# Algebra II : Indirect Proportionality

## Example Questions

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### Example Question #6 : How To Find Inverse Variation

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 #6 : How To Find Inverse Variation

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 #23 : Basic Single Variable Algebra

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 #24 : Basic Single Variable Algebra

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 #3 : 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 #25 : Basic Single Variable Algebra

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 #1 : 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 #27 : Basic Single Variable Algebra

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.

### Example Question #2 : Indirect Proportionality

varies inversely with . If  ,  . What is the value of  if  ?

Explanation:

varies inversely with , so the variation equation can be written as:

can be solved for, using the first scenario:

Using this value for  = 30 and  = 90, we can solve for :

### Example Question #3 : Indirect Proportionality

varies directly with  and inversely with the square root of . Find values for  and  that will give , for a constant of variation .

and

and

All of these answers are correct

and

All of these answers are correct

Explanation:

From the first sentence, we can write the equation of variation as:

We can then check each of the possible answer choices by substituting the values into the variation equation with the values given for  and .

Therefore the equation is true if  and

Therefore the equation is true if  and

Therefore the equation is true if  and

The correct answer choice is then "All of these answers are correct"

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