# Ratios

Freedom of the press has long been a hallmark of liberty in the United States.  However, many of the released SAT essay prompts have to do with balancing public and private lives, knowledge as a burden, and the abundance of information available through better technology.  Here is a current event that relates to all of these ideas:  a judge in the UK is calling for an independent group to regulate the press.  Think carefully about this current event, and decide where you stand on the issues that are raised.  If you decide to use this as one of your five current events, you will need to prepare a list of relevant details about this news story and a list of the broad topics that would let you know that this example relates to your essay prompt.

## Arithmetic: Ratios

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

Use the same method for every math question on the SAT.  Start by reading the question carefully and identifying the bottom line.  Next, assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that it matches the bottom line that you were asked to find.

Miguel is 180 centimeters tall. At 2:00 p.m. one day, his shadow is 60 centimeters long, and the shadow of a nearby fence post is t centimeters long. In terms of t, what is the height, in centimeters, of the fence post?
Bottom line: fence post = ?
Assess your Options:  Collegeboard.org uses a method that includes drawing the person and the post and then creating two right triangles.  That is a waste of time.  All you need to do is recognize that this is a ratio problem and set up your ratios correctly.
Attack the problem:  Set up the labels that you will use in your ratios so that you do not get confused about which number represents the length of the actual person and which number represents the length of his shadow.  You can set it up as actual divided by shadow:
$\frac{actual\: height}{shadow} = \frac{person}{his\: shadow} = \frac{post}{its\: shadow}$

Plug in the values that you know from the problem.  The only value that you do not know is the height of the post.  Leave that as a question mark so that you know which variable you must isolate.

$\frac{actual\: height}{shadow}= \frac{180}{60} = \frac{?}{t}$
All you need to do now is solve for the height of the post.  If you divide 180 by 60, the answer is 3.  You will need to multiply both sides by t to isolate the variable.  (Remember, you are solving for the question mark!)  Then you have your answer.

$3=\frac{?}{t}$

$3t = ?$
Loop Back:  The ? represented the height of the post in your original ratio, so you solved for your bottom line.  Look down at your answer choices.

(A) t + 120
(B) $\frac{t}{3}$

(C) 3t

(D) $3\sqrt{t}$
(E) $(\frac{t}{3})^{2}$

On sat.collegeboard.org, 52% of the responses were correct.

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

# Ratios

The election results are in!  The amount of information after an election day can be overwhelming, but limit yourself to one story from the election, and you will have an excellent current event for your SAT essay.  Many people focus on the presidential election, but there are hundreds of other important issues that were brought before the nation.  One group of United States citizens who currently cannot vote for the president of the United States voted about the possibility of becoming the 51st state.  Read this article about what is happening in Puerto Rico, and think about how this prospective state differs from or is similar to other territories that have become states.  Think of the broad themes raised by this story that could relate this article to SAT essay questions.

## 11/8 Algebra:  Ratios

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

Don’t just read the question; read it carefully.  Make sure you know which labels apply to which numbers.  Identify the bottom line.  Assess your options for solving the problem so that you can choose the most efficient method to attack the problem.  Once you have solved the problem, loop back to make sure that you have solved for the bottom line.

In a class of 80 seniors, there are 3 boys for every 5 girls. In the junior class, there are 3 boys for every 2 girls. If the two classes combined have an equal number of boys and girls, how many students are in the junior class?

Bottom Line:  Number of juniors = ?

Assess your options:  You could work backwards by starting with the answer choices, but it might take you a long time to work through all of the possible answers.  Instead, start turning those ratios into actual numbers of students.

Attack the problem:  You know the most about the seniors, so start with them.  You are given a ratio of 3 boys to 5 girls, and you know that the total number of boys and girls must equal 80.  You know that 3 + 5 = 8, so all you have to do is multiply the 3 and the 5 each by 10 and you will have a total of 80 seniors.  There are 30 senior boys and 50 senior girls.

$\frac{senior\: boys}{senior\: girls}=\frac{3}{5}=\frac{30}{50}$

Now that you know the number of senior boys and senior girls, how does that help you find the number of juniors?  Remember that the two classes combined have an equal number of boys and girls.  That means that the senior boys plus the junior boys must be equal to the senior girls plus the junior girls.

$senior\: boys + junior\: boys = senior\: girls + junior\: girls$

Plug in the numbers that you found for the senior boys and girls.

$30 + junior\: boys = 50 + junior\: girls$

What information do you know about the juniors?  You know that there are 3 boys for every 2 girls.  You do not know the total number of juniors, so use an x to represent this number.  What fraction of the total are the boys?  They are actually three fifths of the total number of juniors because you must add the boys and girls to find the total number of juniors.  That means that the girls are two fifths of the total number of juniors.  Plug this into your formula, remembering that anytime you have “of the total” that means that you must multiply by the unknown total.

$30 + \frac{3}{5}x = 50 + \frac{2}{5}x$

Now solve for x.  Rearrange the equation so that you have like terms on the same sides of the equation, and combine those like terms.  Start by subtracting the two fifths of x from each side.

$30+\frac{1}{5}x = 50$

Get those whole numbers together by subtracting 30 from each side.

$\frac{1}{5}x = 20$

To get rid of the fraction, you will need to multiply both sides by 5.  Your answer is x = 100.

Loop Back:  What does x represent?  It represents the total number of juniors, which matches your bottom line.  You are ready to look down at your answer choices.

(A) 72
(B) 80
(C) 84
(D) 100
(E) 120

On sat.collegeboard.org, 44% of the responses were correct.

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

# Ratios

As always, remember to follow the Knowsys method for math. Read the problem carefully and identify the bottom line (what you are looking for). Then, consider your options. How could you solve it? How should you solve it? Next, attack the problem using the method that you selected. Finally, loop back and verify that your answer matches the bottom line.

In an 8-gram solution of water and alcohol, the ratio by mass of water to alcohol is 3 to 1. If 12 grams of a solution consisting of 2 grams water for each gram of alcohol is added to the 8-gram solution, what fraction by mass of the new solution is alcohol?

This problem is going to test your ability to manipulate ratios. The first thing to remember is that the test makers are always going to try to trick you. When you are given a part-to-part ratio, you will usually need to convert it to a part-to-whole ratio and vice-versa. Before you start, though, make sure that you read the problem carefully and identify the bottom line. You need to know what the final ratio of alcohol to water is when you mix the 8-gram and 12-gram solutions.

Start by finding the amount of alcohol in the 8-gram solution. You know that the ratio of water to alcohol is 3 to 1. That means the ratio of  solution to alcohol is 4 to 1 (notice the conversion from part-to-part to whole-to-part).

$\frac{water}{alchohol}=\frac{3}{1}\therefore \frac{solution}{alchohol}=\frac{4}{1}=\frac{8}{2}$
If the ratio of solution to alcohol is 4 to 1, then there must be a total of 8 grams of solution and 2 grams of alcohol.

Now you need to find out how much alcohol there is in the 12 gram solution. Again, you are given the part-to-part ratio, "2 grams of water for each gram of alcohol." Simply convert it to the whole-to-part ratio.

$\frac{water}{alchohol}=\frac{2}{1}\therefore \frac{solution}{alchohol}=\frac{3}{1}$

You have 3 grams of solution for each gram of alcohol. Since there are 12 grams of solution, there must be 4 grams of alcohol.

Finally, add the two solutions together. You have a total of 20 grams of solution (the 8-gram solution and the 12-gram solution) and you have 6 grams of alcohol (2 grams from the 8-gram solution and 4 grams from the 12-gram solution). That means the ration of alcohol to solution is 6 to 20 or 3 to 10. Now, look at the answers below and select the one that matches your prediction.

(A)
(B)
(C)
(D)
(E)

The correct Answer Choice is (C).

On sat.collegeboard.org 47% of the responses were correct.

For more help with math, 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!

# Rates

College.gov has your starting place for financial aid. One of the major concerns for prospective college students is money; college is expensive, and rising tuition costs combined with less government funding show no end in sight. However, there is money available to help students who need it, and the FAFSA is the best place to start.

## 4/15 Rates

Always remember to follow the Knowsys Method--even in your math classes. Thinking strategically and logically will help you be more efficient far beyond the SAT. First, read carefully to see what the question is actually asking. Then assess your options and select the best one. Attack the problem efficiently, then loop back to make sure that the answer you found matches the question that was originally asked.

A train traveling 60 miles per hour for 1 hour covers the same distance as a train traveling 30 miles per hour for how many hours?

First, note the bottom line.

train 2 hours= ?

Next, look back at the question to determine how you could solve it. You could determine the total distance traveled by train 1 and then calculate the time it would take train 2 to travel the same distance. You could also calculate the times relative to one another. That might sounds odd, but it is actually more efficient than the first method.

The first step is to set up a ratio of  the two rates given. Put the "new" rate, that of train 2, on top because it is the variable you are trying to find. Always reduce ratios to lowest terms.

$\frac{train 2}{train1}=\frac{30}{60}=\frac{1}{2}$

You now have a ratio of the two rates. Here's the cool part: simply flip it over to find a ratio of the two times.

If a car or train travels at twice the planned speed, the trip will take half as long as projected. If it travels at half the planned speed, the trip will take twice as long. This rule applies when traveling 2/3, 5/4, or any fraction of the original rate; the ratio of the times will be the reciprocal of that fraction.

$\frac{train2}{train1}=\frac{2}{1}$

This means that train 2 spent twice as much time as train 1 covering the same amount of ground.

train2 = 2(train1)

Since train 1 traveled for 1 hour, train 2 traveled for 2 hours. Loop back to make sure you answered the right question.

train 2 hours = 2

Good job! Now look at the answer choices.

A) 3

B) 2

C) 1

D) $\frac{1}{2}$

E) $\frac{1}{3}$

On sat.collegeboard.org, 81% of responses were correct.

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

# Equation of a Line

The BBC is a great source for international news or simply a different perspective on American news. Look at this site or other news sites in the last week or two before you take the SAT to find your five current events examples. Even if you don't normally keep up with the news, looking like you do can increase your score!

## 3/25 Equation of a Line

Remember to follow the Knowsys Method and note the bottom line, assess your options, attack the problem, and loop back to check the question before you select your answer.

A line segment containing the points (0,0) and (12,8) will also contain the point

The bottom line here is which answer choice lies on the given line.

There are multiple ways to solve this problem. You could find the equation of the line, or, since the line goes through the origin, you could use ratios to find a point that has the same relationship between its x and y coordinates.

First, reduce the coordinates to lowest terms. You can arrange them in a ratio format if you wish; whether you prefer $\frac{x}{y}$ or $\frac{y}{x}$ does not matter. You might use $\frac{y}{x}=\frac{8}{12}$ since "rise over run" is also the formula for slope. Reduce this to its lowest terms and then check the answer choices for multiples.

$\frac{y}{x}=\frac{8}{12}=\frac{2}{3}$

Now convert it back to (x,y) format. Make sure the x and y go in the right places.

(3,2)

A) (2,3)

B) (2,4)

C) (3,2)

D) (3,4)

E) (4,2)

On sat.collegeboard.org, 58% of responses were correct.

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

# Ratios

As always, remember to read carefully, mark the bottom line, assess your options (by asking "What could I do?" and "What should I do?"), attack the problem, and loop back to ensure that you actually answered the question.

The odometer of a new automobile functions improperly and registers only 2 miles for every 3 miles driven. If the odometer indicates 48 miles, how many miles has the automobile actually been driven?

Now that you have read the problem--I hope you read carefully!--note the bottom line at the top of your scratch work. In this problem, you are trying to figure out how many miles the car has traveled, so the best variable to use is m. At the top of your page, write

m=?

Next, assess your options. What has the problem told you? Are there any hints as to what strategy you could use? You could simply count out to 48, but that would be time-consuming, and on timed tests the long way is the wrong way. In fact the phrase "for every" is a clue; it tells you that the problem has to do with relationships between numbers, and that often means that you will need to use ratios.

When dealing with ratios and proportions, the first step is to assign labels to the various parts of your scratch work. Put the variable you need to find in the numerator and the variable you know the most about in the denominator. Equal to that, put the ratio provided in the problem. Even though this will sometimes give you an improper fraction, is is the fastest way to find the answer.

$\frac{actual}{odometer}=\frac{3}{2}$

Next, add another ratio to create a proportion. Make sure that the numbers you add match the labels on the left. This time, the variable should go in the numerator and the denominator should have a number relating to the label. The problem says that the odometer indicates 48 miles, so write

$\frac{actual}{odometer}=\frac{3}{2}=\frac{m}{48}$

Now you can solve for x. Note that since the variable is in the numerator, you do not need to cross-multiply. Instead, simply multiply both sides by 48.

$\frac{3\ast 48}{2}=\frac{48m}{48}$

Then simplify. Remember to find the fastest and easiest way to simplify; here, a good way to save yourself effort is to divide 48 by 2 instead of multiplying it by 3.

${3\ast 24}=m$

$72=m$

Loop back to make sure you found the correct value. The problem needed to know the total number of miles the car had actually traveled. Based on the ratio, you found exactly that. Now check the answer choices.

A) 144

B) 72

C) 64

D) 32

E) 24

Of 199,722 answers on sat.collegeboard.org, 64% were correct.

Want more help with math? Visit myknowsys.com!

# Mathematics

## Mathematics: Standard Multiple Choice

The SAT instructions are so simplified, they leave out the most important parts of what you need to do. Remember that you need to disregard the answers, carefully read the question, mark the bottom line, and THINK about the most efficient way to solve the problem. Then do the math, check your answer against the bottom line you wrote down, and finally select the correct answer choice.

To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange dye are mixed, what fraction of the new mixture is yellow?

First, note the bottom line at the top of your scratch work. The problem asks for a fraction of the whole mixture, so write:

$\frac{y}{m}=?$

Next, assess your options. Ask "what could I do?" and then "what should I do?" The trap here is to simply add the parts together. Add up all the numbers to give the denominator, then add up the yellow parts to give the numerator. However, what that misses is the word "equal" in the question. The mixture you just added up would not have equal amounts of orange and green dye.

To make the problem more concrete, replace the vague term "parts" with a more specific word like "ounces."

The problem is concerned with relationships between different amounts, which makes it a ratio problem. What ratio will help you find out how much yellow is in the final mixture? Start with each of the dye mixes separately. Each batch of orange dye has 3 ounces of red dye and 2 ounces of yellow, so it is 5 ounces total. The green is made up of 2 ounces of blue dye and 1ounce of yellow, so it is only 3 ounces total.

$\frac{y}{o}=\frac{2}{5}$                                                                 $\frac{y}{g}=\frac{1}{3}$

What is the next step? The problem mentions that equal amounts of green and orange are mixed together. Right now, you have different amounts of green and orange. How do you make them equal? The easiest way is to use the least common multiple (or least common denominator, which is really just the least common multiple on the bottom of a fraction). The least common multiple of 3 and 5 is 15, so put both fractions in terms of that number:

$\frac{y}{o}=\frac{2}{5}=\frac{6}{15}$                                                    $\frac{y}{g}=\frac{1}{3}=\frac{5}{15}$

Now you have everything you need to solve the problem. Remember that you are looking for what fraction of the final mixture is yellow, so find those two values and put them into a fraction.

$5+6=11$              $15+15=30$            $\frac{11}{30}$   doesn't reduce, so it is your answer.

Check that against the answer choices:

$A) \frac{3}{16}$

$B) \frac{1}{4}$

$C) \frac{11}{30}$

$D) \frac{3}{8}$

$E) \frac{7}{12}$