Think Like an Expert: How & When to Work Backwards on GMAT Problem Solving

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Think Like an Expert: How & When to Work Backwards on GMAT Problem Solving

by Céilidh Erickson Apr 21, 2016

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What does it take to be a GMAT expert? It’s not just content knowledge (although of course that’s necessary). A GMAT expert knows how to quickly identify patterns and choose quickly from a variety of strategies. In each of these segments, I’ll show you one of these expert moves and how to use it.

What working backwards is

Working backwards from the answer choices, back-solving, plug-and-chug… no matter what you call it, you’ve probably heard of it before. Many GMAT Problem Solving (PS) questions require laborious algebra to solve, but are much faster and easier to solve by simply plugging the answer choices into the problem to see what fits.

Try this problem:

Ada went to the supermarket with $36, expecting to buy a certain number of energy drinks. However, the store had recently raised the price of energy drinks by $1, causing Ada to purchase 3 fewer energy drinks than expected. How many energy drinks did she originally expect to purchase?

(A) 12

(B) 10

(D) 8

(E) 6

If you did the algebra, you probably ran into some really ugly, messy equations. If you picked numbers, you likely had a much easier time. (Explanations at the end)*

Here’s what I hear from students all the time: “oh, now that you showed me, I can see that working backwards is easier. But I didn’t even think to try it.”I blame high school teachers who made us show all of our work every time! We’re trained to think that algebra is the “right” way to do things, so we jump automatically into creating equations.

So, you’ll have to re-train yourself…

When are you allowed to work backwards?

Here’s the rule, and it’s pretty simple: you’re allowed to work backwards from the answer choices any time a PS question asks you for the value of an unknown (a variable) – in other words, if you could write the question as “x= ?,” you’re allowed to work backwards.

Working backwards usually does not work (or at least not easily) if the question asked for any other kind of information: a sum, a difference, a product, a ratio / proportion, a variable in terms of another variable, etc.

Consider the difference between these two ratio problems, and try working backwards for each:

At a certain animal shelter, the ratio of puppies to kittens on Monday was 4 to 5. During the week, 8 puppies and 7 kittens were adopted and left the shelter. If by Friday the ratio of remaining puppies to remaining kittens was 2 to 3, how many kittens were originally in the shelter on Monday?

(A) 18

(B) 20

(D) 27

(E) 30

(A) 1

(B) 2

(D) 4

(E) 5

Question #1 asks for the original number of kittens, in other words the value of an unknown: k= ? We’re allowed to work backwards! We can easily plug in the answer choices into the original ratio (more on how to do so in a little bit).

Question #2 is asking for the difference between kittens and puppies. In other words, k – p= ? You probably had a lot more trouble working backwards on this one, so algebra probably was the most efficient strategy.

Train yourself to recognize working-backwards-problems

If you’re currently not using the strategy of working backwards because you “didn’t even think about it,” then you have to train yourself to recognize the signals.

Do this right now: grab a copy of the Official Guide. (If you don’t have one, you should definitely get one).

Step 1: Flip open to the first page of the PS section. Glance through the questions, and without solving, just ask yourself which questions you could work backwards on. In other words, which questions ask for the value of a variable? Write the question numbers down.

I’ll give you the first few from OG 2016: #3, #9, #12, #15, #19, #32. Now you practice recognizing the rest!

Step 2: Once you’re confident that you can recognize these problems, you can then go back and solve them by working backwards.

Step 3: Go back and re-solve these same questions, trying algebra this time. Then compare: which strategy was more efficient for you, and on which problems? This might vary from person to person, or topic to topic. Make sure you’re strengthening both muscles!

How to work backwards efficiently

Let’s go back to that first kittens-and-puppies problem. When you’re working backwards, it’s a good idea to create a chart to keep your information organized.

	kittens: 5x	puppies: 4x	kittens – 7	puppies – 8
A	18
B	20
C	25
D	27
E	30

Which answer choice should you start with? You’ll hear differing advice on this one: some people say to start with C, some say to start with B or D. The reasoning is that if you start with B and it’s too big, the answer must be A, and you can avoid testing a 2^nd value. If it’s too small, test D. If D is too small, the answer is E. If D is too large, the answer is C. This way, you’ve tested a maximum of 2 answers.

My recommendation, though, is to start with your intuition, then go with something else in the middle… whatever answer choice seems easiest. Just don’t start with A and test all 5 in a row, because you’ll be doing more work than you need to.

On this problem, intuition should tell you that if the original ratio of puppies to kitten was 4 to 5, the original number of kittens had to be a multiple of 5. We can rule out A and D. Then, I’d start with C, because it’s in the middle of the 3 answers I have left:

	kittens: 5x	puppies: 4x	kittens – 7	puppies – 8
A	18
B	20
C	25	20	18	12
D	27
E	30

If there were 25 kittens, there would have been 20 puppies to create a ratio of 4 to 5. If 7 kittens and 8 puppies leave, then the new ratio is 12 to 18, or 2 to 3. Correct!

Strategies are like muscles: you have to train them

Expert athletes don’t just know the rules of the game; they train themselves specifically to recognize and respond to different plays. If you want to become a GMAT expert, you need more than just knowing the rules: you need to train yourself on each individual skill.

Good luck!

*The algebraic solution:

Let p = expected price and q = expected quantity

pq = 36 and (p + 1)(q – 3) = 36

FOIL: pq – 3p + q – 3 = 36

Isolate: p = 36/q

Substitute: (36/q)q – 3(36/q) + q – 3 = 36

Simplify: 36 – 108/q + q – 3 = 36

Subtract 36 from both sides: – 108/q + q – 3 = 0

Multiply both sides by q: – 108 + q² – 3q = 0

Rearrange: q² – 3q – 108 = 0

Factor: (q – 12)(q + 9) = 0

Solve: q = 12 or -9 ⇒ it must be positive, so q = 12

Hideous and complicated! Let’s try working backwards. We can create a chart:

	q	p	q – 3	p + 1
A	12
B	10
C	9
D	8
E	6

Start with (B): 10 times what would equal 36? Nothing! Rule that out. Now try (C):

	q	p	q – 3	p + 1
A	12
B	10
C	9	4	6	5
D	8
E	6

6 × 5 is not 36, so rule that out. The number was too small, so try a bigger number, (A):

	q	p	q – 3	p + 1
A	12	3	9	4
B	10
C	9	4	6	5
D	8
E	6

9 × 4 = 36, so that works. (A) must be the answer! ?

Want full access to Céilidh’s trove of GMAT knowledge? Try the first class of one of her upcoming GMAT courses absolutely free, no strings attached.

Céilidh Erickson is a Manhattan Prep instructor based on New York City. When she tells people that her name is pronounced “kay-lee,” she often gets puzzled looks. Céilidh is a graduate of Princeton University, where she majored in comparative literature. After graduation, tutoring was always the job that bought her the greatest joy and challenge, so she decided to make it her full-time job. Check out Céilidh’s upcoming GMAT courses (she scored a 760, so you’re in great hands).

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