Avoiding Variables in Work Rate | Arithmetic

Those of you who have studied my approaches and methods know that I do not like to take variables. In GMAT questions, often I do not take any variables, rarely I take one and never do I take two. But I have seen people trying to work with 2-3 variables and then solving a ton of equations simultaneously expending much of their precious in that. Hence comes in the issue of time management.

The common refrain is that given unlimited time, I could solve all the Quant questions but with just 62 mins, I could solve only a fraction and had to guess on others.

Keeping the time constraint in mind, I try to show speed approaches. Different situations will need different approaches but nonetheless the concepts remain the same. Once you understand the concepts, try to evaluate the question holistically, look at the numbers given and the options given before you deep dive into algebra. Often you will find that getting into algebra is not required.

Let me show using a question from GMAT Club here. Its source is not specified but you can find it here.

Question: A and B together can do a piece of work in ${\frac{72}{5}}$ days; B and C together can do a piece of work in 12 days. After A worked for 8 days, B for 12 days C takes up and finishes it alone in 6 days. In how many days will each of them do the work, working alone?

(A) A = 24 days, B = 36 days, C = 18 days.

(B) A = 24 days, B = 36 days, C = 16 days.

(D) A = 24 days, B = 32 days, C = 18 days.

(E) A = 24 days, B = 30 days, C = 18 days.

Solution:

One method is to rely on options. Plug them one by one to see which one satisfies all constraints. But that is usually not my fallback method because it is easy to give options which may not lend themselves to plugging in or which themselves will require a whole lot of work and hence, a lot of time.

Say there are 72 units of work.

A and B together can do a piece of work in $\frac{72}{5}$ days.

Rate of work of A and B together = $\frac{Work}{Time}$ = $\frac{72}{(\frac{72}{5})} = 5$ units every day

B and C together can do a piece of work in 12 days.

Rate of work of B and C together = $\frac{Work}{Time} = \frac{72}{12} = 6$ units every day

After A worked for 8 days, B for 12 days C takes up and finishes it alone in 6 days.

(A worked for 8 days + B worked for 8 days) + (B worked for 4 days + C worked for 4 days) + C worked for 2 days = 72

So (A and B worked for 8 days) + (B and C worked for 4 days) + C worked for 2 days to complete 72 units of work.

$5 \times 8 + 4 \times 6$ + C’s work for 2 days = 72

C’s work for 2 days = 8 units

So C does 4 units of work every day.

Note that in terms of units of work done, it doesn’t matter whether A and B work one after another or simultaneously. They will complete the same units of work. What will vary in these scenarios is the total time taken to complete the work.

C does 4 units of work every day. Then B does 2 units every day (because B and C complete 6 units in a day) and A does 3 units every day (because A and B complete 5 units every day)

Time taken by A = $\frac{Work}{Rate} = \frac{72}{3} = 24$ days,