# Algebra II : Direct Proportionality

## Example Questions

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### Example Question #112 : Sat Subject Test In Math I

Sarah notices her map has a scale of .  She measures between Beaver Falls and Chipmonk Cove.  How far apart are the cities?

Explanation:

is the same as

So to find out the distance between the cities

### Example Question #1 : Direct Proportionality

If an object is hung on a spring, the elongation of the spring varies directly as the mass of the object. A 20 kg object increases the length of a spring by exactly 7.2 cm. To the nearest tenth of a centimeter, by how much does a 32 kg object increase the length of the same spring?

Explanation:

Let  be the mass of the weight and the elongation of the spring. Then for some constant of variation

We can find  by setting  from the first situation:

so

In the second situation, we set  and solve for :

which rounds to 11.5 centimeters.

### Example Question #7 : Other Mathematical Relationships

Sunshine paint is made by mixing three parts yellow paint and one part red paint. How many gallons of yellow paint should be mixed with two quarts of red paint?

(1 gallon = 4 quarts)

Explanation:

First set up the proportion:

x =

Then convert this to gallons:

### Example Question #4 : Other Mathematical Relationships

Sally currently has 192 books. Three months ago, she had 160 books. By what percentage did her book collection increase over the past three months?

Explanation:

To find the percentage increase, divide the number of new books by the original amount of books:

### Example Question #9 : Other Mathematical Relationships

Find  for the proportion .

Explanation:

To find x we need to find the direct proportion. In order to do this we need to cross multiply and divide.

From here we mulitply 100 and 1 together. This gets us 100 and now we divide 100 by 4 which results in

### Example Question #10 : Other Mathematical Relationships

On a map of the United States, Mark notices a scale of    . If the distance between New York City and Los Angeles in real life is  , how far would the two cities be on Mark's map?

Explanation:

If the real distance between the two cities is  , and   =  , then we can set up the proportional equation:

### Example Question #1 : Proportionalities

If  and , find  and .

Explanation:

We cannot solve the first equation until we know at least one of the variables, so let's solve the second equation first to solve for . We therefore get:

With our , we can now find x using the first equation:

We therefore get the correct answer of  and .

### Example Question #2 : Direct Proportionality

If an object is hung on a spring, the elongation of the spring varies directly with the mass of the object. A 33 kilogram object increases the length of a spring by exactly 6.6 centimeters. To the nearest tenth of a kilogram, how much mass must an object posess to increase the length of that same spring by exactly 10 centimeters?

Explanation:

Let  be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation

.

We can find  by setting :

Therefore .

Set  and solve for :

kilograms

### Example Question #73 : Mathematical Relationships

If  is directly proportional to  and when  at , what is the value of the constant of proportionality?

Explanation:

The general formula for direct proportionality is

where  is the proportionality constant. To find the value of this , we plug in  and

Solve for  by dividing both sides by 12

So .

### Example Question #2 : How To Find Direct Variation

The amount of money you earn is directly proportional to the nunber of hours you worked. On the first day, you earned $32 by working 4 hours. On the second day, how many hours do you need to work to earn$48.

Explanation:

The general formula for direct proportionality is

where  is how much money you earned,  is the proportionality constant, and  is the number of hours worked.

Before we can figure out how many hours you need to work to earn $48, we need to find the value of . It is given that you earned$32 by working 4 hours. Plug these values into the formula

Solve for  by dividing both sides by 4.

So . We can use this to find out the hours you need to work to earn $48. With , we have Plug in$48.

Divide both sides by 8

So you will need to work 6 hours to earn \$48.

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