{"id":11356,"date":"2016-04-21T17:15:06","date_gmt":"2016-04-21T17:15:06","guid":{"rendered":"http:\/\/www.manhattanprep.com\/gmat\/?p=11356"},"modified":"2019-09-05T16:00:00","modified_gmt":"2019-09-05T16:00:00","slug":"think-like-an-expert-how-when-to-work-backwards-on-gmat-problem-solving","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gmat\/blog\/think-like-an-expert-how-when-to-work-backwards-on-gmat-problem-solving\/","title":{"rendered":"Think Like an Expert: How & When to Work Backwards on GMAT Problem Solving"},"content":{"rendered":"

\"ManhattanDid you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We\u2019re not kidding! Check out our upcoming courses here<\/a>.
\n<\/strong><\/em><\/p>\n

\n

What does it take to be a GMAT expert? It\u2019s not just content knowledge (although of course that\u2019s necessary). A GMAT expert knows how to quickly identify patterns and choose quickly from a variety of strategies. In each of these segments, I\u2019ll show you one of these expert moves and how to use it.<\/em><\/p>\n

What working backwards is<\/strong><\/p>\n

Working backwards from the answer choices, back-solving, plug-and-chug\u2026 no matter what you call it, you\u2019ve probably heard of it before.\u00a0 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.<\/p>\n

Try this problem:<\/strong><\/p>\n

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?<\/p>\n

(A) 12<\/p>\n

(B) 10<\/p>\n

(C) 9<\/p>\n

(D) 8<\/p>\n

(E) 6<\/p>\n

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)*<\/p>\n

Here\u2019s what I hear from students all the time: \u201coh, now that you showed me, I can see that working backwards is easier. But I didn\u2019t even think to try it.\u201dI blame high school teachers who made us show all of our work every time! We\u2019re trained to think that algebra is the \u201cright\u201d way to do things, so we jump automatically into creating equations.<\/p>\n

So, you\u2019ll have to re-train yourself\u2026<\/p>\n

When are you allowed to work backwards?<\/strong><\/p>\n

Here\u2019s the rule, and it\u2019s pretty simple: you\u2019re allowed to work backwards from the answer choices any time a PS question asks you for the value of an unknown<\/em><\/strong> (a variable) \u2013 in other words, if you could write the question as \u201cx<\/em>= ?,\u201d you\u2019re allowed to work backwards.<\/p>\n

Working backwards usually does\u00a0not<\/em> 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.<\/p>\n

Consider the difference between these two ratio problems, and try working backwards for each:<\/p>\n\n\n\n
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?<\/p>\n
\n

(A) 18<\/p>\n

(B) 20<\/p>\n

(C) 25<\/p>\n

(D) 27<\/p>\n

(E) 30<\/td>\n

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 more kittens than puppies were originally in the shelter on Monday?<\/p>\n
\n

(A) 1<\/p>\n

(B) 2<\/p>\n

(C) 3<\/p>\n

(D) 4<\/p>\n

(E) 5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Question #1 asks for the original number of kittens, in other words the value<\/em> of an unknown: k<\/em>= ? We\u2019re 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).<\/p>\n

Question #2 is asking for the difference<\/em> between kittens and puppies. In other words, k<\/em> \u2013 p<\/em>= ? You probably had a lot more trouble working backwards on this one, so algebra probably was the most efficient strategy.<\/p>\n

Train yourself to recognize working-backwards-problems<\/strong><\/p>\n

If you\u2019re currently not using the strategy of working backwards because you \u201cdidn\u2019t even think about it,\u201d then you have to train yourself<\/em> to recognize the signals.<\/p>\n

Do this right now: grab a copy\u00a0 of the Official Guide<\/a>. (If you don\u2019t have one, you should definitely get one).<\/p>\n

Step 1: Flip open to the first page of the PS section. Glance through the questions, and without solving<\/em>, 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.<\/p>\n

I\u2019ll give you the first few from OG 2016: #3, #9, #12, #15, #19, #32. Now you practice recognizing the rest!<\/p>\n

Step 2: Once you\u2019re confident that you can recognize these problems, you can then go back and solve them by working backwards.<\/p>\n

Step 3: Go back and re-solve<\/em> 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\u2019re strengthening both muscles!<\/p>\n

How to work backwards efficiently<\/strong><\/p>\n

Let\u2019s go back to that first kittens-and-puppies problem. When you\u2019re working backwards, it\u2019s a good idea to create a chart<\/strong> to keep your information organized.<\/p>\n\n\n\n\n\n\n\n\n
\u00a0<\/em><\/td>\nkittens: 5x<\/td>\npuppies: 4x<\/td>\nkittens \u2013 7<\/td>\npuppies \u2013 8<\/td>\n<\/tr>\n
A<\/td>\n18<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
B<\/td>\n20<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
C<\/td>\n25<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
D<\/td>\n27<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
E<\/td>\n30<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Which answer choice should you start with? You\u2019ll 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\u2019s too big, the answer must be A, and you can avoid testing a 2nd<\/sup> value. If it\u2019s 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\u2019ve tested a maximum of 2 answers.<\/p>\n

My recommendation, though, is to start with your intuition<\/em>, then go with something else in the middle\u2026 whatever answer choice seems easiest. Just don\u2019t start with A and test all 5 in a row, because you\u2019ll be doing more work than you need to.<\/p>\n

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\u2019d start with C, because it\u2019s in the middle of the 3 answers I have left:<\/p>\n\n\n\n\n\n\n\n\n
\u00a0<\/em><\/td>\nkittens: 5x<\/td>\npuppies: 4x<\/td>\nkittens \u2013 7<\/td>\npuppies \u2013 8<\/td>\n<\/tr>\n
A<\/span><\/td>\n18<\/span><\/td>\n\u00a0<\/span><\/td>\n\u00a0<\/span><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
B<\/td>\n20<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
C<\/td>\n25<\/td>\n20<\/td>\n18<\/td>\n12<\/td>\n<\/tr>\n
D<\/span><\/td>\n27<\/span><\/td>\n\u00a0<\/span><\/td>\n\u00a0<\/span><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
E<\/td>\n30<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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!<\/p>\n

Strategies are like muscles: you have to train them<\/strong><\/p>\n

Expert athletes don\u2019t 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.<\/p>\n

Good luck!<\/p>\n

*The algebraic solution:<\/p>\n

Let p<\/em> = expected price and q<\/em> = expected quantity<\/p>\n

pq<\/em> = 36\u00a0 and\u00a0 (p<\/em> + 1)(q<\/em> \u2013 3) = 36<\/p>\n

FOIL: \u00a0 pq<\/em> \u2013 3p<\/em> + q<\/em> \u2013 3 = 36<\/p>\n

Isolate: p<\/em> = 36\/q<\/em><\/p>\n

Substitute: (36\/q<\/em>)q<\/em> \u2013 3(36\/q<\/em>) + q<\/em> \u2013 3 = 36<\/p>\n

Simplify:\u00a0 36 \u2013 108\/q<\/em> + q<\/em> \u2013 3 = 36<\/p>\n

Subtract 36 from both sides: \u2013 108\/q<\/em> + q<\/em> \u2013 3 = 0<\/p>\n

Multiply both sides by q<\/em>:\u00a0 \u2013 108 + q<\/em>2<\/sup> \u2013 3q<\/em> = 0<\/p>\n

Rearrange: q<\/em>2<\/sup> \u2013 3q<\/em> \u2013 108 = 0<\/p>\n

Factor:\u00a0 (q<\/em> \u2013 12)(q<\/em> + 9) = 0<\/p>\n

Solve:\u00a0 q<\/em> = 12 or -9 \u21d2 \u00a0it must be positive, so q<\/em> = 12<\/p>\n

Hideous and complicated! Let\u2019s try working backwards. We can create a chart:<\/p>\n\n\n\n\n\n\n\n\n
\u00a0<\/em><\/td>\nq<\/em><\/td>\np<\/em><\/td>\nq<\/em> \u2013 3<\/td>\np<\/em> + 1<\/td>\n<\/tr>\n
A<\/td>\n12<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
B<\/td>\n10<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
C<\/td>\n9<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
D<\/td>\n8<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
E<\/td>\n6<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Start with (B): 10 times what would equal 36? Nothing! Rule that out. Now try (C):<\/p>\n\n\n\n\n\n\n\n\n
\u00a0<\/em><\/td>\nq<\/em><\/td>\np<\/em><\/td>\nq<\/em> \u2013 3<\/td>\np<\/em> + 1<\/td>\n<\/tr>\n
A<\/td>\n12<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
B<\/td>\n10<\/span><\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
C<\/td>\n9<\/td>\n4<\/td>\n6<\/td>\n5<\/td>\n<\/tr>\n
D<\/td>\n8<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
E<\/td>\n6<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

6 \u00d7 5 is not 36, so rule that out. The number was too small, so try a bigger number, (A):<\/p>\n\n\n\n\n\n\n\n\n
\u00a0<\/em><\/td>\nq<\/em><\/td>\np<\/em><\/td>\nq<\/em> \u2013 3<\/td>\np<\/em> + 1<\/td>\n<\/tr>\n
A<\/td>\n12<\/td>\n3<\/td>\n9<\/td>\n4<\/td>\n<\/tr>\n
B<\/span><\/td>\n10<\/span><\/td>\n\u00a0<\/span><\/td>\n\u00a0<\/span><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
C<\/span><\/td>\n9<\/span><\/td>\n4<\/span><\/td>\n6<\/span><\/td>\n5<\/span><\/td>\n<\/tr>\n
D<\/td>\n8<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
E<\/td>\n6<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

9 \u00d7 4 = 36, so that works. (A) must be the answer!\u00a0?<\/p>\n

\n

Want full access to\u00a0C\u00e9ilidh\u2019s trove of GMAT knowledge? Try the first class of one of\u00a0her upcoming GMAT courses<\/a> absolutely free, no strings attached.\u00a0<\/strong><\/p>\n

\n

\"ceilidh-erickson-Manhattan-Prep-GMAT-Instructor\"<\/a>C\u00e9ilidh Erickson<\/a>\u00a0is a Manhattan Prep instructor based on New York City.<\/strong>\u00a0When she tells people that her name is\u00a0pronounced \u201ckay-lee,\u201d she often gets puzzled looks.\u00a0C\u00e9ilidh 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 C\u00e9ilidh\u2019s upcoming GMAT courses<\/a>\u00a0(she scored a 760, so you\u2019re in great hands).<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We\u2019re not kidding! Check out our upcoming courses here. What does it take to be a GMAT expert? It\u2019s not just content knowledge (although of course that\u2019s necessary). A GMAT expert knows how to […]<\/p>\n","protected":false},"author":28,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,873,929,24,8],"tags":[52828,619,52753],"yst_prominent_words":[],"class_list":["post-11356","post","type-post","status-publish","format-standard","hentry","category-manhattan-gmat-blog-algebra","category-for-current-studiers","category-gmat-prep","category-problem-solving","category-quant-on-gmat","tag-back-solving","tag-problem-solving-2","tag-working-backwards"],"_links":{"self":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/11356","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/users\/28"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/comments?post=11356"}],"version-history":[{"count":6,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/11356\/revisions"}],"predecessor-version":[{"id":11737,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/11356\/revisions\/11737"}],"wp:attachment":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/media?parent=11356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/categories?post=11356"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/tags?post=11356"},{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/yst_prominent_words?post=11356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}