8.9 Use Direct and Inverse Variation

If you miss a problem, go back to the section listed and review the material.

Find the multiplicative inverse of $−8$.
If you missed this problem, review the reference.

Solve for $n$: $45 = 20 n$.
If you missed this problem, review the reference.

Evaluate $5 x^{2}$ when $x = 10$.
If you missed this problem, review the reference.

If $y$ varies directly as $x$ and $y = 3 , \text{when} x = 10 .$ find the equation that relates x and y.

If $y$ varies directly as $x$ and $y = 12 \text{when} x = 4$ find the equation that relates x and y.

The number of calories, c, burned varies directly with the amount of time, t, spent exercising. Arnold burned 312 calories in 65 minutes exercising.

1. ⓐ Write the equation that relates c and t.
2. ⓑ How many calories would he burn if he exercises for 90 minutes?

The distance a moving body travels, d, varies directly with time, t, it moves. A train travels 100 miles in 2 hours

ⓐ Write the equation that relates d and t. ⓑ How many miles would it travel in 5 hours?

The distance that Brad travels varies directly with the time spent traveling. Brad travelled 660 miles in 12 hours,

1. ⓐ Write the equation that relates the number of miles travelled to the time.
2. ⓑ How many miles could Brad travel in 4 hours?

The weight of a liquid varies directly as its volume. A liquid that weighs 24 pounds has a volume of 4 gallons.

1. ⓐ Write the equation that relates the weight to the volume.
2. ⓑ If a liquid has volume 13 gallons, what is its weight?

The distance an object falls is directly proportional to the square of the time it falls. A ball falls 144 feet in 3 seconds.

1. ⓐ Write the equation that relates the distance to the time.
2. ⓑ How far will an object fall in 4 seconds?

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.

1. ⓐ Write the equation that relates the area to the radius.
2. ⓑ What is the area of a pizza with a radius of 9 inches?

If $p$ varies inversely with $q$ and $p = 30$ when $q = 12$ find the equation that relates $p$ and $q .$

If $y$ varies inversely with $x$ and $y = 8$ when $x = 2$ find the equation that relates $x$ and $y$.

A car’s value varies inversely with its age. Elena bought a two-year-old car for $20,000.

ⓐ Write the equation of variation. ⓑ What will be the value of Elena’s car when it is 5 years old?

The time required to empty a pool varies inversely as the rate of pumping. It took Lucy 2.5 hours to empty her pool using a pump that was rated at 400 gpm (gallons per minute).

1. ⓐ Write the equation of variation.
2. ⓑ How long will it take her to empty the pool using a pump rated at 500 gpm?

The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees.

1. ⓐ Write the equation of variation.
2. ⓑ How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?

The force needed to break a board varies inversely with its length. Richard uses 24 pounds of pressure to break a 2-foot long board.

1. ⓐ Write the equation of variation.
2. ⓑ How many pounds of pressure is needed to break a 5-foot long board?

Solve.

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

Solve.

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

Solve.

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

Solve.

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

Solve.

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.

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

Solve.

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.

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

Solve.

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

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

Solve.

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

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

Solve.

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

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

Solve.

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.

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

Solve.

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

Solve.

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

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.

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

Write an inverse variation equation to solve the following problems.

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),

1. ⓐ Write the equation that relates the number of hours to the pump rate.
2. ⓑ How long would it take Janet to pump her basement if she used a pump…

Write an inverse variation equation to solve the following problems.

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.

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

Write an inverse variation equation to solve the following problems.

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

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

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

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

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

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?

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?

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.

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?

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

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

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

Determine the values for which the rational expression is undefined.

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

Determine the values for which the rational expression is undefined.

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

Evaluate the rational expressions for the given values.

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

Evaluate the rational expressions for the given values.

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

Simplify.

$\frac{10}{24}$

Simplify.

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

Simplify.

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

Simplify.

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

Multiply.

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

Multiply.

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

Divide.

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

Divide.

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

Divide.

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

Add.

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

Add.

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

Subtract.

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

Subtract.

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

Add and subtract.

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

Add and subtract.

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

Find the LCD.

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

Find the LCD.

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

Rewrite as equivalent rational expressions with the given denominator.

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

Add.

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

Add.

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

Add.

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

Subtract and add.

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

Subtract and add.

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

Subtract and add.

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

Simplify.

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

Simplify.

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

Simplify.

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

Simplify.

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

Solve.

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

Solve.

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

Solve.

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

Solve for the indicated variable.

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

Solve for the indicated variable.

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

Solve.

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

Solve.

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

Solve using proportions.

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

Solve.

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.

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?

Solve.

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?

Solve.

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

Solve.

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?

Solve.

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

Solve.

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

Solve.

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?

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?

Simplify.

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

Perform the indicated operation and simplify.

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

Perform the indicated operation and simplify.

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

Perform the indicated operation and simplify.

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

Solve each equation.

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

Solve each equation.

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

Solve each equation.

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

Solve.

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

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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?