# Algebra II : Proportionalities

## Example Questions

### Example Question #11 : Basic Single Variable Algebra

The quantity x varies directly with y. If x is 26 when y is 100, find x when y is 200.

26

52

104

13

6.5

52

Explanation:

We must set up a proportion. Since x varies directly with y, when y is multiplied by 2, x is also multiplied by 2. 26 times 2 is 52.

Direct variation:

### Example Question #126 : Sat Subject Test In Math I

The number of pizzas is directly proportional to the number of people attending the party. If  pizzas are needed for  people, how many pizzas are needed for  people?

Explanation:

Let  be the number of pizzas and  be the number of people attending. Because  is directly proportional to , we have

and from the problem

.

Therefore, solving for  we get:

Now solving for ,

Thus, for 18 people 9 pizzas are needed.

### Example Question #127 : Sat Subject Test In Math I

Resistance in a wire to the flow of electricity is calculated by  where  is the length of the wire,  is the cross sectional area, and  is the predefined resistivity of the material the wire is made of. Which variable (s) is/are directly proportional to the resistance of the wire?

Explanation:

Direct proportionality results when two quantities increase or decrease at the same time. In the formula for resistance, , as  increases so too does . The same is true for .  It helps to think of these variables in terms of numbers. When the numerator of this formula is increased and the denominator remains the same, the overall quotient would be larger. The opposite is true if the numerator is decreased and the denominator remains the same (quotient would be smaller).

### Example Question #1851 : Algebra Ii

Write the following equation:

k and x are directly proportionate to y.

Explanation:

Direct proportionality means that as one increases so does the other. The equation that is indicative of this statement is the following:

### Example Question #11 : Other Mathematical Relationships

The amount of money earned in a week while working in a snow shoveling business is directly proportional to the number of driveways that the workers shovel. If the business earned $3000 in a week while shoveling 600 driveways how much did they earn for shoveling each driveway? Possible Answers: Correct answer: Explanation: Direct proportionality means you should use the equation The question says that the amount of money earned by the business is directly proportional to the number of driveways shoveled so that means y must be the amount of money earned ($3000) and x must be the amount of driveways shoveled (600). Plugging these values into the equation you should get

Solve for k by doing the opposite of what is being done to k on both sides of the equation. Since k is being multiplied by 600 divide both sides by 600

Each driveway earned the snow shoveling business \$5

### Example Question #130 : Sat Subject Test In Math I

The force from gravity () is directly proportional to the inverse of the square of the distance () you are from an object.  Which formula shows that?

Explanation:

Breaking down the question one small piece at a time:

"The force from gravity () is directly proportional" means that the gravity term is going to be on one side of the proportional symbol, and everything else is going to be on the other side.

"to the inverse...of the distance" means that the distance will be in the denominator.  As distance gets larger, force will get smaller.

"of the square of the distance" means that the distance will be squared.

Putting it all together gives:

### Example Question #11 : Proportionalities

Billy's distance is proportional to his travel time.  If Billy walks 2 miles in five hours, how far would he have traveled if he walked for seven hours?

Explanation:

Write the direct proportionality relationship.

Assume  represents the time, and the  represents the distance traveled.

Substitute the given information to solve for the  constant.

Divide by five on both sides.

The value of  is:

Substitute this value back into the direct proportionality relationship.

Substitute seven into the  and simplify.

### Example Question #1851 : Algebra Ii

A car's distance traveled varies proportionally to the time it had traveled.  If the car travels 75 miles in 2 hours, what is the distance it had traveled in  hours?

Explanation:

Write the formula for direct proportionality.

The distance  is dependent on the time  that the car has traveled.  The variables are directly proportional to each other.

Rewrite the equation in terms of distance and time:

Substitute the known distance and time.

Solve for k by dividing by two.

The equation is:

Substitute the time  hours to determine the distance traveled.

The car will have traveled  in  hours.

### Example Question #13 : Proportionalities

A man wants to find the height of a building.  Being unable to climb the building to measure its height directly, he waits until it's a sunny day and stands so that the top of his shadow lines up with the top of the building's shadow.  He know's that he's  tall, and that he's  from where the shadows stop.  He them measures that he's  from the edge of the building.  What is the building's height?

Explanation:

First, let's make the height of the building .  We're looking for a ratio of the height of the object to the length of its shadow.  For the man, we know both of these things, so we can express the ratio:

We don't know the building's height, so let's just use  as a stand-in for the ratio for now.  Also, the length of the building's shadow is the length of the man's shadow plus the distance he is from the building:

We set these ratios equal to each other, and solve:

### Example Question #14 : Proportionalities

There are two similar buckets of cylindrical shape.  Bucket A has a height of  and a radius of .  Bucket B has a height of .  How much can Bucket B hold?