This chapter covers ways of setting up word problems and solving for the unknowns.

EO 1.5Given a word problem, write equations and SOLVE for the unknown.

Basic Approach to Solving Algebraic Word Problems

Algebra is used to solve problems in science, industry, business, and the home. Algebraic equations can be used to describe laws of motion, pressures of gases, electric circuits, and nuclear facility operations. They can be applied to problems about the ages of people, the cost of articles, football scores, and other everyday matters. The basic approach to solving problems in these apparently dissimilar fields is the same. First, condense the available information into algebraic equations, and, second, solve the equations. Of these two basic steps, the first is frequently the most difficult to master because there are no clearly defined rules such as those that exist for solving equations.

Algebraic word problems should not be read with the objective of immediately determining the answer because only in the simpler problems is this possible. Word problems should be initially read to identify what answer is asked for and to determine which quantity or quantities, if known, will give this answer. All of these quantities are called the unknowns in the problem. Recognizing all of the unknowns and writing algebraic expressions to describe them is often the most difficult part of solving word problems. Quite often, it is possible to identify and express the unknowns in several different ways and still solve the problem. Just as often, it is possible to identify and express the unknowns in several ways that appear different but are actually the same relationship.

In writing algebraic expressions for the various quantities given in word problems, it is helpful to look for certain words that indicate mathematical operations. The words "sum" and "total" signify addition; the word "difference" signifies subtraction; the words "product," "times," and "multiples of" signify multiplication; the words "quotient," "divided by," "per," and "ratio" signify division; and the words "same as" and "equal to" signify equality. When quantities are connected by these words and others like them, these quantities can be written as algebraic expressions.

Sometimes you may want to write equations initially using words. For example, Bob is 30 years older than Joe. Express Bob's age in terms of Joe's.

Bob's age = Joe's age plus 30 years

If we let Bob's age be represented by the symbol B and Joe's age by the symbol J, this becomes B = J + 30 years

Examples:

Equations:

1. The total electrical output of one nuclear facility is 200 megawatts more than that of another nuclear facility.

Let L be the output of the larger facility and S the capacity of the smaller facility. The statement above written in equation form becomes L = 200MW+ S.

2. The flow in one branch of a piping system is one-third that in the other branch.

If B is the flow in the branch with more flow, and b is the flow in the smaller branch, this statement becomes the equation.

3. A man is three times as old as his son was four years ago.

Let M = man's age and S = son's age. Then M = 3 (S-4).

4. A car travels in one hour 40 miles less than twice as far as it travels in the next hour.

Let x_l be the distance it travels the first hour and x₂ the distance it travels the second then, x_l = (2) (x₂) -40.