# Algebra II : Proportionalities

## Example Questions

### Example Question #21 : Basic Single Variable Algebra

Suppose a runner's distance is directly proportional to her time.  If the runner completes 6 miles in 70 minutes, how many minutes did it take the runner to run 4 miles?

Explanation:

Write the equation for direct proportionality.

Substitute the distance and time to solve for the  constant.

Divide by 70 on both sides.

Substitute this value back to .

The equation becomes:

Substitute  to determine how long it took the runner to run 4 miles.

Multiply by  on both sides to isolate the time variable.

Simplify both sides.

### Example Question #22 : Direct Proportionality

The distance of a cyclist is directly proportional to the time he has traveled. Suppose he has traveled 12 miles in 1.5 hours. How far does he travel in a half hour?

Explanation:

Write the equation for direct proportionality.

Substitute the distance and time given to solve for the constant of proportionality, .

Divide by 1.5 on both sides.

Write the equation.

Substitute half an hour for the time to determine the distance the biker has traveled.

The biker traveled four miles in a half hour.

### Example Question #1 : Indirect Proportionality

varies directly with , and inversely with the square root of .

If  and , then .

Find  if  and .

Explanation:

The variation equation can be written as below. Direct variation will put in the numerator, while inverse variation will put in the denominator. is the constant that defines the variation.

To find constant of variation, , substitute the values from the first scenario given in the question.

We can plug this value into our variation equation.

Now we can solve for given the values in the second scenario of the question.

### Example Question #2 : Indirect Proportionality

varies inversely as the square root of . If , then . Find  if  (nearest tenth, if applicable).

Explanation:

The variation equation is  for some constant of variation .

Substitute the numbers from the first scenario to find :

The equation is now .

If , then

### Example Question #2 : Indirect Proportionality

varies inversely with three times the square root of . If , then

Find  if . Round to the nearest tenth if applicable.

Explanation:

In order to find the value of  when , first determine the variation equation based on the information provided:

, for some constant of variation .

Insert the  and  values from the first variance to find the value of :

Now that we know , the variation equation becomes:

or

.

Therefore, when :

### Example Question #4 : Indirect Proportionality

varies directly with two times  and varies indirectly with three times . When

and .

What is the value of  when  and  Round to the nearest tenth if needed.

Explanation:

In order to solve for , first set up the variation equation for   and :

where  is the constant of variation. The  term varies indirectly with  and is therefore in the denominator.

Next, we solve for  based on the initial values of the variables:

Now that we have the value of , we can solve for  in the second scenario:

### Example Question #5 : Indirect Proportionality

The number of slices of pizza you get varies indirectly with the total number of people in the restaurant. If you get  slices when there are  people, how many slices would you get if there are  people?

Explanation:

The problem follows the formula

where P is the number of slices you get, n is the number of people, and k is the constant of variation.

Setting P=3 and n = 16 yields k=48.

Now we substitute 12 in for n and solve for P

Therefore with 12 people, you get 4 slices.

### Example Question #21 : Other Mathematical Relationships

The number of raffle tickets given for a contest varies indirectly with the total number of people in the building. If you get  tickets when there are  people, how many slices would you get if there are  people?

Explanation:

The problem follows the formula

where R is the number of raffle tickets you get, n is the number of people, and k is the constant of variation.

Setting R=20 and n = 150 yields k=3000.

Plugging in 100 for n and solving for R you get:

The answer R is 30 tickets.

### Example Question #7 : Indirect Proportionality

The budget per committee varies indirectly with the total number of committees created. If each committee is allotted  when four committees are established, what would the budget per committee be if there were to be  committees?

Explanation:

The problem follows the formula

where B is the budget per committee, n is the number of committees, and k is the constant of variation.

Setting B=500 and n = 4 yields k=2000.

Now using the following equation we can plug in our n of 2 and solve for B.

The answer of B is \$1000.

### Example Question #8 : Indirect Proportionality

The number of hours needed for a contractor to finish a job varies indirectly with the total number of people the contractor hires. If the job is completed in  hours when there are  people, how many hours would it take if there were  people?

Explanation:

The problem follows the formula

where H is the number of hours to complete the job, n is the number of people hired, and k is the constant of variation.

Setting H=6 and n = 8 yields k=48.

Therefore using the following equation we can plug 16 in for n and solve for H.

Therefore H is 3 hours.