### 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.

## Solve direct variation problems

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula *e* = 0.16*s* tells us her earnings, *e*, come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive.

s, sales prices |
e = 0.16s |
Interpretation |
---|---|---|

$4,600 | e = 0.16(4,600) = 736 |
A sale of a $4,600 vehicle results in $736 earnings. |

$9,200 | e = 0.16(9,200) = 1,472 |
A sale of a $9,200 vehicle results in $1472 earnings. |

$18,400 | e = 0.16(18,400) = 2,944 |
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 below represents the data for Nicole’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 *k* is a nonzero constant greater than zero and is called the **constant of variation**. In this case, *k *= 0.16 and *n *= 1.

### A General Note: Direct Variation

If *x *and *y* are related by an equation of the form

then we say that the relationship is **direct variation** and *y* **varies directly** with the *n*th power of *x*. In direct variation relationships, there is a nonzero constant ratio [latex]k=\frac{y}{{x}^{n}}[/latex], where *k* 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,
*x*, and the output,*y*. - Determine the constant of variation. You may need to divide
*y*by the specified power of*x*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 1: Solving a Direct Variation Problem

The quantity *y* varies directly with the cube of *x*. If *y *= 25 when *x *= 2, find *y* when *x* is 6.

### Q & A

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

*No. Direct variation equations are power functions—they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through (0, 0).*

### Try It

The quantity *y* varies directly with the square of *x*. If *y *= 24 when *x *= 3, find *y* when *x* is 4.

### Try It

## Solve inverse variation problems

Water temperature in an ocean varies inversely to the water’s depth. Between the depths of 250 feet and 500 feet, the formula [latex]T=\frac{14,000}{d}[/latex] gives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F.

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

d, 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 28° F. |

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

250 ft | [latex]\frac{14,000}{250}=56[/latex] | At a depth of 250 ft, the water temperature is 56° 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 graph 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=\frac{k}{x}[/latex] for inverse variation in this case uses *k *= 14,000.

### A General Note: Inverse Variation

If *x* and *y* are related by an equation of the form

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

### Example 2: 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,
*x*, and the output,*y*. - Determine the constant of variation. You may need to multiply
*y*by the specified power of*x*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 3: Solving an Inverse Variation Problem

A quantity *y* varies inversely with the cube of *x*. If *y *= 25 when *x *= 2, find *y* when *x* is 6.

### Try It

A quantity *y* varies inversely with the square of *x*. If *y *= 8 when *x *= 3, find *y* when *x* is 4.

### Try It

## Solve problems involving 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 *c*, cost, varies jointly with the number of students, *n*, and the distance, *d*.

### A General Note: Joint Variation

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

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

### Example 4: Solving Problems Involving Joint Variation

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

### Try It

*x* varies directly with the square of *y* and inversely with *z*. If *x *= 40 when *y *= 4 and *z *= 2, find *x* when *y *= 10 and *z *= 25.

### Key Takeaways

Key Equations

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

Inverse variation | [latex]y=\frac{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
*k*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