# SAT Mathematics : Extrapolating Linear Proportions

## Example Questions

### Example Question #1 : Extrapolating Linear Proportions

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?

Explanation:

Total area of hall

At  person per  square feet,  per person  =  people

### Example Question #2 : Extrapolating Linear Proportions

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?

Explanation:

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 .

### Example Question #2 : Extrapolating Linear Proportions

A mile is  feet. Susan is able to walk a fast pace of  miles per hour.  How many feet will she walk in  minutes?

Explanation:

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

### Example Question #4 : Extrapolating Linear Proportions

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?

Explanation:

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 .

### Example Question #5 : Extrapolating Linear Proportions

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?

Explanation:

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.

### Example Question #1 : Extrapolating Linear Proportions

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?

Explanation:

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.

### Example Question #4 : Extrapolating Linear Proportions

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?

Explanation:

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 .

### Example Question #2 : Extrapolating Linear Proportions

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?

Explanation:

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.

### Example Question #3 : Extrapolating Linear Proportions

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?

Explanation:

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.

### Example Question #4 : Extrapolating Linear Proportions

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?

Explanation:

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:  .