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We each have our strong areas. Some of us are experts at mathematics, choosing to take courses in the maths and sciences in our Plus-2s. Others of us, though, find our strength in languages, and then see the numbers float off the page and do funny dances around us when we’re faced with maths questions. The unfortunate truth is this: to master the SAT and ACT, we need to be able to do well on both the maths and the verbal sections. Thus far, I have given many tips to help those people who might stumble with verbal, but I haven’t done so with maths.
One of my favourite tricks for dealing with some tough maths problem solving questions (i.e. multiple choice) is to Backsolve (there are multiple names for this technique, but “Backsolve” is the most commonly used). The concept behind Backsolving is simple: The test wants you to come up with the answer for the questions it gives you, right? But wait...the test also gives you potential answers. So, why come up with the answer yourself? Instead, take a shortcut and use the choices given to you to determine what’s right.
The strategy for Backsolving is relatively simple:
Okay, perhaps this doesn’t look so simple. But it will make sense when we try it out. But first, we need to figure out
How do we know whether the question can be solved through this technique? Well, there are really only two types of questions suitable. The first is...
We’ve all seen questions like this, where they give us a fairly ugly looking equation and ask us what the variable equals. If the algebra looks easy enough for you to solve, then please go ahead. But if not, then use Backsolving.
Q1. If m+3m−5−m+7m−9=0, what is the value of m?
We have an ugly looking equation that can be solved in two ways. First, we could move the second rational equation (equation in fractions) to the other side of the equal sign, then cross multiply, and then solve. Or, we can recognise that this question type essentially fits the ‘X Equals’ format, thus making it available for Backsolving. So, let’s do this step-by-step:
Understand what the answer choices represent in the question and how they can be plugged in Here, the answer choices simply represent the value of ‘m,’ and thus can simply be plugged into the equation for ‘m.’
Plug in answer choice B or D into the question (choice B/F or C/G on the ACT) I look at choices B and D and ponder which one makes sense to plug in for ‘m.’ For me, at this stage, I figure that it would be easier to calculate 1 rather than -1. So, I start with D.
See if the question works out; if yes, then you have your answer Plugging in 1 for ‘m’ gets me the following: 4−4−8−8. This ends up simplifying to -1−(-1), or 0. Perfect! I don’t need to do the remaining steps, because I have found my answer. I can bubble it in and move on to the next question. Let’s try this technique on one more question. This time, we’ll mix rational and radical equations, thus making the question more complicated.
Q2. What does a equal if a+2−a2=1?
Determine whether the question can be solved through Backsolving This is an ‘X Equals’ style question, so yes. Okay, I suppose I could do the algebra, but it doesn’t look like it’ll be pretty.
Understand what the answer choices represent in the question and how they can be plugged in As before, the answer choices represent the variable in the
equation, so I can just plug them in.
Plug in answer choice B or D into the question (choice B/F or C/G on the ACT) D seems a bit large and thus more difficult to calculate than B. So, I’ll start with B.
See if the question works out; if yes, then you have your answer I immediately see a problem. While 4 divides evenly into 2, the square-root of 6 is not an integer. And in order to get a difference of 1, we need to have only integers. So, we can eliminate B without fully calculating it.
If it doesn’t work out, decide if you need to plug in a larger or smaller number. Honestly, I don’t know. I think it doesn’t really come down to a larger or smaller number, but instead I need a number that, when added to 2, is a perfect square, and when divided by 2, is an integer.
Depending on what you need and how the answers are organised, choose your answer or plug in another choice and then see what happens
So, I know what I need. And I see that the numbers are organised in ascending order, but that doesn’t really matter to me. I look through the answers and can immediately eliminate C and D because they are not multiples of 2. Both A and D will allow me to get a perfect square. Again, it seems that A would be easier to work with, so we plug 2 into the equation, and voilà! We get the right answer!
Is this more work than doing the algebra? Sometimes, yes. Practice both your Backsolving and your algebra skills between now and the exam, and then you will be ready to pick the right strategy based on the question.
As you could see, it seemed that Steps 5 and 6 above did not prove very helpful, and in fact, you may still not fully understand what they are or why we pick B or D instead of other answer choices. Well, next week we’ll look at the second (and more common) situation in which you should use Backsolving. It will make sense then. And with enough practice of this technique, you will be ready by test day to tackle these questions even if you haven’t studied maths for years!
Last week, we started discussing one technique for zipping through Problem Solving maths questions: Use the answers to get the answers. I introduced the first of two reasons to Backsolve. When you have a question that asks for the value of X, you know you can plug in the answer choices and push it through the equation given in the question. Hopefully the answer choice you pick first will give you the solution you seek.
The second and more common usage of Backsolving is with certain types of word problems. But before we talk about these in more detail, let’s review the strategy:
Last time, we didn’t really make use of Steps 5 and 6. But today, we’ll see them become an integral part of our time saving strategy.
Word problems...these tend to trip a lot of us up. Even with people like me who have been doing this test prep stuff for a while, word problems can suck. Undeniably. Definitely. I can get mired down in all the words, trying to figure out how to translate from what’s written to what’s mathematical. Often times, I simply don’t know how to begin. But thankfully, Backsolving allows me to forgo all that and instead gives me an easy starting point.
Let’s prove this with an example.
Q1. A box of widgets contains 1 damaged widget for every 40 widgets in the box. Three out of every 5 damaged widgets are considered not fit to sell, and every widget that is not fit to sell is damaged. If there are 27 widgets not fit to sell in the box, how many widgets are there in the box?
Determine whether the question can be solved through Backsolving Yes, because it’s a word problem with a pretty simple question. This is essentially asking “What is the value of w,” where w represents the number of widgets in a box. So, no problem at all. It wouldn’t work if the question were asking something more complex, like “What is the difference between the total number of” blah blah blah... Automatically, the word “difference” tells us that this won’t be so simple, and so it may be easier to work forward rather than to Backsolve.
Understand what the answer choices represent in the question and how they can be plugged in As I wrote above, the answer choices represent the total number of widgets in a box. I can plug the value into the first sentence. By dividing it by 40, I can determine the number of damaged widgets, which can then be multiplied by 3/5. Hopefully the answer will equal the magic solution of 27!
Plug in answer choice B or D into the question (choice B/F or C/G on the ACT)
Okay, before I actually start to do the calculations, let’s pause for a moment to understand WHY we start with B or D. I want you to look carefully at the five answer choices above. What do you notice about how they’re organised? That’s right! Ascending order. So, let’s say I choose to plug in Choice B and end up with a number larger than 27. What would the only possible answer be? Exactly! It would have to be choice A. Do I need to check? Only if I have time. Otherwise, I pick A and move on.
Essentially, I have a 40% chance of getting the question right with just the first pick. THAT’S why we start with B or D.
So, let’s actually try this out. I’ll start with B – 1680. That represents the number of widgets I have in a box. Given that one of every 40 is defective, I now have 42 bad widgets. And since three of every five defective widgets is not fit to sell, that gives me 25.2.
See if the question works out; if yes, then you have your answer Well, it’s not the 27 we were seeking. Alas, we do not have our answer.
If it doesn’t work out, decide if you need to plug in a larger or smaller number.
Since we got a number of non-sellable widgets that was lower than what we wanted, it’s safe to say that we need a larger number of widgets in a box. Because the answers are written in ascending order, we can automatically eliminate choices A and B (if they were written in descending order, A would be our answer!). If we didn’t have any more time on test day, we’d just pick one of the three remaining answers and move on. I would guess C because the 25.2 from B is not too far off from the coveted 27.
But since we have time, we’ll go all the way!
Depending on what you need and how the answers are organised, choose your answer or plug in another choice and then see what happens.
Okay, we need a larger number. And the choices are written in ascending order. So, we move on to choice D. Plug it into our question and...now you know what to do. What answer do you get? Did you get 29.4? Great!
But is that the answer we seek? No. We now have too many unsellable widgets! That makes D and E wrong. So, what is the only remaining answer? C
Hopefully you didn’t find this process to be too tough. Personally, I love Backsolving with word problems because I don’t have to worry about translating from English to maths, and I don’t have to waste my time finding a starting point in the question itself.
The most important this is to recognise whether a given problem is actually suitable for Backsolving. The answer choices have to easily fit into the question. Or, put another way, if it takes me more than a couple of seconds to figure out how to plug the answer choices into the question, then this is NOT suitable for
Backsolving. For example:
Why don’t these work? It’s simple. They all require the usage of TWO different values, thus making it impossible to know exactly how to plug these choices into the word problem. Now it’s your turn. Try your hand at this question. Yes, you can solve it through normal calculations... but attempt the Backsolving instead. Plug in B or D. Test it out. See if you need to move up or down (based on how the answers are arranged).
Eliminate as you go. Get the answer...
Q2. Square A is inscribed in a circle, which is then inscribed in Square B. If the length of the diagonal of Square B is 8, what is the area of Square A?
Now that you’ve learned how to Backsolve, practice it. As you’re doing questions and questions and questions, whenever you see a Word Problem with a clear unknown, try working backwards from the answer choices. If you keep on doing this, you’ll become an expert in Backsolving and thus save yourself a lot of time and effort on test day.