Math Strategies for Firefighter Exam Study Guide

Backdoor Approaches for Answering Questions That Puzzle You

Remember those word problems you dreaded in high school? Many of them are actually easier to solve by backdoor approaches. The two techniques that follow are terrific ways to solve multiple-choice word problems that you don't know how to solve with a straightforward approach. The first technique, nice numbers, is useful when there are unknowns (like x) in the text of the word problem, making the problem too abstract for you. The second technique, working backward, presents a quick way to substitute numeric answer choices back into the problem to see which one works.

Nice Numbers

1. When a question contains unknowns, like x, plug nice numbers in for the unknowns. A nice number is easy to calculate with and makes sense in the problem, such as 5, 10, 25, or even 1 or 2.
2. Read the question with the nice numbers in place. Then solve it.
3. If the answer choices are all numbers, the choice that matches your answer is the right one.
4. If the answer choices contain unknowns, substitute the same nice numbers into all the answer choices. The choice that matches your answer is the right one. If more than one answer matches, do the problem again with different nice numbers. You will only have to check the answer choices that have already matched.

Example: Judi went shopping with p dollars in her pocket. If the price of shirts was s shirts for d dollars, what is the maximum number of shirts Judi could buy with the money in her pocket?

1. psd
2. 
3. 
4. 

To solve this problem, let's try these nice numbers: p = $100, s = 2; d = $25. Now reread it with the numbers in place:

Judi went shopping with $100 in her pocket. If the price of shirts was 2 shirts for $25, what is the maximum number of shirts Judi could buy with the money in her pocket?

Since 2 shirts cost $25, that means that 4 shirts cost $50, and 8 shirts cost $100. So your answer is 8. Let's substitute the nice numbers into all 4 answers:

1. 100 × 2 × 25 = 5000
2. 
3. 
4. 

The answer is b, because it is the only one that matches your answer of 8.

Working Backward

You can frequently solve a word problem by plugging the answer choices back into the text of the problem to see which one fits all the facts stated in the problem. The process is faster than you think because you will probably only have to substitute one or two answers to find the right one.

This approach works only when:

• All of the answer choices are numbers.
• You are asked to find a simple number, not a sum, product, difference, or ratio.

Here's what to do:

1. Look at all the answer choices and begin with one in the middle of the range. For example, if the answers are 14, 8, 20, and 25, begin by plugging 14 into the problem.
2. If your choice doesn't work, eliminate it. Determine if you need a bigger or smaller answer.
3. Plug in one of the remaining choices.
4. If none of the answers work, you may have made a careless error. Begin again or look for your mistake.

Example: Juan ate of the jellybeans. Maria then ate of the remaining jellybeans, which left 10 jellybeans. How many jellybeans were there to begin with?

1. 60
2. 90
3. 120
4. 140

Starting with one of the middle answers, let's assume there were 90 jellybeans to begin with:

Since Juan ate of them, that means he ate 30 ( × 90 = 30), leaving 60 of them (90 – 30 = 60).Maria then ate of the 60 jellybeans, or 45 of them ( × 60 = 45). That leaves 15 jellybeans (60 – 45 = 15).

The problem states that there were 10 jellybeans left, and you wound up with 15 of them. That indicates that you started with too big a number. Thus, 90, 120, and 140 are all wrong! So, let's try 60:

Since Juan ate of them, that means he ate 20 ( × 60 = 20), leaving 40 of them (60 – 20 = 40).Maria then ate of the 40 jellybeans, or 30 of them ( × 40 = 30). That leaves 10 jellybeans (40 – 30 = 10).

The right answer is a, because this result of 10 remaining jellybeans agrees with the problem.