# Proportions

Read the following SAT test question and then select the correct answer.

The Knowsys Method is to read the question carefully and identify the bottom line.  Think about your options for solving the problem and choose the most efficient method.  Then attack the problem, solve it, and loop back to make sure that you solved for the bottom line.

A gasoline tank on a certain tractor holds 16 gallons of gasoline. If the tractor requires 7 gallons to plow 3 acres, how many acres can the tractor plow with a tankful of gasoline?
First, find the bottom line.  Your bottom line asks how many acres can be plowed with one tankful of gasoline.  A tankful is 16 gallons and you have information about how many gallons it takes to plow a certain number of acres.  Consider your options: you could use basic arithmetic to estimate the answer, you could solve for one gallon of gasoline and then multiply it by 16, or you could use a proportion.  Use a proportion because it only requires you to set up one problem.  Always remember that the long way is the wrong way!  Set up your proportion showing acres per gallon and use an x to mark your bottom line.   Then attack the problem using cross-multiplication to solve for the bottom line.

$\frac{acres}{gallons} = \frac{3}{7} = \frac{x}{16}$

$6\frac{6}{7} = x$

Check back to make sure that the x you solved for matches your bottom line, then look down at your answer choices.

(A)

(B)

(C)

(D)

(E)

On sat.collegeboard.org, 70% of the responses were correct.

For more help with math, visit www.myknowsys.com!

# Proportions

Using current events in your SAT essay will impress your readers with the idea that you are informed about the world around you.  However, you are not limited to current events on this planet.  For example, have you heard the whimsical story about the Mars rover taking pictures of itself?  If not, follow this link to read the article.  This is a cute story, but it also relates to science, planning, creativity, and even self-awareness – topics that have shown up on the SAT.   With just a little more research, you could use the Mars rover as one of your excellent examples for the essay.

## 9/9 Proportions

Read the following SAT test question and then select the correct answer.

Read the question carefully to glean as much information as possible.  Then identify your bottom line, assess your options for solving the problem, and attack that question in the most efficient way possible.  When you have found an answer, loop back to make sure that you have answered the bottom line.

A group of workers can harvest all the grapes from 10 square meters of a vineyard in ½ minute. At this rate, how many minutes will the group need to harvest all the grapes from100 square meters of this vineyard?

Did you make a note of the bottom line?  The bottom line is the number of minutes it takes to harvest 100 square meters.  There are a number of ways to think about this problem, but one of the easiest ways is to set up a proportion.  You could set it up as time over area.  Write the ½ as .5 just so that you do not get confused between the fraction and the proportion.

$\frac{time}{area} = \frac{minutes}{sq. meters} = \frac{.5}{10} = \frac{?}{100}$

When you set up the problem like this, you can easily see that in the denominator you would need to multiply the first 10 by 10 to get the second denominator of 100.  Anything you do to the bottom of a fraction you must also do to the top.  Multiply the .5 by 10 and you will get 5.  You solved for minutes just as the question asked you to do.  Look down at your answer choices.

(A) 5
(B) 10
(C) 20
(D) 50
(E) 60

On sat.collegeboard.org, 69% of the responses were correct.

For more help with the math section of the SAT, visit www.myknowsys.com!

# Ratios, Rates, and Proportions

George Mason University's History News Network is an unusual news site that puts current events in a broad historical context. Normal news stories focus only on what has happened recently, but HNN strives to connect current events to the history that created them.

## 5/12 Ratios, Rates, and Proportions

The c cars in a car service use a total of g gallons of gasoline per week. If each of the cars uses the same amount of gasoline, then, at this rate, which of the following represents the number of gallons used by 5 of the cars in 2 weeks?

First, note the bottom line.

5 cars 2 weeks = ?

Next, assess your options. Since the problem gives so much information about the cars using words rather than numbers, a good place to start is to translate its question into mathematical terms.

c = total number of cars

g = total gallons of gas per week

The gas used in two weeks is easy to find: 2g. The tricky part involves determining how much gas is used by only 5 cars. It is tricky rather than difficult because if you know the trick, this problem is easy. Simply find the gas used by one car over the course of a week and multiply that by 5 cars.

$\frac{g}{c}$ = gas per week for 1 car

$\frac{5g}{c}$ = gas per week for 5 cars

Since the question asks how much gas will be used in 2 weeks, multiply this term by 2. This incorporates the 2g you identified earlier.

$\frac{10g}{c}$ = gas for 5 cars for 2 weeks

Now look at the answer choices.

A) $10cg$

B) $\frac{2g}{5c}$

C) $\frac{5g}{2c}$

D) $\frac{g}{10c}$

E) $\frac{10g}{c}$

On sat.collegeboard.org, 32% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Ratios

Private consulting groups like College Funding Solutions, Inc, and GetCollegeFunding can help you find ways to pay for college. How do you get the best financial aid? Scholarships? Grants? Tuition discounts?Which forms do you fill out, and how? Where should you put your money to make the best impression on government funding groups? Their services may seem expensive, but they can save you enough money in college that you come out far ahead.

## 4/27 Ratios

Always remember to follow the Knowsys Math Method. This may take longer than simply solving the problem at first, as you learn the method, but it will save you time once you begin to use it consistently. Reading carefully will help you make sure you don't miss anything. Identifying the bottom line makes it clear exactly what you are looking for. Stopping to assess your options will help you select the most efficient way to solve every problem and keep you from losing time by spending too much time on a problem. Finally, looping back will ensure that the answer you found matches the question that was asked; if you found the value of m, but the question asked for m + 3, you might get that problem wrong even after doing all the math correctly.

A jar contains only red marbles and green marbles. If a marble is selected at random from the jar, the probability that a red marble will be selected is $\frac{2}{3}$. If there are 36 green marbles in the jar, how many red marbles are in the jar?

When reading carefully, take note of facts that could help you solve the problem. For example, the fact that the jar only has red and green marbles means that this problem will involve only two variables, probably r and g. Later, the value of g is given, and the problem asks how many red marbles there are. The marbles are selected at random; that's good because it means you can rely on the probability given. If you reached into the jar looking for a red marble, the odds of finding one would be extremely high, no matter what the ratio of red marbles to green marbles is.

Next, identify the bottom line. The question asks "how many red marbles are in the jar?" That can be summarized as

r = ?

Now, assess your options. You could try plugging in the answers until you find one that works, but that could take a while. Or you could try setting up a proportion with the red and green marbles to calculate the number of red marbles in the jar. Conveniently, a ratio is already provided! You're halfway done already! So if there are two red marbles for every... Oh wait.

This is an example of why reading carefully is important. The ratio you need to find to solve the problem is r:g, but the ratio the problem gives you is r:a, or the ratio or red marbles to all the marbles in the jar. So, if 2 out of every 3 marbles are red, the remaining 1 must be green. Now you can set up a proportion.

$\frac{red}{green}=\frac{2}{1}=\frac{x}{36}$

It is essential that you always label your scratch work so that it is clear not just what you are doing, but what you did. When you reach the end of a section and begin to work backwards, double-checking problems you're not sure about, labels are invaluable because they show what you did to solve the problem. Now that the proportion is set up, you can solve it easily.

$36(\frac{2}{1})=36(\frac{x}{36})$

36(2) = x

x = 72

Now look at the answer choices:

A) 18

B) 24

C) 54

D) 72

E) 108

On sat.collegeboard.org, 46% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Graphs

Graphs are actually a fairly new form of mathematical analysis, first systematically investigated in the 1930s. This link has a short and, hopefully, informative article on the history of graphs.

## 4/3 Graphs

Always take the time to carefully read the question, identify the bottom line, and assess your options. Then attack the problem and loop back to check your answer. Only after that should you look at the answer choices to select the correct one.

In the figure, the slope of the line through points P and Q is $\frac{3}{2}$. What is the value of k?

First, note the bottom line at the top of your scratch work.

k = ?

Next, assess your options. Getting from a slope to one part of a coordinate pair may take a few steps, so in this case the best way to start is to simply begin with what you know and move toward what you want.

You know that slope is rise over run, and you can calculate the total rise since you have both of the y coordinates for the two points. Using this information, you can set up a proportion.

7 - 1 = 6

$\frac{rise}{run}=\frac{3}{2}=\frac{6}{x}$

From here, you can either cross-multiply and solve for x or use multiples to determine its value (2*3=6, so 3*3=x). Either way, x = 4.

Now that you have solved for x, loop back to the bottom line to see if you are finished.

k = ?

You found the difference between the two x-values, but not the value of k. There is still one step left! Since x, the difference between the x coordinates of P and Q, is 4, simply add the value of the first x coordinate to find the value of k.

4 + 1 = 5

Now look at the answer choices

A) 4

B) 5

C) 6

D) 7

E) 8