### Backsolving: Quantitative Reasoning Strategy

Backsolving is a strategic approach to solve quantitative reasoning questions that are composed of numerical value answer options (and not variables). This is a particularly useful strategy since many such questions on GMAT, GRE and SAT arrange these answer options in ascending or descending order, As a result, the number of tries needed to get to the correct answer is minimised.

## What is Backsolving?

When does backsolving work?

• When the answer options in a question are numbers
• Answer options don’t include variables
• When you can assume that the answer option is true and test the data in the problem statement.

The best approach to solve such questions is to work from the answers. Plug in the value from the answer to the problem statement to test the data. Work with each answer option until the desired result is reached. This can cut down on unnecessary calculation time and effort that the traditional approach would entail.

## Backsolving in action: illustrative example

### Now, give the following question a try

John spent 1/4 of his paycheque to repair his motorcycle, and then paid the registration and insurance, which each cost 1/3 of the remainder of his paycheque. John had \$0 before he received his check. He now has \$231 left. How much was his check for?

(A) \$ 2772
(B) \$ 1622
(C) \$ 924
(D) \$ 870
(E) \$ 693

Trying to arrive at the answer by ‘solving’ the question statement will be a long and annoying task. Instead, let’s understand what the question wants from us and let the answer options do all the heavy lifting.

### Solve strategically

Did you notice that the answer options are arranged in descending order? What this means is that we should be able to pick the right answer without going through too many iterations with the answer options.

What we will do now is plugin the dollar values from the answer options into the question stem and see which one satisfies the conditions (that we are left with 231 dollars after deductions).

But which option do we start with?

Since the answers are in descending (or ascending) order for questions such as this, it would be inefficient to start with the first option (or last one); If the correct value is the last option (or first), we might end up taking a lot of time to get there.

Instead, start with the middle option (C); this way you can eliminate a minimum of 3 answer options in one go. Either the required value needs to be larger than (C) [in which case you eliminate (C) and all option that are smaller] or smaller.

#### Let’s backsolve!

Let’s do the math. Starting with option C, let’s consider John’s paycheque to be \$924.

He spent 1/4 on the repair.

One-quarter of \$924 is \$231.

We subtract that from our original number, leaving us with \$693.

One-third of \$693 is \$231, so we subtract that for the registration, and again for the insurance (693 – 231 – 231).

This leaves us with \$231, this matches exactly with what the data tells us that John is left with. Option C is the correct response!

That was insanely quick wasn’t it? Let’s try a slightly more complex question. Try backsolving.

## Backsolving practice: illustrative example

In a class 1/5th chose Math, 1/4th Biology, ½ History and 10 psychology as optional subject. What is the number of students in the class?

(A) 80
(B) 110
(C) 160
(D) 200
(E) 400

If total number of students in the class is 160, then number of students who choose

Math = ?

Biology = ?

History = ?

Do the math, eliminate answer that cannot fit the requirement; when done check the explanation to see if you’ve applied backsolving correctly.

Solving for (C): If number of students = 160, then…

– Math is 1/5th = 32

– Biology is 1/4th = 40

– History is 1/2 = 80

Adding the 10 psychology students to the total number of math, biology and history students gives a grand total of 162 students. This is larger than the number we assumed when solving for (C).

What does this mean? 10 students is a larger value than what (C) allows. We therefore need to move to an option that is larger in value than (C).

Eliminate (C), (A) and (B)…

Move to (D): If number of students = 200, then…

– Math = 40

– Biology = 50

– History = 100

Totalling these up with 10 psychology students gives us (40 + 50 + 100 + 10) a grand total of 200 which matches the assumed total value of students.

We have a winner! Option D

Sure, we could solve these questions by using algebraic expressions or worse by creating linear equations, yet such manual calculations bring with them a host of troublesome situations where ‘silly’ mistakes can creep in. These ‘silly mistakes’ can be extremely dangerous on the day of the test.