Direct and Inverse Variation

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ACT Math › Direct and Inverse Variation

Questions 1 - 10
1

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:

2

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:

3

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 :

4

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 :

5

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

6

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?

Explanation

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

For our data, this means:

You merely need to solve for :

7

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

8

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?

Explanation

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

For our data, this means:

You merely need to solve for :

9

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:

10

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:

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