Extrapolating Linear Proportions - SAT Math
Card 0 of 40
A cafeteria with 
tables can sit 
people. Some tables can sit 
people and some can sit 
people. What is the ratio of the number of 
-person tables to the number of 
-person tables?
A cafeteria with 
Let 
be the number of 
-person tables, and 
be the number of 
-person tables. Since there are 
tables in the cafeteria, 
. 
represents the number of people sitting at 
-person tables, and 
represents the number of people sitting at 
-person tables. Since the cafeteria can seat 
people, 
. Now we have 
equations and 
unknowns, and can solve the system. To do this, multiply the first equation by 
and subtract it from the second equation. This yields 
; solving for 
tells us there are 
tables that seat 
people. Since 
, 
, so there are 
tables that seat 
people. The ratio of 
is therefore 
.
Let 
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You are planning a party. The maximum number of people the reception hall can hold is 
person for every 
square feet of space. If the hall is 
feet wide and 
feet long, how many people can you invite?
You are planning a party. The maximum number of people the reception hall can hold is 
Total area of hall 
At 
person per 
square feet, 
per person = 
people
Total area of hall
At 
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A mile is 
feet. Susan is able to walk a fast pace of 
miles per hour. How many feet will she walk in 
minutes?
A mile is 
Calculate the number of feet walked in an hour. Then calculate what fraction of an hour 
minutes is.
feet walked in an hour
minutes in an hour, so 
minutes = 
hour 
feet walked in 
minutes
Calculate the number of feet walked in an hour. Then calculate what fraction of an hour 
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A bag contains 
marbles that are either red, blue, or green. The ratio of red to blue to green marbles is 
. If one-third of the red marbles and two-thirds of the green marbles are removed, what fraction of the remaining marbles in the bag will be blue?
A bag contains 
First, we need to figure out how many red, blue, and green marbles are in the bag before any are removed. Let 
represent the number of red marbles. Because the marbles are in a ratio of 
, then if there are 
red marbles, there are 
blue, and 
green marbles. If we add up all of the marbles, we will get the total number of marbles, which is 
.
Because the number of red marbles is 
, there are 
, or 
red marbles. There are 
, or 
blue marbles, and there are 
, or 
green marbles.
So, the bag originally contains 
red, 
blue, and 
green marbles. We are then told that one-third of the red marbles is removed. Because one-third of 
is 
, there would be 
red marbles remaining. Next, two-thirds of the green marbles are removed. Because 
, there would be 
green marbles left after 
are removed.
To summarize, after the marbles are removed, there are 
red, 60 blue, and 
green marbles. The question asks us for the fraction of blue marbles in the bag after the marbles are removed. This means there would be 
blue marbles out of the 
left in the bag. The fraction of blue marbles would therefore be 
, which simplifies to 
.
The answer is 
.
First, we need to figure out how many red, blue, and green marbles are in the bag before any are removed. Let 
Because the number of red marbles is 
So, the bag originally contains 
To summarize, after the marbles are removed, there are 
The answer is 
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Tom wants to buy an aquarium. He has found that he needs one filter for the aquarium for every 
creatures or plants he puts in the tank. The fish he wants to get also requires special plants be present at 
plants for every 
fish. These plants need 
cleaning fish for every 
plants in order to flourish.
If Tom is going to buy 
filters, how many of the original fish he wanted will his aquarium support?
Tom wants to buy an aquarium. He has found that he needs one filter for the aquarium for every 
If Tom is going to buy 
filters can support a total of 
creatures/plants. The fish he wants need 
plants for every 
fish. The plant needs 
cleaning fish per 
plants. Thus for every 
of the fish he wants, he needs 
plants and 
cleaning fish.
This gives us a total of 
creatures. We can complete this number 
times, but then we are left with 
spots open that the filters can support.
This is where the trick arises. We can actually add one more fish in. Since 
plant supports up to 
fish 
, and 2 cleaning fish support up to 
plants, we can add 
fish, 
plant, and 
cleaning fish to get a total of 
creatures. If we attempt to add 
fish, then we must also add the 
plant, but then we don't have enough space left to add the 
cleaning fish necessary to support the remaining plant.
Thus, Tom can buy at most 
of the fish he originally wanted to get.
This gives us a total of 
This is where the trick arises. We can actually add one more fish in. Since 
Thus, Tom can buy at most 
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In the reptile house at the zoo, the ratio of snakes to lizards is 
to 
. After the zoo adds 
more snakes to the exhibit, the ratio changes to 
to 
. How many lizards are in the reptile house?
In the reptile house at the zoo, the ratio of snakes to lizards is 
In order to maintain a proportion, each value in the ratio must be multiplied by the same value:
Before and after the snakes arrive, the number of lizards stays constant.
Before new snakes — Snakes : Lizards = 
After new snakes — Snakes : Lizards = 
Before the new snakes arrive, there are 
snakes. After the 
snakes are added, there are 
snakes. Therefore, 
. Solving for 
gives 
.
There are 
lizards, or 
lizards.
In order to maintain a proportion, each value in the ratio must be multiplied by the same value:
Before and after the snakes arrive, the number of lizards stays constant.
Before new snakes — Snakes : Lizards =
After new snakes — Snakes : Lizards =
Before the new snakes arrive, there are 
There are 
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A small company's workforce consists of store employees, store managers, and corporate managers in the ratio 
. How many employees are either corporate managers or store managers if the company has a total of 
employees?
A small company's workforce consists of store employees, store managers, and corporate managers in the ratio 
Let 
be the number of store employees, 
the number of store managers, and 
the number of corporate managers.
, so the number of store employees is 
.
, so the number of store managers is 
.
, so the number of corporate managers is 
.
Therefore, the number of employees who are either store managers or corporate managers is 
.
Let 
Therefore, the number of employees who are either store managers or corporate managers is 
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The exchange rate in some prehistoric village was 
jagged rocks for every 
smooth pebbles. Also, one shiny rock could be traded for 
smooth pebbles. If Joaquin had 
jagged rocks, what is the maximum number of shiny rocks he could trade for?
The exchange rate in some prehistoric village was 
We can use dimensional analysis to solve this problem. We will create ratios from the conversions given.
Since Joaquin cannot trade for part of a shiny rock, the most he can get is 
shiny rocks.
We can use dimensional analysis to solve this problem. We will create ratios from the conversions given.
Since Joaquin cannot trade for part of a shiny rock, the most he can get is 
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In a flower bed, Joaquin plants 
Begonias for every 
Zinnias, and 
Marigolds for every 
Begonias. What is the ratio of Marigolds to Zinnias planted in the flower bed?
In a flower bed, Joaquin plants 
First, we should write a fraction for each ratio given:
Next, we will multiply these fractions by each other in such a way that will leave us with a fraction that has only Z and M, since we want a ratio of these two flowers only.
So the final answer is 
First, we should write a fraction for each ratio given:
Next, we will multiply these fractions by each other in such a way that will leave us with a fraction that has only Z and M, since we want a ratio of these two flowers only.
So the final answer is
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A lawn can be mowed by 
people in 
hours. If 
people take the day off and do not help mow the grass, how many hours will it take to mow the lawn?
A lawn can be mowed by 
The number of hours required to mow the lawn remains constant and can be found by taking the original 
workers times the 
hours they worked, totaling 
hours. We then split the total required hours between the 
works that remain, and each of them have to work 
and 
hours: 
.
The number of hours required to mow the lawn remains constant and can be found by taking the original 
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A cafeteria with tables can sit people. Some tables can sit people and some can sit people. What is the ratio of the number of -person tables to the number of -person tables?
A cafeteria with
Let be the number of -person tables, and be the number of -person tables. Since there are tables in the cafeteria, . represents the number of people sitting at -person tables, and represents the number of people sitting at -person tables. Since the cafeteria can seat people, . Now we have equations and unknowns, and can solve the system. To do this, multiply the first equation by and subtract it from the second equation. This yields ; solving for tells us there are tables that seat people. Since , , so there are tables that seat people. The ratio of is therefore .
Let
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You are planning a party. The maximum number of people the reception hall can hold is person for every square feet of space. If the hall is feet wide and feet long, how many people can you invite?
You are planning a party. The maximum number of people the reception hall can hold is
Total area of hall
At person per square feet, per person = people
Total area of hall
At
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A mile is feet. Susan is able to walk a fast pace of miles per hour. How many feet will she walk in minutes?
A mile is
Calculate the number of feet walked in an hour. Then calculate what fraction of an hour minutes is.
feet walked in an hour
minutes in an hour, so minutes = hour
feet walked in minutes
Calculate the number of feet walked in an hour. Then calculate what fraction of an hour
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A bag contains marbles that are either red, blue, or green. The ratio of red to blue to green marbles is . If one-third of the red marbles and two-thirds of the green marbles are removed, what fraction of the remaining marbles in the bag will be blue?
A bag contains
First, we need to figure out how many red, blue, and green marbles are in the bag before any are removed. Let represent the number of red marbles. Because the marbles are in a ratio of , then if there are red marbles, there are blue, and green marbles. If we add up all of the marbles, we will get the total number of marbles, which is .
Because the number of red marbles is , there are , or red marbles. There are , or blue marbles, and there are , or green marbles.
So, the bag originally contains red, blue, and green marbles. We are then told that one-third of the red marbles is removed. Because one-third of is , there would be red marbles remaining. Next, two-thirds of the green marbles are removed. Because , there would be green marbles left after are removed.
To summarize, after the marbles are removed, there are red, 60 blue, and green marbles. The question asks us for the fraction of blue marbles in the bag after the marbles are removed. This means there would be blue marbles out of the left in the bag. The fraction of blue marbles would therefore be , which simplifies to .
The answer is .
First, we need to figure out how many red, blue, and green marbles are in the bag before any are removed. Let
Because the number of red marbles is
So, the bag originally contains
To summarize, after the marbles are removed, there are
The answer is
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Tom wants to buy an aquarium. He has found that he needs one filter for the aquarium for every creatures or plants he puts in the tank. The fish he wants to get also requires special plants be present at plants for every fish. These plants need cleaning fish for every plants in order to flourish.
If Tom is going to buy filters, how many of the original fish he wanted will his aquarium support?
Tom wants to buy an aquarium. He has found that he needs one filter for the aquarium for every
If Tom is going to buy
filters can support a total of creatures/plants. The fish he wants need plants for every fish. The plant needs cleaning fish per plants. Thus for every of the fish he wants, he needs plants and cleaning fish.
This gives us a total of creatures. We can complete this number times, but then we are left with spots open that the filters can support.
This is where the trick arises. We can actually add one more fish in. Since plant supports up to fish , and 2 cleaning fish support up to plants, we can add fish, plant, and cleaning fish to get a total of creatures. If we attempt to add fish, then we must also add the plant, but then we don't have enough space left to add the cleaning fish necessary to support the remaining plant.
Thus, Tom can buy at most of the fish he originally wanted to get.
This gives us a total of
This is where the trick arises. We can actually add one more fish in. Since
Thus, Tom can buy at most
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In the reptile house at the zoo, the ratio of snakes to lizards is to . After the zoo adds more snakes to the exhibit, the ratio changes to to . How many lizards are in the reptile house?
In the reptile house at the zoo, the ratio of snakes to lizards is
In order to maintain a proportion, each value in the ratio must be multiplied by the same value:
Before and after the snakes arrive, the number of lizards stays constant.
Before new snakes — Snakes : Lizards =
After new snakes — Snakes : Lizards =
Before the new snakes arrive, there are snakes. After the snakes are added, there are snakes. Therefore, . Solving for gives .
There are lizards, or lizards.
In order to maintain a proportion, each value in the ratio must be multiplied by the same value:
Before and after the snakes arrive, the number of lizards stays constant.
Before new snakes — Snakes : Lizards =
After new snakes — Snakes : Lizards =
Before the new snakes arrive, there are
There are
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A small company's workforce consists of store employees, store managers, and corporate managers in the ratio . How many employees are either corporate managers or store managers if the company has a total of employees?
A small company's workforce consists of store employees, store managers, and corporate managers in the ratio
Let be the number of store employees, the number of store managers, and the number of corporate managers.
, so the number of store employees is .
, so the number of store managers is .
, so the number of corporate managers is .
Therefore, the number of employees who are either store managers or corporate managers is .
Let
Therefore, the number of employees who are either store managers or corporate managers is
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The exchange rate in some prehistoric village was jagged rocks for every smooth pebbles. Also, one shiny rock could be traded for smooth pebbles. If Joaquin had jagged rocks, what is the maximum number of shiny rocks he could trade for?
The exchange rate in some prehistoric village was
We can use dimensional analysis to solve this problem. We will create ratios from the conversions given.
Since Joaquin cannot trade for part of a shiny rock, the most he can get is shiny rocks.
We can use dimensional analysis to solve this problem. We will create ratios from the conversions given.
Since Joaquin cannot trade for part of a shiny rock, the most he can get is
Compare your answer with the correct one above
In a flower bed, Joaquin plants Begonias for every Zinnias, and Marigolds for every Begonias. What is the ratio of Marigolds to Zinnias planted in the flower bed?
In a flower bed, Joaquin plants
First, we should write a fraction for each ratio given:
Next, we will multiply these fractions by each other in such a way that will leave us with a fraction that has only Z and M, since we want a ratio of these two flowers only.
So the final answer is
First, we should write a fraction for each ratio given:
Next, we will multiply these fractions by each other in such a way that will leave us with a fraction that has only Z and M, since we want a ratio of these two flowers only.
So the final answer is
Compare your answer with the correct one above
A lawn can be mowed by people in hours. If people take the day off and do not help mow the grass, how many hours will it take to mow the lawn?
A lawn can be mowed by
The number of hours required to mow the lawn remains constant and can be found by taking the original workers times the hours they worked, totaling hours. We then split the total required hours between the works that remain, and each of them have to work and hours: .
The number of hours required to mow the lawn remains constant and can be found by taking the original
Compare your answer with the correct one above