Step 3: Approach Strategically—Backsolving

Backsolving is just like picking numbers, except instead of coming up with the number yourself, you use the numbers in the answer choices. You’ll literally work backward through the problem, looking for the answer choice that agrees with the information in the question stem. This is a good approach whenever the task of plugging a choice into the question would allow you to confirm its details in a straightforward way.

You want to backsolve systematically, not randomly. Start with either (B) or (D). If the choice you pick isn’t correct, you’ll often be able to figure out whether you need to try a number that’s larger or one that’s smaller. Since numerical answer choices will always be in ascending or descending order, you’ll be able to eliminate several choices at once.

Backsolving can save you a great deal of time. It is also an exceptional approach when you have no idea how to begin a problem.

Backsolving with Word Problems

Example:

  1. Ron begins reading a book at 4:30 p.m. and reads at a steady pace of 30 pages per hour. Michelle begins reading a copy of the same book at 6:00 p.m. If Michelle started 5 pages behind the page that Ron started on and reads at an average pace of 50 pages per hour, at what time would Ron and Michelle be reading the same page?
    1. 7:00 p.m.
    2. 7:30 p.m.
    3. 7:45 p.m.
    4. 8:00 p.m.
    5. 8:30 p.m.

You could perhaps set up a complex system of equations to solve this problem. Even if you knew exactly what those equations would be and how to solve them, that’s not a very efficient use of your time. Backsolving will work better here. Pick an answer choice and see whether Michelle and Ron are on the same page at that time. There’s no compelling reason to prefer one choice to another. So just quickly choose (B) or (D).

Let’s say you choose (B). On what page is Ron at 7:30 p.m.? He started reading at 4:30 p.m., so he’s been reading for 3 hours. His pace is 30 pages per hour. So he’s read 30 × 3, or 90 pages. Since Michelle started 5 pages behind Ron, she’d need to read 95 pages to be at the same place. She’s been reading since 6:00 p.m., so she’s read for 1.5 hours. At 50 pages per hour, she’s read 75 pages. That’s 20 short of what she needs. So (B) is not the right answer.

Since Michelle is reading faster than Ron, she’ll catch up to him with more time. Therefore, they’ll be on the same page sometime later than 7:30 p.m., so you should try an answer choice that gives a later time. The most strategic answer choice to turn to at this point is choice (D). If (D) ends up being correct, you can choose it and move on. If (D) is too late, then (C) must be the answer. If (D) is too early, then the correct choice must be (E).

Let’s try out (D). At 8:00 p.m., Ron has read for 3.5 hours. At a pace of 30 pages per hour, he’s read 30 × 3.5, or 105 pages. Since Michelle started 5 pages behind, she’d need to read 110 pages to be at the same place. She’s been reading for 2 hours at this point; at 50 pages per hour, she’s read 100 pages. That’s still 10 short of what she needs to catch up. So (D) is also not the right answer, and you need a later time than 8:00 p.m.—choice (E) must be correct.

When you start with either (B) or (D), you’ll have picked the right answer 20% of the time. Another 20% of the time, you’ll know the right answer by process of elimination without ever having to test another choice.

Sometimes you may have to test more than one choice. But as you saw above, you should never have to test more than two answer choices. Stick to (B) or (D), and you’ll save valuable time and worry.

Example:

  1. A crate of apples contains 1 bruised apple for every 30 apples in the crate. Three out of every 4 bruised apples are considered not fit to sell, and every apple that is not fit to sell is bruised. If there are 12 apples not fit to sell in the crate, how many apples are there in the crate?
    1. 270
    2. 360
    3. 480
    4. 600
    5. 840

Start backsolving with choice (B). Suppose that there are 360 apples in the crate. Then apples, or 12 apples, are bruised. Then of those 12 apples, or 9 apples, are unsalable. This is too few unsalable apples. So (B) is too small. The answer must be larger than 360, so (A) and (B) are eliminated. Regardless of what you might suspect the answer to be, you should next test (D). If (D) is not right, you’ll know whether it’s too large—in which case (C) would be correct—or too small—in which case (E) would be correct. No matter what, you’ll only have to test one more choice.

Testing (D), suppose that there are 600 apples in the crate. Then apples, or 20 apples, are bruised. Of those 20, , or 15, are unsalable. That’s too many. So (D) and (E) are both out, proving that (C) is correct.

Backsolving works for more than just word problems. You can use it whenever you’re solving for a single variable in the question stem.

Example:

  1. What is the value of x if ?
    1. −2
    2. −1
    3. 0
    4. 1
    5. 2

Since (D) looks easier to work with than (B), start with that choice.

Therefore, (D) is correct, and you don’t have to test any more choices.

Applying the Kaplan Method: Backsolving

Now let’s use the Kaplan Method on a Problem Solving question that lends itself to backsolving:

  1. At a certain zoo, the ratio of sea lions to penguins is 4 to 11. If there are 84 more penguins than sea lions at the zoo, how many sea lions are there?
    1. 24
    2. 36
    3. 48
    4. 72
    5. 132
Step 1: Analyze the Question

A zoo has more penguins than sea lions, and you’re given both a ratio and a numerical difference between them.

Step 2: State the Task

You need to figure out the number of sea lions. You should jot this down in your scratchwork to help you keep track of the task.

Step 3: Approach Strategically

Backsolving is an option whenever you can manageably plug an answer choice into the question stem. After doing so, you merely confirm whether that value is consistent with the other values given in the stem.

Let’s say you start by backsolving choice (B). You’re pretending that there are, in fact, 36 sea lions. So “the ratio of sea lions to penguins is 4 to 11” becomes “the ratio of 36 to penguins is 4 to 11,” or 36:penguins is 4:11. If you don’t see the number of penguins right away, consider rewriting the ratio vertically:

You multiply the proportional value (4) by 9 to get the actual value (36). So the actual number of penguins is 11 × 9 = 99.

Is that consistent with the rest of the information? There should be 84 more penguins than sea lions, but 99 − 36 = 63. If (B) were correct, there would be only 63 more penguins. You can eliminate (B). Do you need more or fewer sea lions? Well, you need to increase the difference between them. Since the animals are in a ratio of 4:11, every time you remove 4 sea lions, you’d remove 11 penguins, shrinking the difference between them by 7. So you definitely need more sea lions—every time you add 4 sea lions, you add 11 penguins, increasing the difference by 7. Eliminate (A) as well and test (D).

You multiply the proportional value (4) by 18 to get the actual value (72). So the actual number of penguins is 11 × 18 = 198. (Or you could more simply say that since you’ve doubled the number of sea lions from what you had in choice (B), you must double the number of penguins as well.)

Is that consistent with the rest of the information? There should be 84 more penguins than sea lions. But 198 − 72 = 126. If (D) were correct, there would be 126 more penguins. That’s too many more, so (D) is eliminated. You need a smaller difference, which means you need a smaller number of sea lions. The answer must be (C).

Step 4: Confirm Your Answer

If you had accidentally answered (E), which is the number of penguins, this step would save you from a wrong answer.

Did you start this problem by adding 84 to the number of sea lions in one of the answer choices and then checking whether it was consistent with the given ratio? That’s fine, too. If you had trouble figuring out whether you needed to increase or decrease the number of sea lions, you could have either checked all five answer choices, stopping when you found the right ratio, or changed your approach to the question.

Practice Set: Backsolving

  1. If  , x =



    4. −2
    5. −3
  2. The length of a rectangle is twice as long as the width, and the rectangle’s area is greater than 72 and less than 200. Which of the following could be the length of the rectangle?
    1. 8
    2. 12
    3. 16
    4. 20
    5. 24
  3. In a certain beanbag game, players toss beanbags at a board with a hole in it. If the beanbag falls into the hole, the player scores 5 points. A beanbag that lands on the board but misses the hole is worth 2 points. If a toss misses the board entirely, 1 point is deducted from the player’s score. Julianna had 20 tosses and missed the board with 3 throws. Her total score was 52. How many times did she toss the beanbag into the hole?
    1. 5
    2. 6
    3. 7
    4. 8
    5. 9