Finding the Right Answer in a World Where So Much is Wrong, or: Estimating Answers on the GMAT

by , Apr 10, 2010

Nate Burke is a Content Developer @ Knewton, where he helps students with their GMAT prep.

Suppose you're midway through your GMAT quant section, and you encounter the following question:

Working alone, Printers X, Y, and Z can do a certain printing job, consisting of a large number of pages, in 12, 15, and 18 hours, respectively. What is the ratio of the time it takes Printer X to do the job, working alone at its rate, to the time it takes Printers Y and Z to do the job, working together at their individual rates?

(A) [pmath]{4/{11}}[/pmath]

(B) [pmath]{1/{2}}[/pmath]

(C) [pmath]{{15}/{22}}[/pmath]

(D) [pmath]{{22}/{15}}[/pmath]

(E) [pmath]{{11}/{4}}[/pmath]

(Taken from GMAC's Quantitative Review, 2nd Ed., p. 78)

You diligently work this problem using the formula:

[pmath](WORK DONE) = (WORK RATE)(TIME)[/pmath]

Since you're looking to find a ratio of two time quantities, you solve for time in this equation:

[pmath](TIME) = {(WORK DONE)}/{(WORK RATE)}[/pmath]

You know that if you can solve for the time it takes printers Y and Z working together to do the job, you can just take this number, call it [pmath]N[/pmath], and find

[pmath]{12}/{N}[/pmath], since [pmath]12[/pmath] is the time that it takes printer X to do the job.

Now, this question can be made a whole lot easier if you recognize that the numbers 12, 15, and 18 are all factors of 180. Since you eventually are going to be adding fractions involving 15 and 18, it might be a good idea to convert everything into quantities involving the number 180:

PRINTER.......................TIME TO DO JOB

X......................................[pmath]{180}/{15}[/pmath]

Y......................................[pmath]{180}/{12}[/pmath]

Z......................................[pmath]{180}/{10}[/pmath]

Now, to figure out how long it will take printers Y and Z to do the job TOGETHER, you can just add their rates. Since they each are doing "one job," you can say that printer Y's rate is: [pmath]{1 job}/{15 hours}[/pmath] and that printer Z's rate is: [pmath]{1 job}/{18 hours}[/pmath]

To add the rates, you need to find a common denominator. Luckily, you have already found 180 as a nice number to work with:

Printer Y: [pmath]{1 job}/{15 hours} = {12 jobs}/{180 hours}[/pmath]

Printer Z: [pmath]{1 job}/{18 hours} = {10 jobs}/{180 hours}[/pmath]

Now that you have [pmath]180[/pmath] as a common denominator, you can just add the rates of printer Y and printer Z:

Printer Y and Z together: [pmath]{12 jobs}/{180 hours} + {10 jobs}/{180hours} = {22 jobs}/{180 hours}[/pmath]

To find the time that it takes both printers to do one printing job, divide [pmath]1 job[/pmath] by this rate: [pmath]{(1 job)}/{({(22 jobs)}/{(180 hours)})} = (1 job)({180 hours}/{22 jobs}) = {(180 hours)}/{22}[/pmath] But note that it's in terms of [pmath]180[/pmath]: you should convert [pmath](12 hours)[/pmath] to be in terms of [pmath]180[/pmath] as well to find the ratio:

[pmath](12 hours) = {(180 hours)}/{15}[/pmath]

The ratio is then [pmath]{({(180 hours)}/{15})}/{({(180 hours)}/{22})} = {22}/{15}[/pmath]

Answer choice D is correct.

Now, all of this works out well contingent on you magically knowing to write each of the rates as factors of 180. What if you had not done that?

Specifically, what if you just took 12 as the time it would take printer X to do the job, and then found the least common multiple of 15 and 18, which is 90, in order to find their combined rate:

[pmath](Rate Y) + (Rate Z) = {1}/{15} + {1}/{18} = {6}/{90} + {5}/{90} = {11}/{90}[/pmath] jobs per hour.

The time it would take the two printers to do the job together, then would be [pmath]{90}/{11}[/pmath] hours per job.

So now you would be looking for the value of the ratio [pmath]{12}/{({90}/{11})}[/pmath]. But this is definitely not one of the answer choices. Moreover, it is not immediately apparent how you should reduce this fraction:

Should you go back and rework the problem?

This is a tough situation to be in, but note that if you had followed this approach and gotten the value [pmath]{12}/{({90}/{11})}[/pmath], you could rule out three of the answer choices: the fraction [pmath]{90}/{11}[/pmath], is roughly equal to [pmath]9[/pmath], so that the fraction [pmath]{12}/{({90}/{11})}[/pmath] is greater than 1. You can eliminate the first three answer choices, because they are all less than 1!

Moreover, if you are confident that [pmath]{12}/{({90}/{11})}[/pmath] is a VERSION of the correct value of the ratio (and you should be!), you can just estimate and see if there is an answer choice that "looks" correct. Looking at [pmath]{12}/{({90}/{11})}[/pmath], you can estimate as before, and just say that [pmath]{90}/{11}[/pmath] is [pmath]9[/pmath] so that [pmath]{12}/{({90}/{11})} = {12}/{9}[/pmath], roughly.

Looking at answer choices D and E, answer choice E is squarely greater than 2, which is not the case for the fraction [pmath]{12}/{9}[/pmath]. Answer choice D must be correct.

Ultimately, the GMAT quantitative section does not exclusively aim to see if you can calculate exact quantities. This is definitely the case with data sufficiency questions, but even on a problem solving question like this, it can often be a better use of time to estimate your way to a correct answer if you're stumped rather than to scrap good work and completely re-solve the question.