprocal
https://gmatclub.com/forum/what-am-i-do ... reciprocal
https://gmatclub.com/forum/gmat-prep-ps ... cal%20rate
https://gmatclub.com/forum/questions-fr ... cal%20rate
https://gmatclub.com/forum/a-good-one-9 ... hilit=rate
https://gmatclub.com/forum/solution-req ... ate%20done
https://gmatclub.com/forum/work-problem ... ate%20done
https://gmatclub.com/forum/hours-to-typ ... ation.%20R

Hope it helps.