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# More Difficult Word Problems Study Guide

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## Introduction to More Difficult Word Problems

Examinations are formidable even to the best prepared, for the greatest fool may ask more than the wisest man can answer.

—CHARLES CALEB COLTON, English writer (1780–1832)

This lesson introduces some "backdoor" techniques you may be able to use for word problems that appear too difficult to solve by a straightforward approach.

Many word problems are actually easier to solve by backdoor—indirect—approaches. These approaches work especially well on multiple-choice tests, but they can sometimes be used to answer word problems that are not presented in that format.

## Nice Numbers

Nice numbers are useful when there are unknowns in the text of the word problem (for example, g gallons of paint) that make the problem too abstract for you. By substituting nice numbers into the problem, you can turn an abstract problem into a concrete one. (See practice problems 1 and 8.)

Here's how to use the nice-numbers technique.

1. When the text of a word problem contains unknown quantities, plug in nice numbers for the unknowns. A nice number is one that is easy to calculate with and makes sense in the context of the problem.
2. Read the problem with the nice numbers in place. Then, solve the question it asks.
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, it's a "do–over" with different nice numbers. You only have to check the answer choices that have already matched.

Here's how to use the technique on a word problem.

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?

a. psd

b.

c.

d.

Solution:

Try these nice numbers:

p = \$100
s = 2
d = \$25

Substitute these numbers for the unknowns in the problem and in all the answer choices. Then reread the new problem and solve the question using your reasoning skills:

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?

a. 100 × 2 × 25 = 5,000

b.

c.

d.

Since 2 shirts cost \$25, that means that 4 shirts cost \$50, and 8 shirts cost \$100. Thus, the answer to our new question is 8. Answer b is the correct answer to the original question because it is the only one that matches our answer of 8.

## Working Backward

Working backward is a relatively quick way to substitute numeric answer choices back into the problem to see which one fits all the facts stated in the problem. The process is much faster than you think because you'll probably only have to substitute one or two answers to find the right one. (See practice problems 4, 14, and 15.)

This approach works only when:

• All of the answer choices are numbers.
• You're 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 the one in the middle of the range. For example, if the answers are 14, 8, 2, 20, and 25, begin by plugging 14 into the problem.
2. If your choice doesn't work, eliminate it. Take a few seconds to try to determine if you need a bigger or smaller answer. Eliminate the answer choices you know won't work because they're too big or too small.
3. Plug in one of the remaining choices.
4. If none of the answers works, you may have made a careless error. Begin again or look for your mistake.

#### Tip

Begin working backwards with the number that is the middle of all the answers. If that gives a number that is too big, work backwards with a smaller answer, or if it yields a number that is too small, begin plugging in the larger answers.

Here's how to solve the jelly bean problem from Lesson 15 by working backward:

Example: Carlos ate of the jelly beans. Maria then ate of the remaining jelly beans, which left10 jelly beans. How many jelly beans were there to begin with?

a. 60

b. 80

c. 90

d. 120

e. 140

Solution:

Start with the middle number: Assume there were 90 jelly beans to begin with.

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

The problem states that there were 10 jelly beans left, and we wound up with 15 of them. That indicates that we started with too big a number. Thus, 120 and 140 are also wrong because they're too big! With only two choices left, let's use common sense to decide which one to try first. The next smaller answer is 80, but it's only a little smaller than 90 and may not be small enough. So, let's try 60:

Since Carlos ate of the jelly beans, that means he ate 20 ( × 60 = 20), leaving 40 jelly beans for Maria (60 – 20 = 40).Maria then ate of the 40 jelly beans, or 30 of them ( × 40 = 30). That leaves 10 jelly beans (40 – 30 = 10). Our result (10 jelly beans left) agrees with the problem. The right answer is a.

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