Proportionalities - Algebra II

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Question

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

Answer

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: $\frac{x_1}{x_2}=\frac{y_1}{y_2}$

$\frac{26}{x_2}=\frac{100}{200}$

$\frac{26}{x_2}=\frac{1}{2}$

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Question

varies inversely with and the square root of . When and , $y=\frac{2}{9}$ . Find when and .

Answer

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

$y = k \cdot \frac{1}{x \cdot \sqrt{z}}$

Next, we substitute the values given in the first scenario to solve for $\small k$ :

$\frac{2}{9} = k \cdot \frac{1}{4 \cdot \sqrt{81}} = k \cdot \frac{1}{4 \cdot 9} = k \cdot \frac{1}{36}$

$\small \rightarrow k = 36 \cdot \frac{2}{9} = \frac{36}{9} \cdot 2 = 4 \cdot 2 = 8$

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

$\small Y = 8 \cdot \frac{1}{8 \cdot \sqrt{4}} = 8 \cdot \frac{1}{8 \cdot 2} = 8 \cdot \frac{1}{16} = \frac{1}{2}$

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Question

Sunshine paint is made by mixing three parts yellow paint and one part red paint. How many gallons of yellow paint should be mixed with two quarts of red paint?

(1 gallon = 4 quarts)

Answer

First set up the proportion:

$\frac{3; parts ; yellow}{1 ; part; red}=\frac{x ; quarts ; yellow}{2 ; quarts ; red}$

x =

Then convert this to gallons:

$6 ; quarts; \cdot \frac{1 ; gallon}{4; quarts}= 1.50; gallons$

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Question

Sarah notices her map has a scale of $\frac{1}{4}; in=1; mile$ . She measures between Beaver Falls and Chipmonk Cove. How far apart are the cities?

Answer

$\frac{1}{4}; in=1; mile$ is the same as

So to find out the distance between the cities

$12.5; in \cdot \frac{4; miles}{1;in }=50; miles$

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Question

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?

Answer

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:

$7.2 = k \cdot 20$

$k = 7.2 \div 20 = 0.36$

so

In the second situation, we set and solve for :

$L = 0.36 \cdot32 =11.52$

which rounds to 11.5 centimeters.

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Question

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?

Answer

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 :

$6.6 = k \cdot 33$

$k = 6.6 \div 33 = 0.2$

Therefore .

Set and solve for :

$M = 10 \div 0.2 = 50$ kilograms

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Question

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

Answer

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

$\frac{24}{12}=\frac{12k}{12}$

So .

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Question

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.

Answer

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.

$\frac{32}{4}=\frac{4k}{4}$

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

$\frac{48}{8}=\frac{8h}{8}$

So you will need to work 6 hours to earn $48.

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Question

Sally currently has 192 books. Three months ago, she had 160 books. By what percentage did her book collection increase over the past three months?

Answer

To find the percentage increase, divide the number of new books by the original amount of books:

She has 32 additional new books; she originally had 160.

$\frac{32}{160} = \frac{16}{80} = 0.20=20%$

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Question

Find for the proportion $\frac{x}{100}=\frac{1}{4}$ .

Answer

To find x we need to find the direct proportion. In order to do this we need to cross multiply and divide.

$\frac{x}{100}=\frac{1}{4}$

From here we mulitply 100 and 1 together. This gets us 100 and now we divide 100 by 4 which results in

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Question

On a map of the United States, Mark notices a scale of $\small in = 250$ . If the distance between New York City and Los Angeles in real life is , how far would the two cities be on Mark's map?

Answer

If the real distance between the two cities is , and = , then we can set up the proportional equation:

$\frac{1 in}{250 miles}=\frac{xin}{2400 miles}$

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Question

If $\small \small \small \frac{x}{108}=\frac{20}{y}$ and $\small \frac{224}{y}=\frac{14}{3}$ , find $\small x$ and $\small y$ .

Answer

We cannot solve the first equation until we know at least one of the variables, so let's solve the second equation first to solve for $\small y$ . We therefore get:

$\small \frac{224}{y}=\frac{14}{3}\rightarrow14y=672\rightarrow y=48$

With our $\small y$ , we can now find x using the first equation:

$\small \frac{x}{108}=\frac{20}{48}\rightarrow48x=2160\rightarrow x=45$

We therefore get the correct answer of $\small x=45$ and $\small y=48$ .

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Question

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?

Answer

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.

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Question

Resistance in a wire to the flow of electricity is calculated by $R=\frac{\rho L}{A}$ where is the length of the wire, is the cross sectional area, and $\rho$ 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?

Answer

Direct proportionality results when two quantities increase or decrease at the same time. In the formula for resistance, $R=\frac{\rho L}{A}$ , as increases so too does . The same is true for $\rho$ . 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).

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Question

Write the following equation:

k and x are directly proportionate to y.

Answer

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

$\small y=kx$

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Question

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?

Answer

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

$\frac{3000}{600}=k$

Each driveway earned the snow shoveling business $5

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Question

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?

Answer

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:

$g\propto \frac{1}{d^{2}}$

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Question

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?

Answer

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.

$\frac{2}{5}=\frac{k(5)}{5}$

The value of is: $k=\frac{2}{5}$

Substitute this value back into the direct proportionality relationship.

$y=\frac{2}{5}x$

Substitute seven into the and simplify.

$y=\frac{2}{5}(7) = \frac{14}{5}$

The answer is: $\frac{14}{5}$

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Question

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?

Answer

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.

$k=\frac{75}{2}$

The equation is: $D=\frac{75}{2}T$

Substitute the time hours to determine the distance traveled.

$D=\frac{75}{2}(3.5) = 131.25$

The car will have traveled in hours.

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Question

Suppose a runner's distance is directly proportional to her time. If the runner completes 6 miles in 70 minutes, how many minutes did it take the runner to run 4 miles?

Answer

Write the equation for direct proportionality.

$y=kx \rightarrow D = kt$

Substitute the distance and time to solve for the constant.

Divide by 70 on both sides.

$\frac{6}{70} =\frac{k(70)}{70}$

$k=\frac{3}{35}$

Substitute this value back to .

The equation becomes:

$D = \frac{3}{35}t$

Substitute to determine how long it took the runner to run 4 miles.

$4 = \frac{3}{35}t$

Multiply by $\frac{35}{3}$ on both sides to isolate the time variable.

$4 \cdot\frac{35}{3}= \frac{3}{35}t\cdot\frac{35}{3}$

Simplify both sides.

The answer is: $\frac{140}{3}$

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