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Use Direct and Inverse Variation

Solve direct variation problems

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

\[s=15h\]

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

Try it.

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

Solution

We’ll list the steps below.

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

Example

Try it.

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.

  1. ⓐ Write the equation that relates c and m.
  2. ⓑ 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.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

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’.

Example

Try it.

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.
Write the equation that relates \(x\) and \(y\).
Substitute in the constant of variation.

Example

Try it.

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

  1. ⓐ Write the equation of variation.
  2. ⓑ 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.
Write the equation that relates \(f\) and \(w\).
Substitute in the constant of variation.

ⓑ

\(\text{Find}\ f\ \text{when}\ w=4030.\)
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.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Chapter 8 Review Exercises

Determine the Values for Which a Rational Expression is Undefined

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

Try it.

\(\frac{2a+1}{3a-2}\)

Solution

\(a\ne \frac{2}{3}\)

Try it.

\(\frac{b-3}{{b}^{2}-16}\)

Try it.

\(\frac{3x{y}^{2}}{5y}\)

Solution

\(y\ne 0\)

Try it.

\(\frac{u-3}{{u}^{2}-u-30}\)

Evaluate Rational Expressions

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

Try it.

\(\frac{4p-1}{{p}^{2}+5}\ \text{when}\ p=-1\)

Solution

\(-\frac{5}{6}\)

Try it.

\(\frac{{q}^{2}-5}{q+3}\ \text{when q}=7\)

Try it.

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

Solution

\(\frac{7}{2}\)

Try it.

\(\frac{{z}^{2}+2}{4z-{z}^{2}}\ \text{when z}=3\)

Simplify Rational Expressions

In the following exercises, simplify.

Try it.

\(\frac{10}{24}\)

Solution

\(\frac{5}{12}\)

Try it.

\(\frac{8{m}^{4}}{16m{n}^{3}}\)

Try it.

\(\frac{14a-14}{a-1}\)

Solution

\(14\)

Try it.

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

Simplify Rational Expressions with Opposite Factors

In the following exercises, simplify.

Try it.

\(\frac{{c}^{2}-c-2}{4-{c}^{2}}\)

Solution

\(-\frac{c+1}{c+2}\)

Try it.

\(\frac{d-16}{16-d}\)

Try it.

\(\frac{7v-35}{25-{v}^{2}}\)

Solution

\(-\frac{7}{5+v}\)

Try it.

\(\frac{{w}^{2}-3w-28}{49-{w}^{2}}\)

Multiply Rational Expressions

In the following exercises, multiply.

Try it.

\(\frac{3}{8}\cdot \frac{2}{15}\)

Solution

\(\frac{1}{20}\)

Try it.

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

Try it.

\(\frac{3{a}^{2}+21a}{{a}^{2}+6a-7}\cdot \frac{a-1}{ab}\)

Solution

\(\frac{3}{b}\)

Try it.

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

Divide Rational Expressions

In the following exercises, divide.

Try it.

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

Solution

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

Try it.

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

Try it.

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

Solution

\(\frac{-1}{11-w}\)

Try it.

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

Try it.

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

Solution

\(\frac{5}{c+4}\)

Try it.

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

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Find the multiplicative inverse of \(-8\).
    If you missed this problem, review .

    Lafunua jibu

    \(-\frac{1}{8}\)

  2. Solve for \(n\): \(45=20n\).
    If you missed this problem, review .

    Lafunua jibu

    \(n=2.25\)

  3. Evaluate \(5{x}^{2}\) when \(x=10\).
    If you missed this problem, review .

    Lafunua jibu

    \(500\)

  4. If y varies directly with x and \(y=20\) when \(x=8\), find the equation that relates x and y.

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

    Lafunua jibu

    \(y=\frac{3}{10}x\)

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

    Lafunua jibu

    \(y=3x\)

  7. 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.

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

    ⓐ

    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.

  8. 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?
    Lafunua jibu

    ⓐ \(c=4.8t\) ⓑ 432 calories

  9. 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?

    Lafunua jibu

    ⓐ \(d=50t\) ⓑ 250 miles

  10. 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.

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

    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.


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

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

  11. 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?
    Lafunua jibu

    ⓐ \(m=55h\) ⓑ 220 miles

  12. 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?
    Lafunua jibu

    ⓐ \(w=6v\) ⓑ 78 pounds

  13. 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.

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

    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.
    Write the equation that relates \(L\) and \(c\).
    Substitute in the constant of variation.


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

  14. 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?
    Lafunua jibu

    ⓐ \(d=16{t}^{2}\) ⓑ 256 feet

  15. 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?
    Lafunua jibu

    ⓐ \(A=3.14{r}^{2}\) ⓑ 254.34 square inches

  16. If y varies inversely with \(x\) and \(y=20\) when \(x=8\), find the equation that relates x and y.

    Lafunua jibu

    Write the formula for inverse variation.
    Substitute the given values for the variables.
    Solve for the constant of variation.
    Write the equation that relates \(x\) and \(y\).
    Substitute in the constant of variation.

  17. If \(p\) varies inversely with \(q\) and \(p=30\) when \(q=12\) find the equation that relates \(p\) and \(q.\)

    Lafunua jibu

    \(p=\frac{360}{q}\)

  18. If \(y\) varies inversely with \(x\) and \(y=8\) when \(x=2\) find the equation that relates \(x\) and \(y\).

    Lafunua jibu

    \(y=\frac{16}{x}\)

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

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

    ⓐ

    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.
    Write the equation that relates \(f\) and \(w\).
    Substitute in the constant of variation.

    ⓑ

    \(\text{Find}\ f\ \text{when}\ w=4030.\)
    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.

  20. 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?

    Lafunua jibu

    ⓐ \(v=\frac{40,000}{a}\) ⓑ $8,000

  21. 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?
    Lafunua jibu

    ⓐ \(t=\frac{1000}{r}\) ⓑ 2 hours

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

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

    ⓐ

    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.
    Write the equation that relates \(f\) and \(L\).
    Substitute in the constant of variation.

    ⓑ

    \(\text{Find}\ f\ \text{when}\ L=20.\)
    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.

  23. 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?
    Lafunua jibu

    ⓐ \(h=\frac{130}{t}\) ⓑ \(1\frac{2}{3}\) hours

  24. 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?
    Lafunua jibu

    ⓐ \(F=\frac{48}{L}\) ⓑ 9.6 pounds

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

    Lafunua jibu

    \(y=\frac{14}{3}x\)

  26. If \(p\) varies directly as \(q\) and \(p=5,\ \text{when}\ q=2\), find the equation that relates \(p\ \text{and}\ q\).

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

    Lafunua jibu

    \(v=3w\)

  28. If \(a\) varies directly as \(b\) and \(a=16,\ \text{when}\ b=4\), find the equation that relates \(a\ \text{and}\ b.\)

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

    Lafunua jibu

    \(p=3.2q\)

  30. If \(y\) varies directly as \(x\) and \(y=12.4,\ \text{when}\ x=4,\) find the equation that relates \(x\ \text{and}\ y\)

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

    Lafunua jibu

    \(a=18b\)

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

  33. 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?
    Lafunua jibu

    ⓐ \(P=10n\) ⓑ \(\text{\$}40\)

  34. 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.

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

  35. 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?
    Lafunua jibu

    ⓐ \(a=4.5p\) ⓑ 27 apples

  36. 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.

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

  37. 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?
    Lafunua jibu

    ⓐ \(p=3.98g\) ⓑ \(\text{\$}59.70\)

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

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

  39. 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?
    Lafunua jibu

    ⓐ \(m=8v\) ⓑ \(\text{16 liters}\)

  40. 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.

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

Symbols used here

f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\neq
not equal
The two sides are different.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Use Direct and Inverse Variation

  1. Solve direct variation problems
  2. Solve inverse variation problems
  3. Write the formula for direct variation.
  4. Substitute the given values for the variables.
  5. Solve for the constant of variation.
  6. Write the equation that relates x and y.
  7. Write the formula for inverse variation.
  8. Substitute the given values for the variables.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Jaribu kufanya mambo yako mwenyewe

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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