# Algebra II : Proportionalities

## Example Questions

### Example Question #1 : 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 #1 : 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"

### Example Question #11 : Indirect Proportionality

varies directly with  and . If  and , then . Find if  and

None of these answers are correct

Explanation:

From the relationship of , , and ; the equation of variation can be written as:

Using the values given in the first scenario, we can solve for k:

Using the value of k and the values of y and z, we can solve for x:

### Example Question #12 : Indirect Proportionality

varies inversely with  and the square root of . When  and . Find  when  and .

None of these answers are correct

Explanation:

First, we can create an equation of variation from the the relationships given:

Next, we substitute the values given in the first scenario to solve for :

Using the value for , we can now use the second values for  and  to solve for :

### Example Question #11 : Indirect Proportionality

varies directly with  and the square root of . If , and  then . Find the value of  if  and .

None of these answers are correct

Explanation:

From the relationship given, we can set up the variation equation

Using the first relationship, we can then solve for

Now using the values from the second relationship, we can solve for x

### Example Question #1 : How To Find Inverse Variation

varies inversely as the square 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 #1871 : Algebra Ii

The current, in amperes, that a battery provides an electrical object is inversely proportional to the resistance, in ohms, of the object.

A battery provides 1.2 amperes of current to a flashlight whose resistance is measured at 20 ohms. How much current will the same battery supply to a flashlight whose resistance is measured at 16 ohms?

Explanation:

If  is the current and  is the resistance, then we can write the variation equation for some constant of variation :

or, alternatively,

To find  , substitute :

The equation is . Now substitute  and solve for :

### Example Question #1 : How To Find Direct Variation

The volume of a fixed mass of gas varies inversely as the atmospheric pressure, as measured in millibars, acting on it, and directly as the temperature, as measured in kelvins, acting on it.

A balloon is filled to a capacity of exactly 100 cubic meters at a time at which the temperature is 310 kelvins and the atmospheric pressure is 1,020 millibars. The balloon is released, and an hour later, the balloon is subject to a pressure of 900 millibars and a temperature of 290 kelvins. To the nearest cubic meter, what is the new volume of the balloon?

Explanation:

If  are the volume, pressure, and temperature, then the variation equation will be, for some constant of variation ,

To calculate , substitute :

The variation equation is

so substitute  and solve for

### Example Question #31 : Other Mathematical Relationships

If  is inversely proportional to  and knowing that  when , determine the proportionality constant.

Explanation:

The general formula for inverse proportionality for this problem is

Given that  when , we can find  by plugging them into the formula.

Solve for  by multiplying both sides by 5

So .

### Example Question #31 : Basic Single Variable Algebra

The number of days needed to construct a house is inversely proportional to the number of people that help build the house. It took 28 days to build a house with 7 people. A second house is being built and it needs to be finished in 14 days. How many people are needed to make this happen?

Explanation:

The general formula of inverse proportionality for this problem is

where  is the number of days,  is the proportionality constant, and  is number of people.

Before finding the number of people needed to build the house in 14 days, we need to find . Given that the house can be built in 28 days with 7 people, we have

Multiply both sides by 7 to find .

So . Thus,

Now we can find the how many people are needed to build the house in 14 days.

Solve for . First, multiply by  on both sides:

Divide both sides by 14

So it will take 14 people to complete the house in 14 days.