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أدلة الدراسة > Prealgebra

Translating Words to Equations and Solving

Learning Outcomes

  • Translate a phrase that contains division into an equation and solve
  • Use the multiplication and division property of equality to solve equations that contain fractions

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,\mathit{\text{a, b,}} and c,\mathit{\text{c,}} if a=ba=b, then a−c=b−ca-c=b-c. Addition Property of Equality: For any real numbers a, b,\mathit{\text{a, b,}} and c,\mathit{\text{c,}} if a=ba=b, then a+c=b+ca+c=b+c.
Division Property of Equality: For any numbers a, b,\mathit{\text{a, b,}} and c,\mathit{\text{c,}} where c≠0\mathit{\text{c}}\ne \mathit{0} if a=ba=b, then ac=bc\frac{a}{c}=\frac{b}{c} Multiplication Property of Equality: For any real numbers a, b,\mathit{\text{a, b,}} and c\mathit{\text{c}} if a=ba=b, then ac=bcac=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: nn divided by 66 is −24-24. Solution:
Translate. .
Multiply both sides by 66 . 6⋅n6=6(−24)\color{red}{6}\cdot\frac{n}{6}=\color{red}{6}(-24)
Simplify. n=−144n=-144
Check: Is −144-144 divided by 66 equal to −24-24 ?
Translate. −1446=?−24\frac{-144}{6}\stackrel{?}{=}-24
Simplify. It checks. −24=−24-24=-24

Try It

  [ohm_question height="270"]146166[/ohm_question]

Example

Translate and solve: The quotient of qq and −5-5 is 7070.

Answer: Solution:

Translate. .
Multiply both sides by −5-5 . 5(q−5)=−5(70)\color{red}{5}(\frac{q}{-5}) = \color{red}{-5}(70)
Simplify. q=−350q=-350
Check: Is the quotient of −350-350 and −5-5 equal to 7070 ?
Translate. −350−5=?70\frac{-350}{-5}\stackrel{?}{=}70
Simplify. It checks. 70=7070=70

Try It

  [ohm_question height="270"]146172[/ohm_question]

Example

Translate and solve: Two-thirds of ff is 1818.

Answer: Solution:

Translate. .
Multiply both sides by 32\frac{3}{2} . 32⋅23f=32⋅18\color{red}{\frac{3}{2}}\cdot\frac{2}{3}f=\color{red}{\frac{3}{2}}\cdot 18
Simplify. f=27f=27
Check: Is two-thirds of 2727 equal to 1818 ?
Translate. 23(27)=?18\frac{2}{3}(27)\stackrel{?}{=}18
Simplify. It checks. 18=1818=18

try It

  [ohm_question height="270"]146180[/ohm_question]

Example

Translate and solve: The quotient of mm and 56\frac{5}{6} is 34\frac{3}{4}.

Answer: Solution:

The quotient of mm and 56\frac{5}{6} is 34\frac{3}{4} .
Translate. m56=34\frac{m}{\frac{5}{6}}=\frac{3}{4}
Multiply both sides by 56\frac{5}{6} to isolate mm . 56(m56)=56(34)\frac{5}{6}\left(\frac{m}{\frac{5}{6}}\right)=\frac{5}{6}\left(\frac{3}{4}\right)
Simplify. m=5⋅36⋅4m=\frac{5\cdot 3}{6\cdot 4}
Remove common factors and multiply. m=58m=\frac{5}{8}
Check:
Is the quotient of 58\frac{5}{8} and 56\frac{5}{6} equal to 34\frac{3}{4} ? 5856=?34\frac{\frac{5}{8}}{\frac{5}{6}}\stackrel{?}{=}\frac{3}{4}
Rewrite as division. 58÷56=?34\frac{5}{8}\div \frac{5}{6}\stackrel{?}{=}\frac{3}{4}
Multiply the first fraction by the reciprocal of the second. 58⋅65=?34\frac{5}{8}\cdot \frac{6}{5}\stackrel{?}{=}\frac{3}{4}
Simplify. 34=34\frac{3}{4}=\frac{3}{4}
Our solution checks.

Try It

  [ohm_question height="270"]146184[/ohm_question]

Example

Translate and solve: The sum of three-eighths and xx is three and one-half.

Answer: Solution:

Translate. .
Use the Subtraction Property of Equality to subtract 38\frac{3}{8} from both sides. 38+x−38=312−38\frac{3}{8}+x-\frac{3}{8}=3\frac{1}{2}-\frac{3}{8}
Combine like terms on the left side. x=312−38x=3\frac{1}{2}-\frac{3}{8}
Convert mixed number to improper fraction. x72−38x\frac{7}{2}-\frac{3}{8}
Convert to equivalent fractions with LCD of 88. x=288−38x=\frac{28}{8}-\frac{3}{8}
Subtract. x=258x=\frac{25}{8}
Write as a mixed number. x=318x=3\frac{1}{8}
We write the answer as a mixed number because the original problem used a mixed number. Check: Is the sum of three-eighths and 3183\frac{1}{8} equal to three and one-half?
38+318=?312\frac{3}{8}+3\frac{1}{8}\stackrel{?}{=}3\frac{1}{2}
Add. 348=?3123\frac{4}{8}\stackrel{?}{=}3\frac{1}{2}
Simplify. 312=3123\frac{1}{2}=3\frac{1}{2}
The solution checks.

try It

  [ohm_question height="270"]146189[/ohm_question] [ohm_question height="270"]146199[/ohm_question]

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