Translating Words Involving Division and Multiplication Into an Algebraic Equation and Solving
Learning Outcomes
Translate a phrase that contains division into an equation and solve
Translate Sentences to Equations and Solve
Recall the four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference. We will use these to solve equations that contain fractions.
Subtraction Property of Equality:
For any real numbers a, b, and c,
if a=b, then a−c=b−c.
Addition Property of Equality:
For any real numbers a, b, and c,
if a=b, then a+c=b+c.
Division Property of Equality:
For any numbers a, b, and c, where c=0
if a=b, then ca=cb
Multiplication Property of Equality:
For any real numbers a, b, and c
if a=b, then ac=bc
When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.
In the next few examples, we’ll translate sentences that contain fractions into equations and then solve the equations. The first property of equality we will use is multiplication.
Example
Translate and solve: n divided by 6 is −24.
Solution:
Translate.
Multiply both sides by 6 .
6⋅6n=6(−24)
Simplify.
n=−144
Check:
Is −144 divided by 6 equal to −24 ?
Translate.
6−144=?−24
Simplify. It checks.
−24=−24✓
Try It
[ohm_question height="270"]146166[/ohm_question]
Example
Translate and solve: The quotient of q and −5 is 70.
Answer:
Solution:
Translate.
Multiply both sides by −5 .
5(−5q)=−5(70)
Simplify.
q=−350
Check:
Is the quotient of −350 and −5 equal to 70 ?
Translate.
−5−350=?70
Simplify. It checks.
70=70✓
Try It
[ohm_question height="270"]146172[/ohm_question]
Example
Translate and solve: Two-thirds of f is 18.
Answer:
Solution:
Translate.
Multiply both sides by 23 .
23⋅32f=23⋅18
Simplify.
f=27
Check:
Is two-thirds of 27 equal to 18 ?
Translate.
32(27)=?18
Simplify. It checks.
18=18✓
try It
[ohm_question height="270"]146180[/ohm_question]
Example
Translate and solve: The quotient of m and 65 is 43.
Answer:
Solution:
The quotient of m and 65 is 43 .
Translate.
65m=43
Multiply both sides by 65 to isolate m .
6565m=65(43)
Simplify.
m=6⋅45⋅3
Remove common factors and multiply.
m=85
Check:
Is the quotient of 85 and 65 equal to 43 ?
6585=?43
Rewrite as division.
85÷65=?43
Multiply the first fraction by the reciprocal of the second.
85⋅56=?43
Simplify.
43=43✓
Our solution checks.
Try It
[ohm_question height="270"]146184[/ohm_question]
Example
Translate and solve: The sum of three-eighths and x is three and one-half.
Answer:
Solution:
Translate.
Use the Subtraction Property of Equality to subtract 83 from both sides.
83+x−83=321−83
Combine like terms on the left side.
x=321−83
Convert mixed number to improper fraction.
x27−83
Convert to equivalent fractions with LCD of 8.
x=828−83
Subtract.
x=825
Write as a mixed number.
x=381
We write the answer as a mixed number because the original problem used a mixed number.
Check:
Is the sum of three-eighths and 381 equal to three and one-half?
We've seen several examples of how to translate a given mathematical relationship from words into equation form in order to solve it. Let's see some examples now that reverse the process. The following video shows examples of translating an equation into words as an aid to solving it. Note that this is different from the written examples on this page because these examples start with the mathematical equation then translate it into words.
https://youtu.be/tubom5d5lxg
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