Use Direct and Inverse Variation

When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation and inverse variation in this section.

Solve Direct Variation Problems

Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation

s = 15 h

Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.

In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find values of y for other values of x.

How to Solve Direct Variation Problems

If y varies directly with x and $y = 20$ when $x = 8$ , find the equation that relates x and y.

Solution

The above image has 3 columns. The table shows the steps to solve direct variation problems. Step one is to write the formula for the direct variation. The direct variation formula is y equals k x. Then we get y equals k times x.

Step two is substitute the given values for the variables. We are given y equals 20 and x equals 8. Then we have 20 equals k times 8.

Step three is to solve for the constant variation. Divide both sides of the equation by 8, then multiply. We now get 20 divided by 8 equals k. K equals 2.5.

Step four is to write the equation that relates x and y. Rewrite the general equation with the value we found k to get y equals 2 and five-tenths times x.

We’ll list the steps below.

Now we’ll solve a few applications of direct variation.

When Raoul runs on the treadmill at the gym, the number of calories, c, he burns varies directly with the number of minutes, m, he uses the treadmill. He burned 315 calories when he used the treadmill for 18 minutes.

ⓐ Write the equation that relates c and m.
ⓑ How many calories would he burn if he ran on the treadmill for 25 minutes?

Solution

ⓐ

	The number of calories, $c$ , varies directly with the number of minutes, $m$ , on the treadmill, and $c = 315$ when $m = 18$ .
Write the formula for direct variation.
We will use $c$ in place of $y$ and $m$ in place of $x$ .
Substitute the given values for the variables.
Solve for the constant of variation.

Write the equation that relates $c$ and $m$ .
Substitute in the constant of variation.

ⓑ

	Find $c$ when $m = 25$ .
Write the equation that relates $c$ and $m$ .
Substitute the given value for $m$ .
Simplify.
	Raoul would burn 437.5 calories if he used the treadmill for 25 minutes.

In the previous example, the variables c and m were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do in most applied problems.

The number of gallons of gas Eunice’s car uses varies directly with the number of miles she drives. Last week she drove 469.8 miles and used 14.5 gallons of gas.

ⓐ Write the equation that relates the number of gallons of gas used to the number of miles driven.
ⓑ How many gallons of gas would Eunice’s car use if she drove 1000 miles?

Solution

	The number of gallons of gas varies directly with the number of miles driven.
First we will name the variables.	Let $g =$ number of gallons of gas. $m =$ number of miles driven
Write the formula for direct variation.
We will use $g$ in place of $y$ and $m$ in place of $x$ .
Substitute the given values for the variables.

Solve for the constant of variation.
We will round to the nearest thousandth.
Write the equation that relates $g$ and $m$ .
Substitute in the constant of variation.

ⓑ

\begin{matrix} Find g when m = 1000 . \\ Write the equation that relates g and m . & g = 0.031 m \\ Substitute the given value for m . & g = 0.031 (1000) \\ Simplify. & g = 31 \\ Eunice’s car would use 31 gallons of gas if she drove it 1,000 miles. \end{matrix}

Notice that in this example, the units on the constant of variation are gallons/mile. In everyday life, we usually talk about miles/gallon.

In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is $y = k x^{2}$ . We solve these applications just as we did the previous ones, by substituting the given values into the equation to solve for k.

The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 4” will support a maximum load of 75 pounds.

ⓐ Write the equation that relates the maximum load to the cross-section.
ⓑ What is the maximum load that can be supported by a beam with diagonal 8”?

Solution

	The maximum load varies directly with the square of the diagonal of the cross-section.
Name the variables.	Let $L =$ maximum load. $c =$ the diagonal of the cross-section
Write the formula for direct variation, where $y$ varies directly with the square of $x$ .
We will use $L$ in place of $y$ and $c$ in place of $x$ .
Substitute the given values for the variables.

Solve for the constant of variation.	$A mathematical equation shows a fraction 75/16 on the left side, equated to a fraction (k * 16)/16 on the right side, indicating an algebraic step to solve for k.$

Write the equation that relates $L$ and $c$ .
Substitute in the constant of variation.

ⓑ

\begin{matrix} Find L when c = 8 . \\ Write the equation that relates L and c . & L = 4.6875 c^{2} \\ Substitute the given value for c . & L = 4.6875 {(8)}^{2} \\ Simplify. & L = 300 \\ \begin{matrix} A beam with diagonal 8” could support \\ a maximum load of 300 pounds. \end{matrix} \end{matrix}

Solve Inverse Variation Problems

Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is $y = \frac{k}{x}$ .

The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is $\frac{1}{x}$ .

We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’.

If y varies inversely with $x$ and $y = 20$ when $x = 8$ , find the equation that relates x and y.

Solution

Write the formula for inverse variation.
Substitute the given values for the variables.

Solve for the constant of variation.	$A mathematical equation is displayed, showing 8 multiplied by 20 on the left side, equaling 8 multiplied by the fraction k over 8 on the right side: 8(20) = 8(k/8).$

Write the equation that relates $x$ and $y$ .
Substitute in the constant of variation.

The fuel consumption (mpg) of a car varies inversely with its weight. A car that weighs 3100 pounds gets 26 mpg on the highway.

ⓐ Write the equation of variation.
ⓑ What would be the fuel consumption of a car that weighs 4030 pounds?

Solution

ⓐ

	The fuel consumption varies inversely with the weight.
First we will name the variables.	Let $f =$ fuel consumption. $w =$ weight
Write the formula for inverse variation.
We will use $f$ in place of $y$ and $w$ in place of $x$ .
Substitute the given values for the variables.

Solve for the constant of variation.	$A mathematical equation is displayed, showing 3100 multiplied by 26 on the left side, which equals 3100 multiplied by the fraction k over 3100 on the right side.$

Write the equation that relates $f$ and $w$ .
Substitute in the constant of variation.

ⓑ

Calculation of car fuel consumption (mpg) based on its weight (pounds) using an inverse relationship.
	$Find f when w = 4030 .$
Write the equation that relates f and w.	$f = \frac{80,600}{w}$
Substitute the given value for w.	$f = \frac{80,600}{4030}$
Simplify.	$f = 20$ A car that weighs 4030 pounds would have fuel consumption of 20 mpg.

The frequency of a guitar string varies inversely with its length. A 26” long string has a frequency of 440 vibrations per second.

ⓐ Write the equation of variation.
ⓑ How many vibrations per second will there be if the string’s length is reduced to 20” by putting a finger on a fret?

Solution

ⓐ

	The frequency varies inversely with the length.
Name the variables.	Let $f =$ frequency. $L =$ length
Write the formula for inverse variation.
We will use $f$ in place of $y$ and $L$ in place of $x$ .
Substitute the given values for the variables.

Solve for the constant of variation.	$A mathematical equation shows 26 multiplied by 440 equals 26 multiplied by the fraction k over 26.$

Write the equation that relates $f$ and $L$ .
Substitute in the constant of variation.	$A mathematical equation shows f equals 11,440 divided by L. The variable 'f' is on the left side of the equation, and a fraction with '11,440' as the numerator and 'L' as the denominator is on the right.$

ⓑ

This table demonstrates the step-by-step calculation of a guitar string's frequency (f) when its length (L) is 20 units.
	$Find f when L = 20 .$
Write the equation that relates f and L.	$f = \frac{11,440}{L}$
Substitute the given value for L.	$f = \frac{11,440}{20}$
Simplify.	$f = 572$ A 20” guitar string has frequency 572 vibrations per second.

Section Exercises

Practice Makes Perfect

Solve Direct Variation Problems

In the following exercises, solve.

If $y$ varies directly as $x$ and $y = 14, when x = 3$ , find the equation that relates $x and y$ .

Solution

$y = \frac{14}{3} x$

If $p$ varies directly as $q$ and $p = 5, when q = 2$ , find the equation that relates $p and q$ .

If $v$ varies directly as $w$ and $v = 24, when w = 8$ , find the equation that relates $v and w .$

Solution

$v = 3 w$

If $a$ varies directly as $b$ and $a = 16, when b = 4$ , find the equation that relates $a and b .$

If $p$ varies directly as $q$ and $p = 9.6, when q = 3$ , find the equation that relates $p and q .$

Solution

$p = 3.2 q$

If $y$ varies directly as $x$ and $y = 12.4, when x = 4,$ find the equation that relates $x and y$

If $a$ varies directly as $b$ and $a = 6, when b = \frac{1}{3}$ , find the equation that relates $a and b .$

Solution

$a = 18 b$

If $v$ varies directly as $w$ and $v = 8, when w = \frac{1}{2}$ , find the equation that relates $v and w .$

The amount of money Sally earns, P, varies directly with the number, n, of necklaces she sells. When Sally sells 15 necklaces she earns $150.

ⓐ Write the equation that relates P and n.
ⓑ How much money would she earn if she sold 4 necklaces?

Solution

ⓐ $P = 10 n$ ⓑ $$ 40$

The price, P, that Eric pays for gas varies directly with the number of gallons, g, he buys. It costs him $50 to buy 20 gallons of gas.

ⓐ Write the equation that relates P and g.
ⓑ How much would 33 gallons cost Eric?

Terri needs to make some pies for a fundraiser. The number of apples, a, varies directly with number of pies, p. It takes nine apples to make two pies.

ⓐ Write the equation that relates a and p.
ⓑ How many apples would Terri need for six pies?

Solution

ⓐ $a = 4.5 p$ ⓑ 27 apples

Joseph is traveling on a road trip. The distance, d, he travels before stopping for lunch varies directly with the speed, v, he travels. He can travel 120 miles at a speed of 60 mph.

ⓐ Write the equation that relates d and v.
ⓑ How far would he travel before stopping for lunch at a rate of 65 mph?

The price of gas that Jesse purchased varies directly to how many gallons he purchased. He purchased 10 gallons of gas for $39.80.

ⓐ Write the equation that relates the price to the number of gallons.
ⓑ How much will it cost Jesse for 15 gallons of gas?

Solution

ⓐ $p = 3.98 g$ ⓑ $$ 59.70$

The distance that Sarah travels varies directly to how long she drives. She travels 440 miles in 8 hours.

ⓐ Write the equation that relates the distance to the number of hours.
ⓑ How far can Sally travel in 6 hours?

The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters.

ⓐ Write the equation that relates the mass to the volume.
ⓑ What is the volume of this liquid if its mass is 128 kilograms?

Solution

ⓐ $m = 8 v$ ⓑ $16 liters$

The length that a spring stretches varies directly with a weight placed at the end of the spring. When Sarah placed a 10 pound watermelon on a hanging scale, the spring stretched 5 inches.

ⓐ Write the equation that relates the length of the spring to the weight.
ⓑ What weight of watermelon would stretch the spring 6 inches?

The distance an object falls varies directly to the square of the time it falls. A ball falls 45 feet in 3 seconds.

ⓐ Write the equation that relates the distance to the time.
ⓑ How far will the ball fall in 7 seconds?

Solution

ⓐ $d = 5 t^{2}$ ⓑ 245 feet

The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 6 inch will support a maximum load of 108 pounds.

ⓐ Write the equation that relates the load to the diagonal of the cross-section.
ⓑ What load will a beam with a 10 inch diagonal support?

The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.

ⓐ Write the equation that relates the area to the radius.
ⓑ What is the area of a personal pizza with a radius 4 inches?

Solution

ⓐ $A = 3.14 r^{2}$ ⓑ $50.24 sq. in.$

The distance an object falls varies directly to the square of the time it falls. A ball falls 72 feet in 3 seconds,

ⓐ Write the equation that relates the distance to the time.
ⓑ How far will the ball have fallen in 8 seconds?

Solve Inverse Variation Problems

In the following exercises, solve.

If $y$ varies inversely with $x$ and $y = 5$ when $x = 4$ find the equation that relates $x$ and $y .$

Solution

$y = \frac{20}{x}$

If $p$ varies inversely with $q$ and $p = 2$ when $q = 1$ find the equation that relates $p$ and $q .$

If $v$ varies inversely with $w$ and $v = 6$ when $w = \frac{1}{2}$ find the equation that relates $v$ and $w .$

Solution

$v = \frac{3}{w}$

If $a$ varies inversely with $b$ and $a = 12$ when $b = \frac{1}{3}$ find the equation that relates $a$ and $b .$

Write an inverse variation equation to solve the following problems.

The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds and gets 33 mpg on the highway.

ⓐ Write the equation that relates the mpg to the car’s weight.
ⓑ What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds?

Solution

ⓐ $g = \frac{92,400}{w}$ ⓑ 16.8 mpg

A car’s value varies inversely with its age. Jackie bought a 10 year old car for $2,400.

ⓐ Write the equation that relates the car’s value to its age.
ⓑ What will be the value of Jackie’s car when it is 15 years old ?

The time required to empty a tank varies inversely as the rate of pumping. It took Janet 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute),

ⓐ Write the equation that relates the number of hours to the pump rate.
ⓑ How long would it take Janet to pump her basement if she used a pump rated at 400 gpm?

Solution

ⓐ $t = \frac{1000}{r}$ ⓑ $2.5 hours$

The volume of a gas in a container varies inversely as pressure on the gas. A container of helium has a volume of 370 cubic inches under a pressure of 15 psi.

ⓐ Write the equation that relates the volume to the pressure.
ⓑ What would be the volume of this gas if the pressure was increased to 20 psi?

On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string on a violin has a frequency of 400 cycles per second.

ⓐ Write the equation that relates the string length to its frequency.
ⓑ What is the frequency of a 10-inch string?

Solution

ⓐ $L = \frac{4,400}{f}$ ⓑ 440 cycles per second

Paul, a dentist, determined that the number of cavities that develops in his patient’s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had 4 cavities when brushing her teeth 30 seconds (0.5 minutes) each night.

ⓐ Write the equation that relates the number of cavities to the time spent brushing.
ⓑ How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?

The number of tickets for a sports fundraiser varies inversely to the price of each ticket. Brianna can buy 25 tickets at $5each.

ⓐ Write the equation that relates the number of tickets to the price of each ticket.
ⓑ How many tickets could Brianna buy if the price of each ticket was $2.50?

Solution

ⓐ $t = \frac{125}{p}$ ⓑ 50 tickets

Boyle’s Law states that if the temperature of a gas stays constant, then the pressure varies inversely to the volume of the gas. Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 psi.

ⓐ Write the equation that relates pressure to volume.
ⓑ If the pressure increases to 330 psi, how much air can Braydon’s tank hold?

Mixed Practice

If $y$ varies directly as $x$ and $y = 5, when x = 3 .$ , find the equation that relates $x and y$ .

Solution

$y = \frac{5}{3} x$

If $v$ varies directly as $w$ and $v = 21, when w = 8 .$ find the equation that relates $v and w .$

If $p$ varies inversely with $q$ and $p = 5$ when $q = 6$ , find the equation that relates $p$ and $q .$

Solution

$p = \frac{30}{q}$

If $y$ varies inversely with $x$ and $y = 11$ when $x = 3$ find the equation that relates $x$ and $y .$

If $p$ varies directly as $q$ and $p = 10, when q = 2 .$ find the equation that relates $p and q$ .

Solution

$p = 5 q$

If $v$ varies inversely with $w$ and $v = 18$ when $w = \frac{1}{3}$ find the equation that relates $v$ and $w .$

The force needed to break a board varies inversely with its length. If Tom uses 20 pounds of pressure to break a 1.5-foot long board, how many pounds of pressure would he need to use to break a 6 foot long board?

Solution

$5 pounds$

The number of hours it takes for ice to melt varies inversely with the air temperature. A block of ice melts in 2.5 hours when the temperature is 54 degrees. How long would it take for the same block of ice to melt if the temperature was 45 degrees?

The length a spring stretches varies directly with a weight placed at the end of the spring. When Meredith placed a 6-pound cantaloupe on a hanging scale, the spring stretched 2 inches. How far would the spring stretch if the cantaloupe weighed 9 pounds?

Solution

$3 inches$

The amount that June gets paid varies directly the number of hours she works. When she worked 15 hours, she got paid $111. How much will she be paid for working 18 hours?

The fuel consumption (mpg) of a car varies inversely with its weight. A Ford Focus weighs 3000 pounds and gets 28.7 mpg on the highway. What would the fuel consumption be for a Ford Expedition that weighs 5,500 pounds? Round to the nearest tenth.

Solution

$15.6 mpg$

The volume of a gas in a container varies inversely as the pressure on the gas. If a container of argon has a volume of 336 cubic inches under a pressure of 2,500 psi, what will be its volume if the pressure is decreased to 2,000 psi?

The distance an object falls varies directly to the square of the time it falls. If an object falls 52.8 feet in 4 seconds, how far will it fall in 9 seconds?

Solution

$267.3 feet$

The area of the face of a Ferris wheel varies directly with the square of its radius. If the area of one face of a Ferris wheel with diameter 150 feet is 70,650 square feet, what is the area of one face of a Ferris wheel with diameter of 16 feet?

Everyday Math

Ride Service It costs $35 for a ride from the city center to the airport, 14 miles away.

ⓐ Write the equation that relates the cost, c, with the number of miles, m.
ⓑ What would it cost to travel 22 miles with this service?

Solution

ⓐ $c = 2.5 m$ ⓑ $55

Road Trip The number of hours it takes Jack to drive from Boston to Bangor is inversely proportional to his average driving speed. When he drives at an average speed of 40 miles per hour, it takes him 6 hours for the trip.

ⓐ Write the equation that relates the number of hours, h, with the speed, s.
ⓑ How long would the trip take if his average speed was 75 miles per hour?

Writing Exercises

In your own words, explain the difference between direct variation and inverse variation.

Solution

Answers will vary.

Make up an example from your life experience of inverse variation.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “solve direct variation problems.”, the next row reads “solve direct variation problems.” The remaining columns are blank.

ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

Chapter 8 Review Exercises

Simplify Rational Expressions

Determine the Values for Which a Rational Expression is Undefined

In the following exercises, determine the values for which the rational expression is undefined.

$\frac{2 a + 1}{3 a - 2}$

Solution

$a \neq \frac{2}{3}$

$\frac{b - 3}{b^{2} - 16}$

$\frac{3 x y^{2}}{5 y}$

Solution

$y \neq 0$

$\frac{u - 3}{u^{2} - u - 30}$

Evaluate Rational Expressions

In the following exercises, evaluate the rational expressions for the given values.

$\frac{4 p - 1}{p^{2} + 5} when p = −1$

Solution

$- \frac{5}{6}$

$\frac{q^{2} - 5}{q + 3} when q = 7$

$\frac{y^{2} - 8}{y^{2} - y - 2} when y = 1$

Solution

$\frac{7}{2}$

$\frac{z^{2} + 2}{4 z - z^{2}} when z = 3$

Simplify Rational Expressions

In the following exercises, simplify.

$\frac{10}{24}$

Solution

$\frac{5}{12}$

$\frac{8 m^{4}}{16 m n^{3}}$

$\frac{14 a - 14}{a - 1}$

Solution

$14$

$\frac{b^{2} + 7 b + 12}{b^{2} + 8 b + 16}$

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

$\frac{c^{2} - c - 2}{4 - c^{2}}$

Solution

$- \frac{c + 1}{c + 2}$

$\frac{d - 16}{16 - d}$

$\frac{7 v - 35}{25 - v^{2}}$

Solution

$- \frac{7}{5 + v}$

$\frac{w^{2} - 3 w - 28}{49 - w^{2}}$

Multiply and Divide Rational Expressions

Multiply Rational Expressions

In the following exercises, multiply.

$\frac{3}{8} \cdot \frac{2}{15}$

Solution

$\frac{1}{20}$

$\frac{2 x y^{2}}{8 y^{3}} \cdot \frac{16 y}{24 x}$

$\frac{3 a^{2} + 21 a}{a^{2} + 6 a - 7} \cdot \frac{a - 1}{a b}$

Solution

$\frac{3}{b}$

$\frac{5 z^{2}}{5 z^{2} + 40 z + 35} \cdot \frac{z^{2} - 1}{3 z}$

Divide Rational Expressions

In the following exercises, divide.

$\frac{t^{2} - 4 t - 12}{t^{2} + 8 t + 12} \div \frac{t^{2} - 36}{6 t}$

Solution

$\frac{6 t}{(t + 6)^{2}}$

$\frac{r^{2} - 16}{4} \div \frac{r^{3} - 64}{2 r^{2} + 8 r + 32}$

$\frac{11 + w}{w - 9} \div \frac{121 - w^{2}}{9 - w}$

Solution

$\frac{−1}{11 - w}$

$\frac{3 y^{2} - 12 y - 63}{4 y + 3} \div (6 y^{2} - 42 y)$

$\frac{\frac{c^{2} - 64}{3 c^{2} + 26 c + 16}}{\frac{c^{2} - 4 c - 32}{15 c + 10}}$

Solution

$\frac{5}{c + 4}$

$\frac{8 m^{2} - 8 m}{m - 4} \cdot \frac{m^{2} + 2 m - 24}{m^{2} + 7 m + 10} \div \frac{2 m^{2} - 6 m}{m + 5}$

Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

In the following exercises, add.

$\frac{3}{5} + \frac{2}{5}$

Solution

$1$

$\frac{4 a^{2}}{2 a - 1} - \frac{1}{2 a - 1}$

$\frac{p^{2} + 10 p}{p + 5} + \frac{25}{p + 5}$

Solution

$p + 5$

$\frac{3 x}{x - 1} + \frac{2}{x - 1}$

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

$\frac{d^{2}}{d + 4} - \frac{3 d + 28}{d + 4}$

Solution

$d - 7$

$\frac{z^{2}}{z + 10} - \frac{100}{z + 10}$

$\frac{4 q^{2} - q + 3}{q^{2} + 6 q + 5} - \frac{3 q^{2} + q + 6}{q^{2} + 6 q + 5}$

Solution

$\frac{q - 3}{q + 5}$

$\frac{5 t^{2} + 4 t + 3}{t^{2} - 25} - \frac{4 t^{2} - 8 t - 32}{t^{2} - 25}$

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add and subtract.

$\frac{18 w}{6 w - 1} + \frac{3 w - 2}{1 - 6 w}$

Solution

$\frac{15 w + 2}{6 w - 1}$

$\frac{a^{2} + 3 a}{a^{2} - 16} - \frac{3 a + 8}{16 - a^{2}}$

$\frac{2 b^{2} + 3 b - 15}{b^{2} - 49} - \frac{b^{2} + 16 b - 1}{49 - b^{2}}$

Solution

$\frac{3 b^{2} + 19 b - 16}{(b - 7) (b + 7)}$

$\frac{8 y^{2} - 10 y + 7}{2 y - 5} + \frac{2 y^{2} + 7 y + 2}{5 - 2 y}$

Add and Subtract Rational Expressions With Unlike Denominators

Find the Least Common Denominator of Rational Expressions

In the following exercises, find the LCD.

$\frac{4}{m^{2} - 3 m - 10}, \frac{2 m}{m^{2} - m - 20}$

Solution

$(m + 2) (m - 5) (m + 4)$

$\frac{6}{n^{2} - 4}, \frac{2 n}{n^{2} - 4 n + 4}$

$\frac{5}{3 p^{2} + 19 p + 6}, \frac{2 p}{3 p^{2} + 25 p + 8}$

Solution

$(3 p + 1) (p + 6) (p + 8)$

Find Equivalent Rational Expressions

In the following exercises, rewrite as equivalent rational expressions with the given denominator.

Rewrite as equivalent rational expressions with denominator $(m + 2) (m - 5) (m + 4)$ :

\frac{4}{m^{2} - 3 m - 10}, \frac{2 m}{m^{2} - m - 20} .

Rewrite as equivalent rational expressions with denominator $(n - 2) (n - 2) (n + 2)$ :

\frac{6}{n^{2} - 4 n + 4}, \frac{2 n}{n^{2} - 4} .

Solution

$\frac{6 n + 12}{(n - 2) (n - 2) (n + 2)},$
$\frac{2 n^{2} - 4 n}{(n - 2) (n - 2) (n + 2)}$

Rewrite as equivalent rational expressions with denominator $(3 p + 1) (p + 6) (p + 8)$ :

\frac{5}{3 p^{2} + 19 p + 6}, \frac{7 p}{3 p^{2} + 25 p + 8}

Add Rational Expressions with Different Denominators

In the following exercises, add.

$\frac{2}{3} + \frac{3}{5}$

Solution

$\frac{19}{15}$

$\frac{7}{5 a} + \frac{3}{2 b}$

$\frac{2}{c - 2} + \frac{9}{c + 3}$

Solution

$\frac{11 c - 12}{(c - 2) (c + 3)}$

$\frac{3 d}{d^{2} - 9} + \frac{5}{d^{2} + 6 d + 9}$

$\frac{2 x}{x^{2} + 10 x + 24} + \frac{3 x}{x^{2} + 8 x + 16}$

Solution

$\frac{5 x^{2} + 26 x}{(x + 4) (x + 4) (x + 6)}$

$\frac{5 q}{p^{2} q - p^{2}} + \frac{4 q}{q^{2} - 1}$

Subtract Rational Expressions with Different Denominators

In the following exercises, subtract and add.

$\frac{3 v}{v + 2} - \frac{v + 2}{v + 8}$

Solution

$\frac{2 (v^{2} + 10 v - 2)}{(v + 2) (v + 8)}$

$\frac{−3 w - 15}{w^{2} + w - 20} - \frac{w + 2}{4 - w}$

$\frac{7 m + 3}{m + 2} - 5$

Solution

$\frac{2 m - 7}{m + 2}$

$\frac{n}{n + 3} + \frac{2}{n - 3} - \frac{n - 9}{n^{2} - 9}$

$\frac{8 d}{d^{2} - 64} - \frac{4}{d + 8}$

Solution

$\frac{4}{d - 8}$

$\frac{5}{12 x^{2} y} + \frac{7}{20 x y^{3}}$

Simplify Complex Rational Expressions

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify.

$\frac{\frac{5 a}{a + 2}}{\frac{10 a^{2}}{a^{2} - 4}}$

Solution

$\frac{a - 2}{2 a}$

$\frac{\frac{2}{5} + \frac{5}{6}}{\frac{1}{3} + \frac{1}{4}}$

$\frac{x - \frac{3 x}{x + 5}}{\frac{1}{x + 5} + \frac{1}{x - 5}}$

Solution

$\frac{(x + 2) (x - 5)}{2}$

$\frac{\frac{2}{m} + \frac{m}{n}}{\frac{n}{m} - \frac{1}{n}}$

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

$\frac{6 + \frac{2}{q - 4}}{\frac{5}{q + 4}}$

Solution

$\frac{2 (3 q - 11) (q + 4)}{5 (q - 4)}$

$\frac{\frac{3}{a^{2}} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b^{2}}}$

$\frac{\frac{2}{z^{2} - 49} + \frac{1}{z + 7}}{\frac{9}{z + 7} + \frac{12}{z - 7}}$

Solution

$\frac{z - 5}{21 z + 21}$

$\frac{\frac{3}{y^{2} - 4 y - 32}}{\frac{2}{y - 8} + \frac{1}{y + 4}}$

Solve Rational Equations

Solve Rational Equations

In the following exercises, solve.

$\frac{1}{2} + \frac{2}{3} = \frac{1}{x}$

Solution

$\frac{6}{7}$

$1 - \frac{2}{m} = \frac{8}{m^{2}}$

$\frac{1}{b - 2} + \frac{1}{b + 2} = \frac{3}{b^{2} - 4}$

Solution

$\frac{3}{2}$

$\frac{3}{q + 8} - \frac{2}{q - 2} = 1$

$\frac{v - 15}{v^{2} - 9 v + 18} = \frac{4}{v - 3} + \frac{2}{v - 6}$

Solution

$no solution$

$\frac{z}{12} + \frac{z + 3}{3 z} = \frac{1}{z}$

Solve a Rational Equation for a Specific Variable

In the following exercises, solve for the indicated variable.

$\frac{V}{l} = h w for l$

Solution

$l = \frac{V}{h w}$

$\frac{1}{x} - \frac{2}{y} = 5 for y$

$x = \frac{y + 5}{z - 7} for z$

Solution

$z = \frac{y + 5 + 7 x}{x}$

$P = \frac{k}{V} for V$

Solve Proportion and Similar Figure Applications Similarity

Solve Proportions

In the following exercises, solve.

$\frac{x}{4} = \frac{3}{5}$

Solution

$\frac{12}{5}$

$\frac{3}{y} = \frac{9}{5}$

$\frac{s}{s + 20} = \frac{3}{7}$

Solution

$15$

$\frac{t - 3}{5} = \frac{t + 2}{9}$

In the following exercises, solve using proportions.

Rachael had a 21 ounce strawberry shake that has 739 calories. How many calories are there in a 32 ounce shake?

Solution

$1126 calories$

Leo went to Mexico over Christmas break and changed $525 dollars into Mexican pesos. At that time, the exchange rate had $1 US is equal to 16.25 Mexican pesos. How many Mexican pesos did he get for his trip?

Solve Similar Figure Applications

In the following exercises, solve.

∆ABC is similar to ∆XYZ. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides.

This image shows two triangles. The large triangle is labeled A B C. The length from A to B is labeled 8. The length from B to C is labeled 7. The length from C to A is labeled b. The smaller triangle is triangle x y z. The length from x to y is labeled 2 and two-thirds. The length from y to z is labeled x. The length from x to z is labeled 3.

Solution

$b = 9; x = 2 \frac{1}{3}$

On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is 700 miles, find the distance from

ⓐ Paris to Rome
ⓑ Paris to Vienna

This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Paris”, the next vertex is labeled “Vienna”, and the next vertex is labeled “Rome”. The distance from Paris to Vienna is 7.7 centimeters. The distance from Vienna to Rome is 7 centimeters. The distance from Rome to Paris is 8.9 centimeters.

Tony is 5.75 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.

Solution

23 feet

The height of a lighthouse in Pensacola, Florida is 150 feet. Standing next to the statue, 5.5 foot tall Natalie cast a 1.1 foot shadow How long would the shadow of the lighthouse be?

Solve Uniform Motion and Work Applications Problems

Solve Uniform Motion Applications

In the following exercises, solve.

When making the 5-hour drive home from visiting her parents, Lisa ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles in the bad weather. How fast did she drive when the weather was bad?

Solution

$45 mph$

Mark is riding on a plane that can fly 490 miles with a headwind of 20 mph in the same time that it can fly 350 miles against a tailwind of 20 mph. What is the speed of the plane?

John can ride his bicycle 8 mph faster than Luke can ride his bike. It takes Luke 3 hours longer than John to ride 48 miles. How fast can John ride his bike?

Solution

$16 mph$

Mark was training for a triathlon. He ran 8 kilometers and biked 32 kilometers in a total of 3 hours. His running speed was 8 kilometers per hour less than his biking speed. What was his running speed?

Solve Work Applications

In the following exercises, solve.

Jerry can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together?

Solution

$\frac{4}{5} hour$

Lisa takes 3 hours to mow the lawn while her cousin, Barb, takes 2 hours. How long will it take them working together?

Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone?

Solution

$12 days$

Sue and Deb work together writing a book that takes them 90 days. If Sue worked alone it would take her 120 days. How long would it take Deb to write the book alone?

Use Direct and Inverse Variation

Solve Direct Variation Problems

In the following exercises, solve.

If $y$ varies directly as $x$ , when $y = 9$ and $x = 3$ , find $x$ when $y = 21 .$

Solution

$7$

If $y$ varies inversely as $x$ , when $y = 20$ and $x = 2$ find $y$ when $x = 4 .$

If $m$ varies inversely with the square of $n$ , when $m = 4$ and $n = 6$ find $m$ when $n = 2 .$

Solution

$36$

Vanessa is traveling to see her fiancé. The distance, d, varies directly with the speed, v, she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?

If the cost of a pizza varies directly with its diameter, and if an 8” diameter pizza costs $12, how much would a 6” diameter pizza cost?

Solution

$$ 9$

The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph?

Solve Inverse Variation Problems

In the following exercises, solve.

The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for $6 each, how many tickets can Madelyn afford to buy if the price increased to $8?

Solution

$9 tickets$

On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an 11-inch string on a violin has a frequency of 360 cycles per second, what frequency does a 12 inch string have?

Practice Test

In the following exercises, simplify.

$\frac{3 a^{2} b}{6 a b^{2}}$

Solution

$\frac{a}{2 b}$

$\frac{5 b - 25}{b^{2} - 25}$

In the following exercises, perform the indicated operation and simplify.

$\frac{4 x}{x + 2} \cdot \frac{x^{2} + 5 x + 6}{12 x^{2}}$

Solution

$\frac{x + 3}{3 x}$

$\frac{5 y}{4 y - 8} \cdot \frac{y^{2} - 4}{10}$

$\frac{4}{p q} + \frac{5}{p}$

Solution

$\frac{4 + 5 q}{p q}$

$\frac{1}{z - 9} - \frac{3}{z + 9}$

$\frac{\frac{2}{3} + \frac{3}{5}}{\frac{2}{5}}$

Solution

$\frac{19}{6}$

$\frac{\frac{1}{m} - \frac{1}{n}}{\frac{1}{n} + \frac{1}{m}}$

In the following exercises, solve each equation.

$\frac{1}{2} + \frac{2}{7} = \frac{1}{x}$

Solution

$\frac{14}{11}$

$\frac{5}{y - 6} = \frac{3}{y + 6}$

$\frac{1}{z - 5} + \frac{1}{z + 5} = \frac{1}{z^{2} - 25}$

Solution

$\frac{1}{2}$

$\frac{t}{4} = \frac{3}{5}$

$\frac{2}{r - 2} = \frac{3}{r - 1}$

Solution

In the following exercises, solve.

If $y$ varies directly with $x,$ and $x = 5$ when $y = 30,$ find $x$ when $y = 42 .$

If $y$ varies inversely with $x$ and $x = 6$ when $y = 20,$ find $y$ when $x = 2 .$

Solution

If $y$ varies inversely with the square of $x$ and $x = 3$ when $y = 9,$ find $y$ when $x = 4 .$

The recommended erythromycin dosage for dogs, is 5 mg for every pound the dog weighs. If Daisy weighs 25 pounds, how many milligrams of erythromycin should her veterinarian prescribe?

Solution

$125 mg$

Julia spent 4 hours Sunday afternoon exercising at the gym. She ran on the treadmill for 10 miles and then biked for 20 miles. Her biking speed was 5 mph faster than her running speed on the treadmill. What was her running speed?

Kurt can ride his bike for 30 miles with the wind in the same amount of time that he can go 21 miles against the wind. If the wind’s speed is 6 mph, what is Kurt’s speed on his bike?

Solution

$34 mph$

Amanda jogs to the park 8 miles using one route and then returns via a 14-mile route. The return trip takes her 1 hour longer than her jog to the park. Find her jogging rate.

An experienced window washer can wash all the windows in Mike’s house in 2 hours, while a new trainee can wash all the windows in 7 hours. How long would it take them working together?

Solution

$1 \frac{5}{9} hours$

Josh can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Josh’s dad working alone to split the logs?

The price that Tyler pays for gas varies directly with the number of gallons he buys. If 24 gallons cost him $59.76, what would 30 gallons cost?

Solution

$$ 74.70$

The volume of a gas in a container varies inversely with the pressure on the gas. If a container of nitrogen has a volume of 29.5 liters with 2000 psi, what is the volume if the tank has a 14.7 psi rating? Round to the nearest whole number.

The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio, as shown on the figure below, that gives the map distances between these cities in inches.

This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Dayton”, the next vertex is labeled “Columbus”, and the next vertex is labeled “Cincinnati”. The distance from Dayton to Columbus is 3.2 inches. The distance from Columbus to Cincinnati is 5.3 inches. The distance from Cincinnati to Dayton is 2.4 inches.

The actual distance from Dayton to Cincinnati is 48 miles. What is the actual distance between Dayton and Columbus?

Solution

64 miles