5.4 Application i - translating from verbal to mathetical expressions

Basic mathematics review Page 1 / 2

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules (<link document="m21980"/>) and (<link document="m21979"/>)). Objectives of this module: be able to translate from verbal to mathematical expressions.

Overview

Translating from Verbal to Mathematical Expressions

Translating from verbal to mathematical expressions

To solve a problem using algebra, we must first express the problem algebraically. To express a problem algebraically, we must scrutinize the wording of the problem to determine the variables and constants that are present and the relationships among them. Then we must translate the verbal phrases and statements to algebraic expressions and equations.

To help us translate verbal expressions to mathematics, we can use the following table as a mathematics dictionary.

Mathematics dictionary
Word or Phrase	Mathematical Operation
Sum, sum of, added to, increased by, more than, plus, and	$+$
Difference, minus, subtracted from, decreased by, less, less than	$-$
Product, the product of, of, muitiplied by, times	$\cdot$
Quotient, divided by, ratio	$\div$
Equals, is equal to, is, the result is, becomes	$=$
A number, an unknown quantity, an unknown, a quantity	$x$ (or any symbol)

Sample set a

Translate the following phrases or sentences into mathematical expressions or equations.

$\underset{6 + x}{\underset{︸}{\underset{6}{\underset{︸}{six}} \underset{+}{\underset{︸}{more than}} \underset{x}{\underset{︸}{a number}}}} .$

$\underset{15 - x}{\underset{︸}{\underset{15}{\underset{︸}{Fifteen}} \underline{\underset{︸}{minus}} \underset{x}{\underset{︸}{a number}}}} .$

$\underset{y - 8}{\underset{︸}{\underset{y}{\underset{︸}{A quantity}} \underline{\underset{︸}{less}} \underset{8}{\underset{︸}{eight}}}} .$

$\underset{2 x = 10}{\underset{︸}{\underset{2 \cdot}{\underset{︸}{Twice}} \underset{x}{\underset{︸}{a number}} \underset{=}{\underset{︸}{is}} \underset{10}{\underset{︸}{ten .}}}}$

$\underset{\frac{1}{2} z = 20}{\underset{︸}{\underset{\frac{1}{2}}{\underset{︸}{One half}} \underset{\cdot}{\underset{︸}{of}} \underset{z}{\underset{︸}{a number}} \underset{=}{\underset{︸}{is}} \underset{20}{\underset{︸}{twenty .}}}}$

$\underset{3 y = 5 + 2 y}{\underset{︸}{\underset{3}{\underset{︸}{Three}} \underset{\cdot}{\underset{︸}{times}} \underset{y}{\underset{︸}{a number}} \underset{=}{\underset{︸}{is}} \underset{5}{\underset{︸}{five}} \underset{+}{\underset{︸}{more than}} \underset{2 \cdot}{\underset{︸}{twice}} \underset{y}{\underset{︸}{the same number .}}}}$

Practice set a

Translate the following phrases or sentences into mathematical expressions or equations.

Eleven more than a number.

$11 + x$

Nine minus a number.

$9 - x$

A quantity less twenty.

$x - 20$

Four times a number is thirty two.

$4 x = 32$

One third of a number is six.

$\frac{x}{3} = 6$

Ten times a number is eight more than five times the same number.

$10 x = 8 + 5 x$

Sometimes the structure of the sentence indicates the use of grouping symbols.

Sample set b

Translate the following phrases or sentences into mathematical expressions or equations.

$\underset{\frac{x}{5} - 10 = 15}{\underset{︸}{\underset{(x \div 5)}{\underset{︸}{A number divided by five,}} \underline{\underset{︸}{minus}} \underset{10}{\underset{︸}{ten,}} \underset{=}{\underset{︸}{is}} \underset{15}{\underset{︸}{fifteen .}}}}$

Commas set off terms.

$\underset{The wording indicates this is to be considered as a single quantity.}{\underset{︸}{\underset{8}{\underset{︸}{Eight}} \underset{\div}{\underset{︸}{divided by}} \underset{(5 + x)}{\underset{︸}{five more than a number}} \underset{=}{\underset{︸}{is}} \underset{10}{\underset{︸}{ten}}}}$

$\frac{8}{5 + x} = 10$

$\underset{x (10 + x) = 20}{\underset{︸}{\underset{x}{\underset{︸}{A number}} \underset{\cdot}{\underset{︸}{multiplied by}} \underset{(10 + x)}{\underset{︸}{ten more than itself}} \underset{=}{\underset{︸}{is}} \underset{20}{\underset{︸}{twenty .}}}}$

A number plus one is divided by three times the number minus twelve and the result is four.
$\begin{matrix} (x + 1) \div (3 \cdot x - 12) & = & 4 \\ \frac{x + 1}{3 x - 12} & = & 4 \end{matrix}$
Notice that since the phrase "three times the number minus twelve" does not contain a comma, we get the expression $3 x - 12$ . If the phrase had appeared as "three times the number, minus twelve," the result would have been
$\frac{x + 1}{3 x} - 12 = 4$

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Source: OpenStax, Basic mathematics review. OpenStax CNX. Jun 06, 2012 Download for free at http://cnx.org/content/col11427/1.2

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