# Algebra 1 : How to find direct variation

## Example Questions

### Example Question #2 : Direct Proportionality

If an object is hung on a spring, the elongation of the spring varies directly as the mass of the object. A 20 kg object increases the length of a spring by exactly 7.2 cm. To the nearest tenth of a centimeter, by how much does a 32 kg object increase the length of the same spring?

Explanation:

Let  be the mass of the weight and the elongation of the spring. Then for some constant of variation

We can find  by setting  from the first situation:

so

In the second situation, we set  and solve for :

which rounds to 11.5 centimeters.

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

varies directly with the square root of . If , then  . What is the value of  if ?

None of these answers are correct.

Explanation:

If  varies directly with the square root of , then for some constant of variation

If , then ; therefore, the equation becomes

or

.

Divide by 5 to get , making the equation

.

If , then .

### Example Question #8 : Direct Proportionality

If an object is hung on a spring, the elongation of the spring varies directly with the mass of the object. A 33 kilogram object increases the length of a spring by exactly 6.6 centimeters. To the nearest tenth of a kilogram, how much mass must an object posess to increase the length of that same spring by exactly 10 centimeters?

Explanation:

Let  be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation

.

We can find  by setting :

Therefore .

Set  and solve for :

kilograms

### Example Question #9 : Direct Proportionality

If  is directly proportional to  and when  at , what is the value of the constant of proportionality?

Explanation:

The general formula for direct proportionality is

where  is the proportionality constant. To find the value of this , we plug in  and

Solve for  by dividing both sides by 12

So .

### Example Question #10 : Direct Proportionality

The amount of money you earn is directly proportional to the nunber of hours you worked. On the first day, you earned $32 by working 4 hours. On the second day, how many hours do you need to work to earn$48.

Explanation:

The general formula for direct proportionality is

where  is how much money you earned,  is the proportionality constant, and  is the number of hours worked.

Before we can figure out how many hours you need to work to earn $48, we need to find the value of . It is given that you earned$32 by working 4 hours. Plug these values into the formula

Solve for  by dividing both sides by 4.

So . We can use this to find out the hours you need to work to earn $48. With , we have Plug in$48.

Divide both sides by 8

So you will need to work 6 hours to earn $48. ### Example Question #101 : Mathematical Relationships 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? Possible Answers: Correct answer: 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 #1 : Equations The monthly cost to insure your cars varies directly with the number of cars you own. Right now, you are paying$420 per month to insure 3 cars, but you plan to get 2 more cars, so that you will own 5 cars. How much does it cost to insure 5 cars monthly?

Explanation:

The statement, 'The monthly costly to insure your cars varies directly with the number of cars you own' can be mathematically expressed as . M is the monthly cost, C is the number of cars owned, and k is the constant of variation.