### Learning Outcomes

- Solve direct variation problems.
- Solve inverse variation problems.
- Solve problems involving joint variation.

A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section we will look at relationships, such as this one, between earnings, sales, and commission rate.

## Direct Variation

In the example above, Nicoleâ€™s earnings can be found by multiplying her sales by her commission. The formula [latex]e = 0.16s[/latex] tells us her earnings, [latex]e[/latex], come from the product of 0.16, her commission, and the sale price of the vehicle, [latex]s[/latex].Â If we create a table, we observe that as the sales price increases, the earnings increase as well.

[latex]s[/latex], sales prices | [latex]e = 0.16s[/latex] | Interpretation |
---|---|---|

$4,600 | [latex]eÂ = 0.16(4,600) = 736[/latex] |
A sale of a $4,600 vehicle results in $736 earnings. |

$9,200 | [latex]eÂ = 0.16(9,200) = 1,472[/latex] |
A sale of a $9,200 vehicle results in $1472 earnings. |

$18,400 | [latex]eÂ = 0.16(18,400) = 2,944[/latex] |
A sale of a $18,400 vehicle results in $2944 earnings. |

Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called **direct variation**. Each variable in this type of relationship **varies directly **with the other.

The graph belowÂ represents the data for Nicoleâ€™s potential earnings. We say that earnings vary directly with the sales price of the car. The formula [latex]y=k{x}^{n}[/latex] is used for direct variation. The value [latex]k[/latex] is a nonzero constant greater than zero and is called the **constant of variation**. In this case, [latex]k=0.16[/latex]Â and [latex]n=1[/latex].

### A General Note: Direct Variation

If [latex]x[/latex]*Â *and [latex]y[/latex]Â are related by an equation of the form

[latex]y=k{x}^{n}[/latex]

then we say that the relationship is **direct variation** and [latex]y[/latex]Â **varies directly** with the [latex]n[/latex]th power of [latex]x[/latex]. In direct variation relationships, there is a nonzero constant ratio [latex]k=\dfrac{y}{{x}^{n}}[/latex], where [latex]k[/latex]Â is called the **constant of variation**, which help defines the relationship between the variables.

### How To: Given a description of a direct variation problem, solve for an unknown.

- Identify the input, [latex]x[/latex], and the output, [latex]y[/latex].
- Determine the constant of variation. You may need to divide [latex]y[/latex]Â by the specified power of [latex]x[/latex]Â to determine the constant of variation.
- Use the constant of variation to write an equation for the relationship.
- Substitute known values into the equation to find the unknown.

### Example: Solving a Direct Variation Problem

The quantity [latex]y[/latex]Â varies directly with the cube of [latex]x[/latex]. If [latex]y=25[/latex]Â when [latex]x=2[/latex], find [latex]y[/latex]Â when [latex]x[/latex]Â is 6.

### Q & A

**Do the graphs of all direct variation equations look like Example 1?**

*No. Direct variation equations are power functionsâ€”they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through *[latex](0, 0)[/latex]*.*

### Try It

The quantity [latex]y[/latex]Â varies directly with the square of [latex]y[/latex]. If [latex]y=24[/latex]Â when [latex]x=3[/latex], find [latex]y[/latex]Â when [latex]x[/latex]Â is 4.

Watch this video to see a quick lesson in direct variation. Â You will see more worked examples.

## Inverse and Joint Variation

Water temperature in an ocean varies inversely to the waterâ€™s depth. Between the depths of 250 feet and 500 feet, the formula [latex]T=\Dfrac{14,000}{d}[/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earthâ€™s surface. Consider the Atlantic Ocean, which covers 22% of Earthâ€™s surface. At a certain location, at the depth of 500 feet, the temperature may be 28Â°F.

If we create a tableÂ we observe that, as the depth increases, the water temperature decreases.

d[/latex], depth | [latex]T=\frac{\text{14,000}}{d}[/latex] | Interpretation |
---|---|---|

500 ft | [latex]\frac{14,000}{500}=28[/latex] | At a depth of 500 ft, the water temperature is 28Â° F. |

350 ft | [latex]\frac{14,000}{350}=40[/latex] | At a depth of 350 ft, the water temperature is 40Â° F. |

250 ft | [latex]\frac{14,000}{250}=56[/latex] | At a depth of 250 ft, the water temperature is 56Â° F. |

We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be **inversely proportional** and each term **varies inversely** with the other. Inversely proportional relationships are also called **inverse variations**.

For our example, the graphÂ depicts the **inverse variation**. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formula [latex]y=\dfrac{k}{x}[/latex] for inverse variation in this case uses [latex]k=14,000[/latex].

### A General Note: Inverse Variation

If [latex]x[/latex] and [latex]y[/latex]Â are related by an equation of the form

[latex]y=\dfrac{k}{{x}^{n}}[/latex]

where [latex]k[/latex]Â is a nonzero constant, then we say that [latex]y[/latex]Â **varies inversely** with the [latex]n[/latex]th power of [latex]x[/latex]. In **inversely proportional** relationships, or **inverse variations**, there is a constant multiple [latex]k={x}^{n}y[/latex].

### Example: Writing a Formula for an Inversely Proportional Relationship

A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.

### How To: Given a description of an indirect variation problem, solve for an unknown.

- Identify the input, [latex]x[/latex], and the output, [latex]y[/latex].
- Determine the constant of variation. You may need to multiply [latex]y[/latex]Â by the specified power of [latex]x[/latex]Â to determine the constant of variation.
- Use the constant of variation to write an equation for the relationship.
- Substitute known values into the equation to find the unknown.

### Example: Solving an Inverse Variation Problem

A quantity [latex]y[/latex]Â varies inversely with the cube of [latex]x[/latex]. If [latex]y=25[/latex]Â when [latex]x=2[/latex], find [latex]y[/latex]Â when [latex]x[/latex]Â is 6.

### Try It

A quantity [latex]y[/latex]Â varies inversely with the square of [latex]x[/latex]. If [latex]y=8[/latex]Â when [latex]x=3[/latex], find [latex]y[/latex]Â when [latex]x[/latex]Â is 4.

The following video presents a short lesson on inverse variation and includes more worked examples.

## Joint Variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called **joint variation**. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable [latex]c[/latex], cost, varies jointly with the number of students, [latex]n[/latex], and the distance, [latex]d[/latex].

### A General Note: Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if [latex]x[/latex]Â varies directly with both [latex]y[/latex]Â and [latex]z[/latex], we have [latex]x=kyz[/latex]. If [latex]x[/latex]Â varies directly with [latex]y[/latex]Â and inversely with [latex]z[/latex], we have [latex]x=\dfrac{ky}{z}[/latex]. Notice that we only use one constant in a joint variation equation.

### Example: Solving Problems Involving Joint Variation

A quantity [latex]x[/latex]Â varies directly with the square of [latex]y[/latex]Â and inversely with the cube root of [latex]z[/latex]. If [latex]x=6[/latex]Â when [latex]y=2[/latex]Â and [latex]z=8[/latex], find [latex]x[/latex]Â when [latex]y=1[/latex]Â and [latex]z=27[/latex].

### Try It

[latex]x[/latex] varies directly with the square of [latex]y[/latex]Â and inversely with [latex]z[/latex]. If [latex]x=40[/latex]Â when [latex]y=4[/latex]Â and [latex]z=2[/latex], find [latex]x[/latex]Â when [latex]y=10[/latex]Â and [latex]z=25[/latex].

The following video provides another worked example of a joint variation problem.

## Key Equations

Direct variation | [latex]y=k{x}^{n},k\text{ is a nonzero constant}\\[/latex]. |

Inverse variation | [latex]y=\dfrac{k}{{x}^{n}},k\text{ is a nonzero constant}\\[/latex]. |

## Key Concepts

- A relationship where one quantity is a constant multiplied by another quantity is called direct variation.
- Two variables that are directly proportional to one another will have a constant ratio.
- A relationship where one quantity is a constant divided by another quantity is called inverse variation.
- Two variables that are inversely proportional to one another will have a constant multiple.
- In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation.

## Glossary

**constant of variation**- the non-zero value [latex]k[/latex] that helps define the relationship between variables in direct or inverse variation

**direct variation**- the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other

**inverse variation**- the relationship between two variables in which the product of the variables is a constant

**inversely proportional**- a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases

**joint variation**- a relationship where a variable varies directly or inversely with multiple variables

**varies directly**- a relationship where one quantity is a constant multiplied by the other quantity

**varies inversely**- a relationship where one quantity is a constant divided by the other quantity