Proportionalities - Algebra II
Card 0 of 20
The quantity x varies directly with y. If x is 26 when y is 100, find x when y is 200.
The quantity x varies directly with y. If x is 26 when y is 100, find x when y is 200.
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: 
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:
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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)
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)
First set up the proportion:
x = 
Then convert this to gallons:
First set up the proportion:
x =
Then convert this to gallons:
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Sarah notices her map has a scale of 
. She measures 
between Beaver Falls and Chipmonk Cove. How far apart are the cities?
Sarah notices her map has a scale of
is the same as 
So to find out the distance between the cities
So to find out the distance between the cities
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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?
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?
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.
Let
We can find
so
In the second situation, we set
which rounds to 11.5 centimeters.
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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?
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?
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
Let
We can find
Therefore
Set
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If 
is directly proportional to 
and when 
at 
, what is the value of the constant of proportionality?
If
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 
.
The general formula for direct proportionality is
where
Solve for
So
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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.
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.
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.
The general formula for direct proportionality is
where
Before we can figure out how many hours you need to work to earn $48, we need to find the value of
Solve for
So
Plug in $48.
Divide both sides by 8
So you will need to work 6 hours to earn $48.
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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?
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?
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.
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.
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Find 
for the proportion 
.
Find
To find x we need to find the direct proportion. In order to do this we need to cross multiply and divide.
From here we mulitply 100 and 1 together. This gets us 100 and now we divide 100 by 4 which results in
To find x we need to find the direct proportion. In order to do this we need to cross multiply and divide.
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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On a map of the United States, Mark notices a scale of 
. 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?
On a map of the United States, Mark notices a scale of
If the real distance between the two cities is 
, and 
= 
, then we can set up the proportional equation:
If the real distance between the two cities is
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If 
and 
, find 
and 
.
If
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 
. We therefore get:
With our 
, we can now find x using the first equation:
We therefore get the correct answer of 
and 
.
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
With our
We therefore get the correct answer of
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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?
The number of pizzas is directly proportional to the number of people attending the party. If
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.
Let
and from the problem
Therefore, solving for
Now solving for
Thus, for 18 people 9 pizzas are needed.
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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?
Resistance in a wire to the flow of electricity is calculated by
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).
Direct proportionality results when two quantities increase or decrease at the same time. In the formula for resistance,
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Write the following equation:
k and x are directly proportionate to y.
Write the following equation:
k and x are directly proportionate to y.
Direct proportionality means that as one increases so does the other. The equation that is indicative of this statement is the following:
Direct proportionality means that as one increases so does the other. The equation that is indicative of this statement is the following:
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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?
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?
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
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
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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?
The force from gravity (
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:
Breaking down the question one small piece at a time:
"The force from gravity (
"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:
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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?
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?
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.
The answer is: 
Write the direct proportionality relationship.
Assume
Substitute the given information to solve for the
Divide by five on both sides.
The value of
Substitute this value back into the direct proportionality relationship.
Substitute seven into the
The answer is:
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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?
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
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.
Write the formula for direct proportionality.
The distance
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
The car will have traveled
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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?
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?
Write the equation for direct proportionality.
Substitute the distance and time to solve for the 
constant.
Divide by 70 on both sides.
Substitute this value back to 
.
The equation becomes:
Substitute 
to determine how long it took the runner to run 4 miles.
Multiply by 
on both sides to isolate the time variable.
Simplify both sides.
The answer is: 
Write the equation for direct proportionality.
Substitute the distance and time to solve for the
Divide by 70 on both sides.
Substitute this value back to
The equation becomes:
Substitute
Multiply by
Simplify both sides.
The answer is:
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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?
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
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:
First, let's make the height of the building
We don't know the building's height, so let's just use
We set these ratios equal to each other, and solve:
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