# 8.7 Applications  (Page 3/4)

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## Practice set e

It takes person A 4 hours less than person B to complete a certain task. Working together, both can complete the task in $\frac{8}{3}$ hours. How long does it take each person to complete the task working alone?

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Person A, 4 hr to complete the task; person B, 8 hr complete the task.

## Sample set f

The width of a rectangle is $\frac{1}{3}$ its length. Find the dimensions (length and width) if the perimeter is 16 cm.

Step 1:  Let $x$ = length. Then, $\frac{x}{3}=$ width.

Step 2:  Make a sketch of the rectangle.

The perimeter of a figure is the total length around the figure.
$\begin{array}{l}\begin{array}{ccccccc}\hfill & \hfill & x+\frac{x}{3}+x+\frac{x}{3}\hfill & =\hfill & 16\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 2x+\frac{2x}{3}& =\hfill & 16\hfill & \hfill & \hfill \\ \text{Step\hspace{0.17em}3:}\hfill & \hfill & \hfill 2x+\frac{2x}{3}& =\hfill & 16.\hfill & \hfill & \text{The\hspace{0.17em}LCD\hspace{0.17em}is\hspace{0.17em}3}\text{.}\hfill \\ \hfill & \hfill & 3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2x+3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2x}{3}\hfill & =\hfill & 3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}16\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 6x+2x& =\hfill & 48\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 8x& =\hfill & 48\hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill x& =\hfill & 6\hfill & \hfill & \text{Check\hspace{0.17em}this\hspace{0.17em}potential\hspace{0.17em}solution}\text{.}\hfill \\ \text{Step\hspace{0.17em}4:}\hfill & \hfill & 6+\frac{6}{3}+6+\frac{6}{3}\hfill & =\hfill & 16\hfill & \hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & 6+2+6+2\hfill & =\hfill & 16\hfill & \hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 16& =\hfill & 16\hfill & \hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct.}\hfill \\ \hfill & \hfill & \hfill \text{Since\hspace{0.17em}}x& =\hfill & 6,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{x}{3}=\frac{6}{3}=2\hfill & \hfill & \hfill \end{array}\hfill \\ \text{Step\hspace{0.17em}5}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{The\hspace{0.17em}length\hspace{0.17em}=\hspace{0.17em}6\hspace{0.17em}cm\hspace{0.17em}and\hspace{0.17em}the\hspace{0.17em}width\hspace{0.17em}=\hspace{0.17em}2\hspace{0.17em}cm}\text{.}\hfill \end{array}$

## Practice set f

The width of a rectangle is $\frac{1}{12}$ its length. Find the dimensions (length and width) if the perimeter is 78 feet.

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

length = 36 ft, width = 3 ft.

## Exercises

For the following problems, solve using the five-step method.

When the same number is added to both the numerator and denominator of the fraction $\frac{3}{7},$ the result is $\frac{2}{3}.$ What is the number?

When the same number is added to both the numerator and denominator of the fraction $\frac{5}{8},$ the result is $\frac{3}{4}.$ What is the number?

When the same number is added to both the numerator and denominator of the fraction $\frac{3}{8},$ the result is $\frac{1}{6}.$ What is the number?

The number added is $-2.$

When the same number is added to both the numerator and denominator of the fraction $\frac{7}{9},$ the result is $\frac{2}{3}.$ What is the number?

When the same number is subtracted from both the numerator and denominator of $\frac{1}{10},$ the result is $\frac{2}{3}.$ What is the number?

The number subtracted is $-17.$

When the same number is subtracted from both the numerator and denominator of $\frac{3}{4},$ the result is $\frac{5}{6}.$ What is the number?

One third of a number added to the reciprocal of number yields $\frac{13}{6}.$ What is the number?

$x=\frac{1}{2},6$

Four fifths of a number added to the reciprocal of number yields $\frac{81}{10}.$ What is the number?

One half of a number added to twice the reciprocal of the number yields 2. What is the number?

2

One fourth of a number added to four times the reciprocal of the number yields $\frac{-10}{3}.$ What is the number?

One inlet pipe can fill a tank in 8 hours. Another inlet pipe can fill the tank in 5 hours. How long does it take both pipes working together to fill the tank?

$3\frac{1}{13}\text{\hspace{0.17em}hours}$

One pipe can drain a pool in 12 hours. Another pipe can drain the pool in 15 hours. How long does it take both pipes working together to drain the pool?

A faucet can fill a bathroom sink in 1 minute. The drain can empty the sink in 2 minutes. If both the faucet and drain are open, how long will it take to fill the sink?

two minutes

A faucet can fill a bathtub in $6\frac{1}{2}$ minutes. The drain can empty the tub in $8\frac{1}{3}$ minutes. If both the faucet and drain are open, how long will it take to fill the bathtub?

An inlet pipe can fill a tank in 5 hours. An outlet pipe can empty the tank in 4 hours. If both pipes are open, can the tank be filled? Explain.

No. $x=-20$ hours.

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