Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

Example 2.20

Translate and solve: The number 143 is the product of −11 and \(y\).

Solution

Begin by translating the sentence into an equation.
Translate.

Translate.

\( \underbrace{\text{The number 143}}_{143} \underbrace{\text{is}}_{=} \underbrace{\text{the product of -11 and y}}_{-11y} \)
Divide by −11. \(\frac{143}{-11} = \frac{-11y}{-11}\)
Simplify. \(-13=y\)

Check:

\(143=−11y \)

\(143 \stackrel {?} {=} −11(−13)\)

\(143=143\) ✓    

Try It 2.39

Translate and solve: The number 132 is the product of −12 and \(y\).

Try It 2.40

Translate and solve: The number 117 is the product of −13 and \(z\).

Example 2.21

Translate and solve: \(n\) divided by 8 is −32.

Solution

Begin by translating the sentence into an equation.

Translate.

\(  \underbrace{\text{n divided by 8}}_{\frac{n}{8}} \underbrace{\text{is}}_{=} \underbrace{-32}_{-32} \)
Multiple both sides by 8. \(8 \cdot \frac{n}{8} = 8(-32)\)
Simplify. \(n=-256\)

Check:

 Is \(n\) divided by 8 equal to −32?
Let \(n=−256\).
Is −256 divided by 8 equal to −32?
Translate. \(\frac{-256}{8} \stackrel {?} {=}-32\)
 Simplify. \(−32=−32\) ✓
Try It 2.41

Translate and solve: n divided by 7 is equal to −21.

Try It 2.42

Translate and solve: n divided by 8 is equal to −56.

Example 2.22

Translate and solve: The quotient of y and −4 is 68.

Solution
Translate.

\(  \underbrace{\text{The quotient of y and -4}}_{\frac{y}{-4}} \underbrace{\text{is}}_{=} \underbrace{68}_{68} \)
Multiple both sides by -4. \(-4 (\frac{y}{-4}) = -4(68)\)
Simplify. \(y=-272\)

Check:

 Is the quotient of \(y\) and −4 equal to 68?
Let \(y=−272\).
Is the quotient of −272 and −4 equal to 68?
Translate. \(\frac{-272}{-4} \stackrel {?} {=}68\)
 Simplify. \(68=68\) ✓

Try It 2.43

Translate and solve: The quotient of \(q\) and −8 is 72.

Try It 2.44

Translate and solve: The quotient of \(p\) and −9 is 81.

Example 2.23

Translate and solve: Three-fourths of \(p\) is 18.

Solution

Begin by translating the sentence into an equation. Remember, "of" translates into multiplication.

Translate.  

\( \underbrace{\text{Three-fourths of p}}_{\frac{3}{4}p} \underbrace{\text{is}}_{=} \underbrace{18}_{18} \)
Multiple both sides by \(\frac{4}{3}\). \(\frac{4}{3} \cdot \frac{3}{4}p = \frac{4}{3} \cdot 18\)
Simplify. \(p=24\)

Check:

 Is three-fourths of \(p\) equal to 18?
Let \(p=24\).
Is three-fourths of 24 equal to 18?
Translate. \(\frac{3}{4} \cdot 24 \stackrel {?} {=}18\)
 Simplify. \(18=18\) ✓


Try It 2.45

Translate and solve: Two-fifths of \(f\) is 16.

Try It 2.46

Translate and solve: Three-fourths of \(f\) is 21.

Example 2.24

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

Solution

Begin by translating the sentence into an equation.

Translate.  

\( \underbrace{\text{The sum of three-eighths and x}}_{\frac{3}{8}} \underbrace{\text{is}}_{=} \underbrace{\frac{1}{2}}_{\frac{1}{2}} \)
Subtract \(\frac{3}{8}\) from both sides. \( \frac{3}{8} - \frac{3}{8}+x = \frac{1}{2}- \frac{3}{8} \)
Simplify and rewrite fractions with common denominators. \(x=\frac{4}{8}-\frac{3}{8}\)
Simplify.
 \(x=\frac{1}{8}\)

Check:

 Is the sum of three-eighths and \(x\) equal to one-half?
Let \(x=\frac{1}{8}\).
Is the sum of three-eighths and one-eighth equal to one-half?
Translate. \(\frac{3}{8} + \frac{1}{8} \stackrel {?} {=}\frac{1}{2}\)
Simplify. \(\frac{4}{8}\stackrel {?} {=}\frac{1}{2}\)
Simplify.
 \(\frac{1}{2}=\frac{1}{2}\) ✓

Try It 2.47

Translate and solve: The sum of five-eighths and \(x\) is one-fourth.

Try It 2.48

Translate and solve: The sum of three-fourths and \(x\) is five-sixths.

Back to '2.3: Multiplication/Division Property of Equations'