Estimation and working from the answer choices

The choices are information

A problem-solving question is multiple choice, and only one choice is right. That is useful: you need enough work to tell the right choice from the others, and sometimes far less than a full solution. GMAT Quantitative Reasoning gives you 45 minutes for 21 questions without a calculator, so a method that replaces two minutes of arithmetic with thirty seconds of reasoning is worth learning on purpose.

This lesson covers three such methods: estimating, testing the choices, and picking numbers. Each is a real mathematical method, not a guess, and each has a way to go wrong.

Estimation: let the spread set the precision

Before estimating, look at how far apart the choices are.

• Choices differ by powers of ten (0.4, 4, 40, 400): you only need the size of the answer. Round hard.
• Choices are close (3.8, 4.0, 4.2): rounding will not separate them. Compute properly, or estimate with care about direction.

Round to numbers that are easy to handle, such as $49.6 \approx 50$, $0.198 \approx 0.2$, $2.47 \approx 2.5$, and keep track of which way each rounding moves the result. If you round one factor up, rounding another down partly cancels it; rounding everything the same way piles up the error.

For powers of ten, write each number in scientific notation and handle the powers separately:

$$\frac{6 \times 10^4 \times 5 \times 10^{-4}}{3 \times 10^{-2}} = \frac{30}{3 \times 10^{-2}} = 10 \times 10^{2} = 1000$$

Testing the choices (back-solving)

When the question asks for a single number and the choices are numbers, you can put each choice into the conditions of the problem and see which one works. Two habits make it fast:

• Start with the middle value. When the choices are in order, a middle choice that comes out too big or too small tells you which direction to move, so you rarely test more than two or three.
• Test against the original wording, not against an equation you may have set up wrongly.

Back-solving shines when setting up the equation is harder than checking a value: quadratics with integer answers, age problems, problems with conditions such as "consecutive odd integers".

Picking numbers

When the choices are expressions in variables, pick simple values for the variables, compute the answer from the story, then evaluate each choice with the same values. The choice that matches is the answer.

Good picks are small, different from each other, and make the arithmetic clean. Avoid 0 and 1, which make many expressions equal (any power of 1 is 1; anything times 0 is 0), and avoid giving two variables the same value. Respect every condition in the stem: if $x$ must be an odd integer, pick 3, not 2.

Test every choice. If two choices match your numbers, the pick was unlucky: pick new numbers and test only those two again.

Picking numbers also handles percent and ratio questions with no actual amounts: start from 100 for a percent question, or from a common multiple of the denominators for a fraction question.

Quick elimination

Even when you plan to compute, a few checks can strike out choices first:

• Sign: must the answer be positive?
• Size: is it more or less than a value in the stem? A discounted price cannot exceed the original.
• Type: must it be an integer, or even, or divisible by 3?
• Units digit: the units digit of a product depends only on the units digits of the factors.

When not to use these methods

If the algebra takes one line, do the algebra. These methods are tools for when the direct route is long or error-prone, not a replacement for it. And an estimate is never safe when two choices sit within the error of your rounding.