Direct Variation

Introduction

Hours worked	1	2	3	4
Dollars Earned	10	20	30	40

If John earns $10.00 an hour, the above table shows the relationship between hours worked and dollars earned. From this table, we can see that dollars earned is equal to hours worked multiplied by 10.

Quarts of Milk	1	2	3	4
Cups of Milk	4	8	12	16

If a quart of milk provides four cups, the above table shows the relationship between the number of quarts of milk and the number of cups they provide. From this table, we can see that cups of milk is equal to quarts of milk multiplied by 4.

Number of Dollars	1	2	3	4
Number of Cookies that can be purchased	5	10	15	20

If a cookie costs 20 cents, the above table shows the relationship between the number of dollars and the number of cookies they can purchase. From this table, we can see that number of cookies is equal to number of dollars multiplied by 5.

In all of these relations, one variable is equal to another variable multiplied by a non-zero constant. Such relations are referred to as direct proportions or direct variations.

Definition

Given two variables x and y, the statements

varies directly as

is directly proportional to

is proportional to

all mean that y = kx, for some fixed nonzero real number k. The constant k is called the constant of variation or constant of proportionality.

Finding the Formula for Direct Variation Relations

Direct variation relations have the form y = kx where x and y are variables and k is a non-zero constant. The formula for a direct variation relationship can be found with the following steps.

Step 1: Translate the statement into a direct variation formula.

y is directly proportional to x means

y =k·x

Step 2: Substitute known values to find k.

Step 3: Substitute k and write the formula.

Examples

Example 1

u is proportional to v. If u = 16 when v = 4, then write the formula for the relation between u and v.

Step 1: Translate the statement into a direct variation formula.

u is proportional to v means

u =k·v

Step 2: Substitute known values to find k.

16 = k· 4

$k = \frac{16}{4} = 4$

Step 3: Substitute k and write the formula.

u =4v

Conclusion: the formula to express the relationship between u and v is u =4v

Example 2

p varies directly as z. If p = 210 when z = 200, then write the formula for the relation between p and z.

Step 1: Translate the statement into a direct variation formula.

p varies directly as z means

p = k·z

Step 2: Substitute known values to find k.

210 = k· 200

$k = \frac{210}{200} = 1.05$

Step 3: Substitute k and write the formula.

p =1.05z

Conclusion: the formula to express the relationship between p and z is

p = 1.05z

Steps to Solve Simple Direct Variation Problems

Simple direct variation problems consist in find a formula and verify what happen with a quantity when we change the other quantity of the problem.

Step 1: Find formula.

Step 2: Identify known variables and substitute into formula.

Step 3: Solve for Unknown.

Examples

Example 1

w is directly proportional to m. If w = 42 when m = 6, then find the value of m when w = 140

Step 1:
Find formula

1: Translate the statement into a direct variation formula.

w is directly proportional to m means

w = k·m

2: Substitute known values to find k.

42 = k· 6

$k = \frac{42}{6} = 7$

3: Substitute k and write the formula.

w = 7m

Step 2: Identify known variables and substitute into formula.

140 = 7·m

Step 3: Solve for Unknown.

$m = \frac{140}{7} = 20$

Conclusion: m = 20 when w = 140

Example 2

A varies directly as b. If A = 3 when b = 8, then find the value of A when b = 1000

Step 1:

Find formula.

1: Translate the statement into a direct variation formula

A varies directly as b means

A = k·b

2: Substitute known values to find k.

3 = k· 8

$k = \frac{3}{8} = 0.375$

3: Substitute k and write the formula.

A = 0.375b

Step Identify known variables and substitute into formula.

A = 0.375·1000

Step 3: Solve for Unknown

A = 375

Conclusion: A = 375 when b = 1000

Steps to Solve Word problems of Direct Variation

Word problems of direct variation are problems where the involved quantities have names, but the way to solve them is not different from the last section. The steps to solve then are the same

Step 1: Find formula.

Step 2: Identify known variables and substitute into formula.

Step 3: Solve for Unknown

Examples

Example 1

The amount of sale taxes on a new car is direct proportional to purchase price of the car, if a $25000 car pays $1750 in sales taxes. What is the purchase price of a new car which has a $3500 sale taxes?

Step 1: Find formula

1: Translate the statement into a direct variation formula.

The amount of sale taxes on a new car is direct proportional to purchase price of the car means

sale taxes = k· purchase price

2: Substitute known values to find k.

1750 = k 25000

$k = \frac{1750}{25000} = 0.07$

3: Substitute k and write the formula.

sale taxes = 0.07· purchase price

Step 2: Identify known variables and substitute into formula.

3500 = 0.07· purchase price

Step 3: Solve for Unknown

purchase price = 50000

Conclusion: A new car which pays $3,500 sale taxes has a purchase price of $50,000

Example 2

The cost of a House in Florida is proportional to the size of the house. A 2850-square-foot house cost $182400, then what is the cost of a 3640-square-foot house?

Step 1: Find formula

1: Translate the statement into a direct variation formula.

The cost of a House in Florida is proportional to the size of the house means

cost =k· size

2: Substitute known values to find k.

182400 = k 2850

$k = \frac{182400}{2850} = 64$

3: Substitute k and write the formula.

cost = 64 size

Step 2: Identify known variables and substitute into formula.

cost = 64· 3640

Step 3: Solve for Unknown

cost = 232960

Conclusion: The cost of a 3640-square-foot house in Florida is $232,960