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One of the most common questions I hear from GMAT tutoring students is some variant of “When should I backsolve?”
I think I drive my students batty sometimes by not giving a clear, one-size-fits-all answer to that question, but there is some method to the madness – the ‘backsolve’ escape hatch is something that different students will use at different times, when facing different problems. Much like the question, “When should I ask for directions when I get lost on vacation?” a certain level of context about the issue at hand – as well as the person asking the question -- is simply required before it can be answered.
Backsolving, of course, is an approach to thorny quant problems that involves jamming the given answer choices into the question to see if one sticks. It only works on questions for which there are numbers in the answer choices (as opposed to variables). A beginning GMAT math student could use it on a question like this:
John’s age today is 20 years less than five times his current age. How old will John be in two years?
(A) 1
(B) 2
(C) 5
(D) 7
(E) 15
Without even attempting to throw some algebra behind this, a test-taker could start checking the different answer choices, eliminating choices that didn’t fit the constraints until finding an answer. He could remember that on some particular cloudy day in Boston, his GMAT tutor told him to always begin backsolving from the middle (because GMAT answer choices are always written in ascending numerical order, testing “C” first is a great way to get a sense of whether to go up or down from there).
In the case of any stsandardized test preparation, backsolving is never a first resort, though, due to time constraints. An example like the one shown above lends itself to quick testing (or even quicker algebra, to say that x + 20 = 5x, and then remembering to add two more years to the solution to select ‘D’). However, a set-up with a bit more complexity might require more time to be expended checking particular answers.
A question like the following would take longer than the above example to backsolve:
The ratio of owners to renters in a building is 20:1. If ten more renters (but no new owners) suddenly moved into the building, the ratio would become 16:1. How many owners currently live in the building?
(A) 200
(B) 300
(C) 400
(D) 600
(E) 800
Yes, this could be backsolved, but it might take several minutes to test the various solutions. Without a calculator to double-check one’s work, and given the stressful experience of GMAT quant, this could lead to a scrambled effort and wasted time. A far faster, cleaner way to approach this would be to set up an equation that looked like 20x/(1x+10) = 16/1 (Sam, can you format that so the fractions are vertical? Thanks). That would quickly yield the answer E, or 800.
When algebra can be used, use it. When you can translate a wordy setup into symbols and numbers, do it. However, when those options fail, just remember that you do have a nice card to play – one of the five numbers in the answer choices MUST be the correct answer, and you can cleverly insert it into the problem to find the solution when you are feeling algebraically-challenged.
Knowing when to backsolve is all about knowing your comfort zone. Every GMAT test taker has a different level of fluency with algebra, and each will have a unique point at which she says, “Okay, it’s time to pull the emergency cord here and just jam through this with the answer choices.” The only way to find this unique point is do so many sample problems that you begin to develop a feel for which set-ups are manageable, and which require that option of last resort.
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