By the end of this section, you will be able to:

- Solve direct variation problems
- Solve inverse variation problems

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.

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

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*.

We’ll list the steps below.

- Step 1. Write the formula for direct variation.
- Step 2. Substitute the given values for the variables.
- Step 3. Solve for the constant of variation.
- Step 4. Write the equation that relates x and y.

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

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.

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

Many applications involve two variable that *vary inversely*. As one variable increases, the other decreases. The equation that relates them is *y*=*k/x*.

The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of *x* is 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’.

- Step 1. Write the formula for inverse variation.
- Step 2. Substitute the given values for the variables.
- Step 3. Solve for the constant of variation.
- Step 4. Write the equation that relates x and y.

**Solve Direct Variation Problems**

463.If y varies directly as x and y=14,whenx=3, find the equation that relates xandy.

464.If p varies directly as q and p=5,whenq=2, find the equation that relates pandq.

465.If v varies directly as w and v=24,whenw=8, find the equation that relates vandw.

466.If a varies directly as b and a=16,whenb=4, find the equation that relates aandb.

467.If p varies directly as q and p=9.6,whenq=3, find the equation that relates pandq.

468.If y varies directly as x and y=12.4,whenx=4, find the equation that relates xandy

469.If a varies directly as b and a=6,whenb=13, find the equation that relates aandb.

470.If v varies directly as w and v=8,whenw=12, find the equation that relates vandw.

471.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?

472.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?

473.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?

474.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?

475.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?

476.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?

477.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?

478.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?

479.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?

480.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?

481.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?

482.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.

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

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

485.If v varies inversely with w and v=6 when w=1/2 find the equation that relates v and w.

486.If a varies inversely with b and a=12 when b=1 / 3 find the equation that relates a and b.

Write an inverse variation equation to solve the following problems.

487.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?

488.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 ?

489.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?

490.

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?

491.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?

492.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?

493.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?

494.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?

495.If y varies directly as x and y=5,whenx=3., find the equation that relates xandy.

496.If v varies directly as w and v=21,whenw=8. find the equation that relates vandw.

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

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

499.If p varies directly as q and p=10,whenq=2. find the equation that relates pandq.

500.If v varies inversely with w and v=18 when w=13 find the equation that relates v and w.

501.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?

502.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?

503.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?

504.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?

505.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.

506.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?

507.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?

508.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?

495.If y varies directly as x and y=5,whenx=3., find the equation that relates xandy.

496.If v varies directly as w and v=21,whenw=8. find the equation that relates vandw.

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

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

499.If p varies directly as q and p=10,whenq=2. find the equation that relates pandq.

500.If v varies inversely with w and v=18 when w=13 find the equation that relates v and w.

501.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?

502.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?

503.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?

504.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?

505.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.

506.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?

507.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?

508.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?

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

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

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

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

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