GMAT Math : Work and Time

Work and time questions crop up from time to time in the GMAT Quantitative section.

Let’s start with a simple example :

Q) If John can complete a piece of work in 4 days and if Mary can complete it in 2 days. How many days will it take for them to complete the work together?

A. 1/2

B. 3/4

C. 5/3

D. 4/3

E. 7/4

Readers are encouraged to take some time to solve the question 🙂

Invert It!

The question is a good example to introduce the concept of “invert it” which is an invaluable technique to solve work and time questions.  Let’s highlight it with the previous example :

The proportion of work John can do in 1 day = 1/4 which is the inverse of the total number of days to complete the piece of work. Think about this for a second. If it takes him 4 days to complete the entire work, in 1 day he will be done with only one fourth (1/4) of the work.

Similarly, the proportion of work Mary can do in 1 day = 1/2.

Total amount of work they can do together in 1 day = proportion John completes + proportion Mary completes = 1/4 +1/2 = 3/4. Therefore if in 1 day they can together complete three fourth of the work, how many days will it take for them to complete the entire work together? The answer is the inverse of the proportion i.e., 4/3 or 1.33 days. Take a moment to go through this example again if unclear and internalize the concept of inverting.

Hence, we go with option (D)

Let’s do a few more variations of these questions :

Q) A alone can do a piece of work in 16 days, B alone can do a piece of work in 12 days. When A,B,C worked together the work was completed in 4 days. How long would it take for C to complete the work alone?

A. 9.6

B. 9.2

C. 8.6

D. 8.5

E. 9.8

Solution

Let C take c days to complete the work alone.

Therefore proportion of work completed by C in 1 day = 1/c (inverting days here)

Similarly proportion of work completed by  B in 1 day = 1/12

Similarly proportion of work completed by  A in 1 day = 1/16

Together, the proportion of work they can complete in 1 day = 1/c + 1/12 + 1/16

Now we know together they completed the entire work in 4 days. Hence proportion of work they completed in 1 day = 1/4

Equating the above equations => 1/c + 1/12 + 1/16 = 1/4.

We get c = 9.6 days and we go with option (A)

Q) A, B and C can complete a piece of work individually in 10, 20 and 30 days. What is the closest number to the number of days it will take for A to complete the piece of work if he is being helped by B and C every second day?

A. 9

B. 7

C. 10

D. 4

E. 8

Solution :

Since B and C work only every second day, their effective work period doubles individually ie 20×2 = 40 days and 30×2 = 60 days. Think about it, if these guys work only every second day, it will definitely take longer to complete the piece of work. By how much? Twice as much as earlier.

Now, having found the effective work periods for B and C, let’s use the familiar concept of inverting it to get to the answer.

Therefore 1/a + 1/b(effective) + 1/c(effective) = total proportion of work completed in 1 day.

Therefore, 1/10 + 1/40 + 1/60 = 0.141666 of the work. Now inverting this again to find total days taken to complete entire work we get 1/0.141666 ~= about 7 days or option (B)

Now let’s throw in a slight twist by adding an element of payment for the work done :

Q) A, B can complete a piece of work individually in 7 and 10 days. If the contract is for $2000, and together with C, the duo is able to complete the work in 2 days. How much shall C be paid of the contract amount?

A. 1742

B. 1320

C. 1200

D. 1030

E. 1100

Solution :

As always, let’s use the concept of invert it. Let C complete the task alone in c days.

We know 1/7 + 1/10 + 1/c = 1/2 ( Think about why this statement is true – Hint : read previous examples)

Therefore c = 3.88 days.

Next, amount paid to C = Contract amount x (proportion contribution of work done by C in a day/proportion of work done in a day)

Now, in each given day, A contributes 1/7 to the work, B contributes 1/10 to the work and C contributes 1/3.88 of the work.

Therefore amount to be paid to C = $2000 x ((1/3.88)/(1/3.88+1/7+1/10)) = $1030 or option (E)

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