21

Signal Words and Phrases

Algebra is an efficient tool for solving word problems. You model the problems using algebraic symbolism in which variables represent unknown quantities. Then you use the rules and procedures of algebra to find solutions to the problems. To be successful in the modeling process, you need skills in translating everyday language into algebraic language.

Recall that you usually express a variable as an upper or lowercase letter (such as x, y, z, A, B, or C.)

This chapter presents common signal words and phrases that give clues to the translation of verbal expressions and statements into algebraic shorthand. From previous chapters, particularly Chapter 6, you have some familiarity with algebraic expressions. This chapter builds on what you already know (for example, that 5x means 5 times x) so that you can apply that knowledge in a practical setting.

Common Signal Words and Phrases for Addition

Examples of common signal words and phrases that indicate addition are shown in the following table. Just as in arithmetic, the plus sign (+) is the symbol for addition.

Problem   Translate each verbal expression into algebraic symbolism.

a. The sum of 5x and 40

b. 9 more than 6y

c. The total of 4x, 3y, and 7b

d. C increased by 120

e. 400 increased by 50x

Solution

a. The sum of 5x and 40

Step 1.   Identify the signal word or phrase.

The word sum indicates addition.

Step 2.   Translate the verbal expression.

5x + 40

b. 9 more than 6y

Step 1.   Identify the signal word or phrase.

The phrase more than indicates addition.

Step 2.   Translate the verbal expression.

6y + 9

c. The total of 4x, 3y, and 7b

Step 1.   Identify the signal word or phrase.

The phrase total of indicates addition.

Step 2.   Translate the verbal expression.

4x + 3y + 7b

d. C increased by 120

Step 1.   Identify the signal word or phrase.

The phrase increased by indicates addition.

Step 2.   Translate the verbal expression.

C + 120

e. 400 increased by 50x

Step 1.   Identify the signal word or phrase.

The phrase increased by indicates addition.

Step 2.   Translate the verbal expression.

400 + 50x

Common Signal Words and Phrases for Subtraction

Examples of common signal words and phrases that indicate subtraction are shown in the following table. Just as in arithmetic, the minus sign (−) is the symbol for subtraction. Notice in the table that the order of the terms in subtraction is important.

Problem   Translate each verbal expression into algebraic symbolism.

a. The difference of 50x and 75y

b. 98 decreased by 4w

c. 17b subtracted from 200

d. C less than 165

e. 600 reduced by 22A

Solution

a. The difference of 50x and 75y

Step 1.   Identify the signal word or phrase.

The word difference indicates subtraction.

Step 2.   Translate the verbal expression.

50x − 75y

b. 98 decreased by 4w

Step 1.   Identify the signal word or phrase.

The phrase decreased by indicates subtraction.

Step 2.   Translate the verbal expression.

98 − 4w

c. 17b subtracted from 200

Step 1.   Identify the signal word or phrase.

The phrase subtracted from indicates subtraction.

Step 2.   Translate the verbal expression.

200 − 17b

d. C less than 165

Step 1.   Identify the signal word or phrase.

The phrase less than indicates subtraction.

Step 2.   Translate the verbal expression.

165 − C

e. 600 reduced by 22A

Step 1.   Identify the signal word or phrase.

The phrase reduced by indicates subtraction.

Step 2.   Translate the verbal expression.

600 − 22A

Common Signal Words and Phrases for Multiplication

Examples of common signal words and phrases that indicate multiplication are shown in the following table. Three ways to denote multiplication in algebraic expressions are juxtaposition (that is, side-by-side placement), parentheses, and the dot multiplication symbol (⋅). Commonly, juxtaposition and parentheses are used when the factors involve variables. Use the dot multiplication symbol or parentheses when the factors are constants.

The times sign (×) is not used in algebraic expressions.

Use the word quantity to make your meaning clear, as in the quantity (z + 4).

Problem:   Translate each verbal expression into algebraic symbolism.

a. The product of 50x and 20

b. 56 times 8

c. 14x multiplied by 3y

d. 250 of z

e. 25% of 40B

Solution

a. The product of 50x and 20

Step 1.   Identify the signal word or phrase.

The word product indicates multiplication.

Step 2.   Translate the verbal expression.

(50x)(20)

Never use juxtaposition to show multiplication between constant factors. Writing the factors 56 and 8 side by side looks like 568, instead of the product, 56 times 8.

b. 56 times 8

Step 1.   Identify the signal word or phrase.

The word times indicates multiplication.

Step 2.   Translate the verbal expression.

56 ⋅ 8 or (56)(8)

c. 14x multiplied by 3y

Step 1.   Identify the signal word or phrase.

The phrase multiplied by indicates multiplication.

Step 2.   Translate the verbal expression.

(14x)(3y)

d. 250 of z

Step 1.   Identify the signal word or phrase.

The word of between two numerical quantities indicates multi­plication.

Step 2.   Translate the verbal expression.

250z

e. 25% of 40B

Step 1.   Identify the signal word or phrase.

The word of between two numerical quantities indicates multiplication.

Step 2.   Translate the verbal expression.

25%(40B)

Common Signal Words and Phrases for Division

Examples of common signal words and phrases that indicate division are shown in the following table. In algebraic expressions, the fraction bar is used to indicate division, as shown here: Observe in the table that the order of the parts in a division expression is important.

Neither of the division symbols, ÷ or is used in algebraic expressions.

Problem   Translate each verbal expression into algebraic symbolism.

Caution: Division by zero is undefined. For example, has no meaning.

a. The ratio of 600x to 125y

b. 34p divided by 17

c. 2,500 per 1,000K

d. The quotient of C and d

e. The quantity (2x + 1) divided by 4

Solution

a. The ratio of 600x to 125y

Step 1.   Identify the signal word or phrase.

The word ratio indicates division.

Step 2.   Translate the verbal expression.

b. 34p divided by 17

Step 1.   Identify the signal word or phrase.

The phrase divided by indicates division.

Step 2.   Translate the verbal expression.

c. 2,500 per 1,000K

Step 1.   Identify the signal word or phrase.

The word per indicates division.

Step 2.   Translate the verbal expression.

d. The quotient of C and d

Step 1.   Identify the signal word or phrase.

The word quotient indicates division.

Step 2.   Translate the verbal expression.

e. The quantity (2x + 1) divided by 4

Step 1.   Identify the signal word or phrase.

The phrase divided by indicates division.

Step 2.   Translate the verbal expression.

Common Signal Words and Phrases for Equality

Examples of common signal words and phrases that indicate equality are shown in the following table. Just as in arithmetic, the equal sign (=) is the symbol for equality.

Problem   Translate each verbal statement into algebraic symbolism.

Do not put periods at the end of algebraic symbolism for verbal statements.

a. 2w + 2(w + 5) yields 78.

b. (3n − 10) is equal to (2n + 5).

c. 15x + 6(300 − x) results in 9(300).

d.

e. C is πd.

Solution

a. 2w + 2(w + 5) yields 78.

Step 1.   Identify the signal word or phrase.

The word yields indicates equality.

Step 2.   Translate the verbal expression.

2w + 2(w + 5) = 78

b. (3n − 10) is equal to (2n + 5).

Step 1.   Identify the signal word or phrase.

The phrase is equal to indicates equality.

Step 2.   Translate the verbal statement.

(3n − 10) = (2n + 5)

c. 15x + 6(300 − x) results in 9(300).

Step 1.   Identify the signal word or phrase.

The phrase results in indicates equality.

Step 2.   Translate the verbal statement.

15x + 6(300 − x) = 9(300)

d.

Step 1.   Identify the signal word or phrase.

The word equals indicates equality.

Step 2.   Translate the verbal statement.

e. C is πd.

Step 1.   Identify the signal word or phrase.

The word is indicates equality.

Step 2.   Translate the verbal statement.

C = πd

A final note: The signal words and phrases in this chapter are by no means all-inclusive. However, they are representative of the kinds of words and phrases that typically occur in word problems.

Exercise 21

For 1—10, translate each verbal expression into algebraic symbolism.

1. The sum of 80x and 500

2. 125 increased by 40%x

3. The difference of P and 2w

4. K less than 100

5. 420 reduced by 5y

6. The product of 50x and 20

7. of 25M

8. The quotient of 525 and r

9. The ratio of 10d to 135

10. The quantity (5x − 3) divided by 5

For 11−15, translate each verbal statement into algebraic symbolism.

11. 6%B yields 57.60.

12. 0.25x + 0.10(42 − x) is equal to 5.55.

13. 55t + 65t results in 624.

14. c² equals 8² + 15².

15. (L + 13) is P.