# ACT Math : Direct and Inverse Variation

## Example Questions

### Example Question #1 : Direct And Inverse Variation

The price of silver varies directly as the square of the mass. If 3.6 g of silver is worth $64.80, what is the value of 7.5 g of silver? Possible Answers:$135.00

$281.25$301.75

$178.50$215.25

$281.25 Explanation: This is a direct variation problem of the form y = kx2 The first set of data 3.6 g and$64.80 is used to calculate the proportionality constant, k.  So 64.80 = k(3.6)2 and solving the equation gives k = 5.

Now we move to the new data, 7.5 g and we get y = 5(7.5)2 to yield an answer of $218.25.$135.00 is the answer obtained if using proportions.  This is an error because it does not take into consideration the squared elements of the problem.

### Example Question #2 : Variables

The diameter of a specific brand of candy wrapper is  longer than half the volume of the candy itself. Find the expression for the diameter, , in terms of the volume, .

Explanation:

The question asks for an equation that can relate  and  to each other, based on the information given. We are told that half the volume +  determines the total diameter.

This gives us:

### Example Question #1 : How To Use The Direct Variation Formula

An instrument reads two values,  and  daily. The values directly vary with respect to each other. If on Monday the value of  was  and  was , which of the following could be the values for  and  on Wednesday?

Explanation:

An instrument reads two values,  and  daily. The values directly vary with respect to each other. If on Monday the value of  was  and  was , which of the following could be the values for  and  on Wednesday?

Direct variation means that any pairing of the related values will always have the same ratio, thus we know that for any other values  and , those values will be equal according to the following equation:

Thus, for our information, we know:

This means that the new values of  and , when divided must be equal to . Therefore, the only possible answer is

### Example Question #2 : Direct And Inverse Variation

In an experimental setting, the pressure and the temperature of gas in a container are directly proportional. In the first experiment, the pressure of the gas was  and the temperature of the gas was , where  is a constant value not equal to . If a second experiment found a temperature of , which of the following represents the pressure of the gas?

It cannot be determined.

Explanation:

For direct variation of related variables, we know that the following equation holds:

For our data, this would be:

To start simplifying and solving this, first factor the top of the left fraction:

Cancel the s:

Next, multiply by :

Since  does not equal , cancel the s:

Simplify:

### Example Question #1 : Direct And Inverse Variation

In a given solution, the proportion of water to apple juice is directly proportional. If the first batch of the solution contained  gallons of apple juice and  gallons of water, how many gallons of apple juice will be needed for a solution containing  gallons of apple juice?

Explanation:

This problem is easily solved by setting up a ratio. For directly proportional amounts, we know:

For our data, this would be:

This is first simplified as being:

Next, multiply by  on both sides to solve for :

Finally:

### Example Question #4 : Direct And Inverse Variation

The length, in inches, of a box is . Which of the following gives the length,  inches, in terms of the width,  inches, of the box?

Explanation:

Since the question asks for the length in terms of width, I like to start by setting the problem up as , then add the conditions off the problem. I start at the first condition with  than the width and go ahead and add a  after , then it say  less than TWICE the width, so I add my  in front of the . This gives you ,  essentially you're working the problem piece by piece.

### Example Question #1 : How To Use The Inverse Variation Formula

varies inversely with . When , . What does  equal when

Explanation:

1. Use the given values of  and  to solve for :

2. Solve for  when  in the above equation:

### Example Question #6 : Direct And Inverse Variation

If , what is the value of ?

Explanation:

To solve this algebraic equation, subtract  from both sides, and then subtract  from both sides.

We end up with the equation , for which the solution is:

### Example Question #7 : Direct And Inverse Variation

In a given set of experiments, the values of two variables are always inversely proportional. If in the first experiment the first variable was  and the second was , what could you expect the second variable to be if the first is  in a later experiment?

Explanation:

Recall that inverse variation means that when one variable increases, the other decreases. This gives you the following equation:

Now, for our data, we know:

You merely have to solve for :

Divide by :

### Example Question #8 : Direct And Inverse Variation

Throughout a party, the number of joyful non-philosophers in a room is always inversely proportional to the number of philosophers in the room. The room begins with  people,  of whom are philosophers. Later in the day, there are  philosophers in the room. How many joyful non-philosophers are at the party at the later time?