# Writing Equations

As you prepare for college, one of the best things that you can do for yourself, outside of studying, is to build good relationships with your teachers.  Learning the proper way to ask for help from your teachers can mean the difference between finally understanding a concept and getting written off as a whiner.  Read this article and think about how you can use the given advice not just in the future, but in your classes right now.

## Algebra: Writing Equations

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

Always read each question carefully and make a note of the bottom line.  Assess your options for finding the bottom line and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

A florist buys roses at $0.50 a piece and sells them for$1.00 a piece. If there are no other expenses, how many roses must be sold in order to make a profit of $300? Bottom Line: # roses = ? Assess your Options: You could find the profit from a single rose and then start plugging in answer choices, but that is not the fastest way to solve this problem. A better way to solve this problem is to simply write an equation. You could also solve this problem in a few seconds by using logic. Attack the Problem: Writing an equation will not take you much time. Start by finding the profit from a single rose:$0.50.  (You know that the florist spends $0.50 to make each dollar, so$1.00 - $0.50 =$0.50.)

If each rose brings in a profit of $0.50, then how many must you sell to get$300?  Start by writing the fifty cents, and then use x to represent the unknown number of roses.  Each rose costs the same, so multiply the two numbers.  Together they must all equal $300.$0.50x = $300. (Just divide 300 by .5 to isolate the variable.) x = 600 Loop back: The x represented roses so you found your bottom line. Look down at your answer choices. (A) 100 (B) 150 (C) 200 (D) 300 (E) 600 The correct answer is (E). Alternatively: You can solve this problem in a few seconds. Think about it logically; if you get less than$1 for each rose and you need $300, can you sell 300 roses and get the profit you need? No! You need more than$300 roses.  There is only one answer choice that works.

On sat.collegeboard.org, 71% of the responses were correct.

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

# Sentence Completions

The cost of college has become a major topic of debate in the news.  Are you worried about financing higher education?  Here is an article stating that the problem might not be as big as the media makes it seem.  Do you agree with this writer’s perspective?

Choose the word or set of words that, when inserted in the sentence, best fits the meaning of the sentence as a whole.

Cover up your answers and read the sentence carefully with an open mind.  Make a prediction to fill the blank, and then match that prediction to the correct answer choice.  Eliminate any answer choice that does not match.  Be sure to look at all of your options, even if the first answer seems correct.

The show’s host was usually genial, but he had a reputation for turning ------- when provoked by guests who challenged his opinions.

This sentence is easy if you know what the word genial means; all you have to do is predict the opposite.  However, you can still use logic to fill in the blank if you have no idea what that word means.  How do you feel when you are provoked and challenged?  Not good. Probably angry.  Use a word such as “upset” as your prediction and look down at the answer choices.

(A) surly
(B) intrusive
(C) lenient
(D) convincing
(E) giddy

(A)  You may know that the Latin root “sur” means over, but that is not going to help you much here.  Overly what?  If you don’t know what this word means, keep it.

(B)  You may know a word related to intrusive, intrusion.  People often say the words, “Forgive my intrusion,” so you know that this is a negative word.  However, an intrusion is what causes people to be upset, not a synonym for the word “upset.”  It does not match your prediction, so eliminate it.

(C)  You want people to be lenient when you have done wrong, so the word "lenient" is positive.  You are looking for something negative.  Eliminate this answer choice.

(D)  The word convincing is positive, and who would turn convincing when provoked?  That doesn’t make any sense.  Eliminate this answer choice.

(E)  Giddy just does not sound like an angry word.  Eliminate it.

This is a good time to talk about the limitations of Latin and Greek roots.  Not all words come from Latin.  The word surly actually comes from the Middle English word “sirly” meaning like a lord (a sir).  Over centuries the meaning changed from lordly to domineering to arrogant to bad-tempered.  Latin roots can help you improve your SAT score, but it is always better to know the definition of a particular word.  This is why the Knowsys program includes common Latin roots, but focuses on commonly tested words.

Words used in this SC:
Genial: friendly and cheerful
Intrusive: the quality of being unwelcome
Lenient: lax, tolerant, not strict
Convincing: persuasive
Giddy: dizzy or lighthearted

On sat.collegeboard.org, 59% of the responses were correct.

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

# Functions

Many schools in the United States participate in campaigns to keep children from smoking.  However, there are countries that are taking even greater measures to make smoking unattractive.  Cigarette packaging in Australia will no longer display colorful logos, but instead will display images depicting the dangers of smoking.  As you read this article, think about whether or not you agree with these measures, and then think about the themes that might relate this current event to an SAT essay topic.

Also, if you are a senior who dreads the college application process, take a look at this checklist and remember to breathe in the next few months!

## 8/16 Functions

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

Read each math question carefully so that you can identify exactly what is being asked of you.  Once you have identified the bottom line, assess your options to find an efficient way to solve the problem.  Finally, attack the problem, solve it, and loop back to make sure that your answer addresses the bottom line that you were asked to find.

Which of the following could be the equation of the function graphed in the xy-plane above?

You have been given a graph, and you must find the equation that has been graphed.  You could plug all of the answer choices into your calculator, but that would take a long time and you risk making a typo.  Instead, break the graph down into its most basic components.  What shape that you have often seen does this graph most resemble?  It looks like a parabola opening upwards, so you know that f(x) = x² will be part of your equation.

Picture the f(x) = x² parabola in your mind.  It passes through the origin at (0,0).  However, the graph in this problem would extend past the point (0,0) into the negative numbers if you continued the basic curve of the parabola.  To translate the function down on the graph, you would need to subtract a number from the original function.  Now you have f(x) = x² - n, where n = any number.

There is one more step.  The basic curve of the normal parabola has been reflected across the x-axis in this problem so that all the values of the parabola are now positive.  What can you do to make sure that all of the numbers in a function are positive?  Take the absolute value of the function.  Now you have f(x) = |x² - n|.  Look down at your answer choices.

(A) y =  (-x)² + 1
(B) y = -x² + 1
(C) y = |x² + 1|
(D) y = |x² - 1|
(E) y = |(x – 1)²|

(A), (B), and (C) cannot be the answers because they all add to the equation and would result in a parabola that has been shifted above the x-axis.  (E) will not be symmetric to the y-axis, and the graph that you have remains symmetric to the y-axis; it has not been shifted to the right or the left. The (x – 1)² part of the equation in (E) shifts the entire parabola away from its original position on the y-axis.  (D) is the only answer that matches the equation you wrote for this graph.

On sat.collegeboard.org, 39% of the answers were correct.

For more help, visit www.myknowsys.com!

# Percents

College loans may seem like the only way to get through your education, but they can be risky. Ben Bernanke, chairman of the Federal Reserve, warns that over-investing in college loans can lead to economic ruin. Instead of loans that you will have to pay back, look for scholarships, grants, and federal aid money that other agencies are willing to pay because they see it as an investment in their future. Scholarships and awards are given out for a dizzying variety of reasons, so make your search thorough! Do you have good grades? Do you have a hobby or sport that you're good at? Is there a cause you've volunteered for? Do you or does someone in your family have a rare disease? Some scholarships are restricted to members of certain racial or ethnic minorities; others are based on religious affiliation. Check if your parents' employers have any kind of scholarship fund. Sites like fastweb.com and findtherightscholarship.com can help in your scholarship search.

## 8/7 Percents

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

When you work a percent problem, it is especially important to read the problem carefully because small words like “of” make all the difference in describing a percentage.  Once you have understood the question and identified the bottom line, assess your options and choose a method to attack the problem.  After you have finished the problem, loop back to verify that the answer addresses the bottom line.

The population of Norson, the largest city in Transitania, is 50 percent of the rest of the population of Transitania. The population of Norson is what percent of the entire population of Transitania?

Your bottom line is a percentage: the number of people in Norson as a percent of the entire population of the country of Transitania.  At the top of your scratch work, write N = ?%  Next, assess your options.  You don’t know how many people are in the country of Transitania, so solving the problem algebraically would be challenging. Instead, pick a number so that the problem will be more concrete.  When picking numbers on a percent problem, you should always pick 100 because 100 is the easiest number to use; the answer you find is already out of 100 so you will never need an extra step to find the correct percentage. Now you're ready to attack the problem.

If you assume that there are 100 people total in all of Transitania, you still do not know how many people are in the city of Norson.  Use the variable “N” to represent the people in the city.  The problem tells you that the population of this city “is 50% of the rest of the population of Transitania.”  You know that Norson has half as many people as the rest of the population.  Think about it this way: for every person in Norson, there must be 2 people outside of the city.  Write an equation representing this knowledge and solve it.

N + 2N = 100

3N = 100

N = 33.33

You found that 33.33 people out of 100 live in the city of Norson, so you know that the percentage of people who live in Norson is 33.33%.  This matches the bottom line that you needed to find.

(A)               20%

(B)               25%

(C)               30%

(D)               33 1/3%

(E)                50%

On sat.collegeboard.org, 38% of answers were correct.

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

# Subject-Verb Agreement

Teachers and schools have gone back and forth on the value of rote memorization. Some claim that forcing students to endlessly repeat facts and processes only kills their creativity. Others argue that practice is essential to make certain information and tasks automatic--freeing up mental energy for higher-level applications. One value of memorization and repetition is that it can be highly effective as a way to learn facts on your own, without a teacher or classroom. If you've ever used vocabulary flashcards or a multiplication app, you've experienced rote learning! Learn more about the value of memorization here, and sign up for the Knowsys online vocabulary flashcards here

## 7/31 Improving Sentences

Part or all of the following sentence is underlined; beneath the sentence are five ways of phrasing the underlined material. Select the option that produces the best sentence. If you think the original phrasing produces a better sentence than any of the alternatives, select choice A.

Read the following sentence to yourself, listening for errors.  Evaluate the underlined portion using the Big 8 Grammar Rules and predict a correction for any errors that you find.  Remember that after an improving sentences question, choice (A) will always be the same as the original sentence.  Eliminate any answer choices that do not help you towards your goal of creating a clear and precise sentence.

Rote learning, the process of memorizing by repetition, is how many young children learn the alphabet or the multiplication tables.

The underlined portion of this sentence includes a verb.  The third Big 8 Grammar Rule concerns subject and verb agreement.  Think back to this section in your Knowsys book.  You learned that the exam writers use 5 subject/verb tricks to make identifying a subject/verb agreement error more difficult.  One of these tricks is to separate subjects from verbs.  In this sentence, the subject is separated from the verb by an appositive phrase. An appositive phrase is a defining phrase that is set off by commas.  Because this phrase merely defines the subject, you should be able to ignore that portion of the sentence and still have a complete sentence.  You have now simplified the sentence to read: “Rote learning is how many young children learn the alphabet or the multiplication tables.”

You can now easily see that the subject matches the verb.  You can also see that for this to be a complete sentence, the very first word in the underlined portion must be a verb.  You want your subject and verb to be as close as possible, so you can eliminate any answer choices that do not begin with a verb.

(A) is how
(B) is used for when
(C) this is how
(D) the way
(E) which is used when

You have now eliminated (C), (D), and (E).  You did not find an error in the original sentence, which matches (A), but you must still examine (B) to see if it improves the sentence.  Notice that (B) adds more words to the sentence without making the message any clearer.  The Knowsys method tells you to lean towards choices that are short and concise.  It also tells you not to be afraid to pick a response indicating that there is no error.

On sat.collegeboard.org, 68% of responses were correct.

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

# Pronoun Antecedent Agreement

As you are preparing for the SAT, it's a good idea to start thinking about which colleges you want to apply to. It's important to set goals and know what scores you want to get. US News and World Report has a great list of the top colleges here. The list includes a variety of different categories including "best value schools" and "A+ Schools for B students."

## 6/25 Improving Sentences

Part or all of the following sentence is underlined; beneath the sentence are five ways of phrasing the underlined material. Select the option that produces the best sentence. If you think the original phrasing produces a better sentence than any of the alternatives, select choice A.

"Improving Sentences" questions are very straightforward. All you need to do is look at the underlined portion. If there is an error or some obvious fault in the underlined portion, make a prediction about how you could improve the sentence. Then look at the answer choices. Remember that the correct answer choice will not always match your prediction exactly (since there are often several ways to improve a sentence). Even so, predicting will help you to analyze why the sentence is less than ideal as it stands. Also, don't forget that sometimes the sentence is best the way it is (in which case you will select answer choice (A)).

The well-preserved, 121-million-year-old fossilized bird embryo on display at the museum has several features that suggest that its young could move about and feed themselves very soon after they hatched.

As you read the sentence, make sure to carefully analyze what the sentence is implying. In this case, it sounds as thought it is the "121-million-year-old fossilized bird embryo on display at the museum" that have young that "could move about and feed themselves . . . ." Clearly, a 121-million-year old fossilized embryo cannot have young. Instead, it is the type of bird that comes from the embryo whose "young could move about and feed themselves . . ." You need to select the answer that makes this clear.

(A) its
(B) her
(C) their
(D) the species'
(E) for this species

As the sentence is written, it isn't clear (and in fact, it's misleading) so you can eliminate answer choice (A). (B) uses an inappropriate pronoun (you wouldn't refer to a 121-million-year-old fossilized embryo as "her"). (C) is also wrong. Remember that "their" should never be used as a neuter gender singular pronoun. (D) Fits your prediction well. It specifies that it is the young of the species of the fossilized embryo that "could move about . . . " (E) uses the word "this" which is unnecessary in this case.

The correct answer choice is (D)

On sat.collegeboard.org 49% of the responses were correct.

Want more help with grammar? Visit www.myknowsys.com.

# Idioms

With the ever increasing number of students graduating from college in debt each year, it's becoming more and more important for you to understand the consequences of student loans. Remember that even if the average salary in your future field of work is $60,000 a year, you probably won't earn that much at your first job. The average college graduate finishes with around$25,000 in debt and paying that off within 2 years is much more challenging than you would think. You can learn more about how to manage student loans (or even better, avoid them altogether) here

## 6/22 Identifying Sentence Errors

The following sentence contains either a single error, or no error at all. If the sentence contains an error, select the one underlined part that must be changed to make the sentence correct. If the sentence contains no error, select choice E.

When you are working on an Identifying Sentence Errors question, always read the sentence and focus on anything that stands out and sounds wrong or strange. Then determine wether that part of the sentence actually has an error. Always double-check the other sections of the sentence to verify that they are correct as they are, and finally mark the correct answer.

Today, also representing nations and other political entities, flags are used to represent youth groups, athletic competitions, and international bodies. No error

If no part of the sentence stands out as clearly wrong, it is best to go step by step through each section and look for errors. (A) should sound a little strange. In this case there is an idiomatic error. Remember that idioms are just rules for how to use words and phrases (for example, you should say "less money" or "fewer dollars" but not vice versa). Instead of saying "Today, also representing nations . . . " the sentence should read "Today, in addition to representing nation . . ." Now that you have identified the error, don't forget to double check the other sections. There is nothing wrong with (B). (C) has the correct verb tense. (D) is also error free. (E) cannot be the correct answer because you have already identified an error.

The correct answer choice is (A).

on sat.collegeboard.org 74% of the responses wore correct.

Want more help with grammar? Visit myknowsys.com

# Subject Verb Agreement

Flashcards are one of the most important tools you have when you are studying for the SAT (or almost any test for that matter). You will need to commit math formulas, idioms, and vocabulary words to memory before test day. There are lots of different websites were you can create, review, print and even share digital flashcards.  One of the best is flashcardexchange.com.

## 6/16 Identifying Sentence Errors

The following sentence contains either a single error or no error at all. If that sentence contains an error, select the one underlined part that must be changed to make the sentence correct. If the sentence contains no error, select choice E.

For Identifying Sentence Errors questions, always read the sentence and focus on anything that sounds strange or wrong. Determine wether that part of the sentence is actually incorrect, then identify the part of the sentence that, if changed, would correct the error. Double-check that the other sections of the sentence are correct as they are, and finally, mark the correct answer on your answer sheet.

In order to prepare for the speech he was given to all of the parents and teachers at the school, George practiced speaking in front of a group of his friendsNo error

In this sentence, B should stand out right away. As the sentence stands right now, the verb "given" is in the wrong form. If you only read the sentence up to B it almost looks like it could be correct. You have "In order to prepare for the speech he was given . . ." and that doesn't seem like a mistake. However, if you continue reading, the sentence becomes "In order to prepare for the speech he was given to the parents and teachers . . . " Now it should be obvious there is something wrong. Clearly, the speech "George" is preparing for is a speech he "was giving . . . " or "would be giving. . . "  Once you have identified the error, you are finished with the problem. Remember that you should always try to predict how you would fix the sentence, it will help you to be confident that the underlined portion that you choose is really an error.

The correct answer choice is (B).

Want more help with grammar? Visit myknowsys.com!

# Subject Verb Agreement

Listening to audiobooks can be a great way to multitask while you are doing something else (such as mowing the lawn or sitting in the bus on the way to school). If you are interested in listening to some audiobooks (which, by the way, is a great way to improve your vocabulary and find some "excellent examples" for your essay) you can download a huge variety of different audio books here for free.

## 6/7 Improving Sentences

Part or all of the following sentence is underlined; beneath the sentence are five ways of phrasing the underlined material. Select the option that produces the best sentence. If you think the original phrasing produces a better sentence than any of the alternatives, select choice A.

On Improving Sentences questions, always remember to read the sentence and make a prediction about what changes would correct any errors you find. This prediction will help you eliminate wrong answers and quickly zero in on the correct choice.

Listening to good storybooks sharpen children’s awareness and appreciation for the sounds of spoken language

Since the first underlined word you see is a verb, you should start off by identifying the subject and the verb in this sentence. In this case, the subject is "listening." The verb is "sharpen." If you have trouble seeing that "listening" is the verb, try thinking about what is performing the action. What is it that is "sharpenen[s]?" It must be the "listening" that is performing the action so the subject of the sentence is "listening.At this point you should notice that the verb sounds wrong. You wouldn't say "listening sharpen", you would say "listening sharpens." The test makers are trying to trick you by separating the subject and the verb . You may look at it and think that "storybooks sharpen" sounds correct. The problem is that it is not the "storybooks" that "sharpen children's awareness . . . " It is "Listening" that "sharpens children's awareness . . . " Now that you have a prediction, it's time to take a look at the answer choices below.

(A) sharpen children’s awareness
(B) sharpens children’s awareness of
(C) are what sharpens the awareness of children
(D) sharpens the awareness of children
(E) is to sharpen children’s awareness

As you can see, answer choice (B) matches your prediction exactly. Note that "children" is possessive because it is the "children's awareness and appreciation" that is sharpened by listening. (A) is clearly wrong because it is simply what was originally in the underlined portion. (C) is incorrect because "listening are what sharpens . . ." is clearly not grammatically correct (once again, the problem is subject-verb agreement). (D) is incorrect because of the parallel structure in the non-underlined portion of the sentence. (D) would read "Listening . . . sharpens the awareness of children and appreciation for sounds of spoken language." (E) is incorrect because it changes the meaning of the sentence (even if answer choice (A) is incorrect, the implied meaning of the sentence is still correct and must be preserved in the correct answer choice).

The correct answer choice is (B).

On sat.collegeboard.com 57% of the responses were correct.

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

# Equations and Factoring

Extracurricular activities are an essential part of a great college application. They can really help you to stand out in the eyes of an admissions officer. While it is important to demonstrate that you do have a life outside of school, some activities are viewed more favorably than others. It is also essential that you present your activities in the proper way to ensure that you don't come across as either desperate or uninterested. You can learn more about extracurricular activities and their effect on college applications here.

## 6/5 Equations and Factoring

Add  to  and then subtract  from the sum. If  is a positive integer, the result must be an integer multiple of

This problem is a great chance to put the Knowsys method for math to use. Remember, the first step is to read the problem carefully. If you started reading this question and felt a sinking feeling in your stomach when you saw 8x or the words "positive integer" stop and take a moment to reread the problem. It's really not as difficult as it first appears to be. Step 2 of the Knowsys method is to identify the bottom line. In this case we are looking for what the result must be a multiple of.  Next we assess our options. We could write out a formula and try to simplify it or we could pick numbers. In this case it is probably faster to write out a formula but we will work the problem using both methods so that you know how to use either one. Step 4 is to attack the problem. Believe it or not, attitude makes a big difference. Relax, trust in the method, and go step by step. Don't worry if you aren't sure exactly how the problem will work out.
If we write out the formula described in the problem we get the following
$8x+2x-5=$

We can simplify the equation like so

$8x+2x-5=10x-5=5(2x-1)$

If we look at the fully factored form of the equation we can see that we are going to multiply whatever we have in the parenthesis by 5. That means that our result must be a multiple of 5. Now we look at the answers below.

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

You can see that answer (B) matches our prediction exactly. As I mentioned earlier, there is one other way to solve this problem. We can pick numbers for X. The easiest number to pick for X is 1. If we do that the formula becomes

$8x+2x-5=8(1)+2(1)-5=8+2-5=5$

In this case, our formula gives us the number 5. Since we need an answer that must be true, we can eliminate any answers that do not fit and in this case only (B) works (don't forget that step 5 of the Knowsys math method is to loop back and verify that your answer matches the bottom line). Picking numbers can be a great strategy if you get stuck on a problem but it is generally faster to solve the problem by writing out the formula.

On sat.collegeboard.com 68% of the responses were correct.

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

# Sentence Completions

Did you know that changing the location you study at can help you to learn? Studies like the one mentioned in this New York Times article have shown that as we learn, our brain makes associations between the material we are studying and the sights, sounds, and even smells of our surroundings. By studying in different locations with different characteristics you can make it easier for your brain to remember new material. This doesn't mean that you should study on your couch with the TV on, but it does mean that every once in a while moving from your room to the back porch, or even a coffee shop  might not be a bad idea.

## 5/31 Sentence Completions

Choose the word or set of words that, when inserted in the sentence, best fits the meaning of the sentence as a whole.

Remember that the answer choices on most SAT questions are there to confuse you. Out of the five listed, four of them are wrong! Use the Knowsys Method: look at the sentence, make a prediction without reading the answer choices, and then compare the answer choices to your prediction to find the one that matches most closely.

His inclination to succumb to flattery made him ------- to the ------- of people who wished to take advantage of him.

The first step is to decide which blank is easier to predict. In this case the first blank is definitely easier to predict than the second blank. Since this man has an "inclination to succumb to flattery" we can predict that that inclination made him "vulnerable"  to something. We now look at the answer choices below and see which choices we can eliminate.

(A) immune . . predilection
(B) prejudicial . . intentions
(C) susceptible . . cajolery
(D) resistant . . blandishments
(E) amenable . . rejection

We can eliminate (A) because it is the exact opposite of our prediction. (B) also will not work because "prejudicial" does not mean "vulnerable". (C) fits our prediction very well. (D) can be eliminated because again, it is the opposite of our prediction. Unfortunately, since (E) also matches our prediction fairly well we will have to now predict the second black and check (C) and (E) again. Based on the sentence we know that this man is "vulnerable" to flattery so we can predict that this makes him "vulnerable to the control" or "manipulation of the people who wished to take advantage of him". Now, even if we are not sure what "cajolery" means, we can eliminate (E) because "rejection" does not match our prediction. You can see that by following the Knowsys method, we were able to get to the correct answer without having to know the definitions for all of the words.

Words tested in this SC:
immune: Exempt; not subject to
predilection: Condition of favoring or liking; tendency towards
prejudicial: Exhibiting prejudice or bias
intentions: Courses of action that a person intends to follow
susceptible:  Likely to be affected by something
resistant: That which makes resistance or offers opposition
blandishments: Flattery intended to coax or cajole
amenable:  Willing to respond to persuasion or suggestions
rejection: The act of rejecting or the state of being rejected

On sat.collegeboard.com 63% of the responses were correct.

Need to build your SAT Vocab?  Visit www.myknowsys.com to check out the Knowsys Vocabulary Builder Program.

# Algebra: Substitution

Since you are preparing for the SAT by reading this blog, now is a great time to start thinking about where you might like to go to college. You can search for colleges based on your preferences and learn more about your options here.

## 5/30 Algebra: Substitution

Always follow the Knowsys Method for math problems. It will save you time not only on the SAT, but also on math tests and quizzes in your school classes. Read the question carefully and identify the bottom line. Assess your options. Ask what you could do and then what you should do, and solve the problem quickly and efficiently once you have decided on a strategy. Finally, loop back to double check that you answered the question correctly

If A,B, and C are numbers such that  and , then  is equal to which of the following?

At first this problem may seem to be challenging but it is important that we tackle it step by step. Most of the more difficult problems on the SAT consist of 3 or 4 steps, each of which is fairly easy on their own. As long as we focus on the next step and don't panic, most of the "hard" problems will actually turn out to be fairly straightforward. The first step is always to identify the "bottom line". In this problem, we are looking for what  is equal to.
Since the equation currently has 3 different variables we cannot solve it as it is. In order to eliminate some of the variables we will need to use substitution. We start with first equation

$\frac{a}{b}=3$

We can manipulate this equation to give us

$a=3b$.

We then substitute this into the original equation to get

$\frac{a+b}{b+c} = \frac{3b+b}{b+c}=\frac{4b}{b+c}$

Now we work on eliminating c. By solving the second equation we get the following.

$\frac{b}{c}=7 \therefore \frac{b}{7}=c$

Substituting this into our equation now gives us

$\frac{4b}{b+c}=\frac{4b}{b+\frac{b}{7}}=\frac{4b}{\frac{8b}{7}}$

At this point, you may be starting to panic because the equations look so complicated. If so, take a deep breath and relax, we are almost done. Remember that with fractions, dividing is the same as multiplying by a reciprocal. We can now solve our equation since we only have one variable.

$\frac{4b}{\frac{8b}{7}} = \frac{4b}{1}\times \frac{7^{}}{8b}=\frac{28b}{8b}=\frac{7}{2}$

Now all that's left to do is look at the answers and find which one matches our solution.

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

The answer is A and that matches our prediction exactly!

On sat.collegeboard.org, 42% of the responses were correct.

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

# Subject Verb Agreement

How many of you like to read? Did you know that reading makes you smarter? It might seem obvious that reading is good for you but did you know that it has other benefits besides increasing your knowledge? Reading has numerous benefits from the fairly obvious ones like improved critical thinking and memory, to the surprising ones like stress reduction or improved social skills (tell that to someone the next time they call you a bookworm). You can learn more about the benefits of reading here. If you don't have extra money to go see a movie this summer remember, the public library is free (and air conditioned too)!

## 5/26 Improving Sentences

Part or all of the following sentence is underlined; beneath the sentence are five ways of phrasing the underlined material. Select the option that produces the best sentence. If you think the original phrasing produces a better sentence than any of the alternatives, select choice A.

All of the ingredients for Pablo’s secret chocolate chip cookie recipe, which included the special dark chocolate, was available in his mother’s pantry.

As always, we want to predict our answer before we look at the answer choices. Doing so will help us to feel more confident in our answer when we see a choice that matches our prediction. In this case, the test makers are trying to trick us by separating the subject and the verb. The subject is the ingredients not the recipe and, therefore, the correct verb is were not was. This is a classic trick that is used on the SAT. By separating the subject and the verb, the test makers try to confuse you. It would be correct to say "the recipe was available in . . . " but in this case, it the "ingredients" that "were available in . . . ". Additionally, if we look closely we can see that the word "which" is ambiguous. We don't know if it is the "recipe" or the "ingredients" that "included the special dark chocolate". We can predict that the correct answer will probably have the verb "were", and also fix the ambiguity brought about by the improper use of the word "which". Now it's time to look at the answer choices and see which one has the proper subject and verb agreement (and eliminates the ambiguity).

(A) which included the special dark chocolate, was

(B) which included the special dark chocolate, they were

(C) including the special dark chocolate that was

(D) including the special dark chocolate, being

(E) including the special dark chocolate, were

Remember that choice (A) is exactly the same as the sentence originally appeared in the question. Since we already know there is an error we do not need to read choice (A) and we can immediately cross it off. (B) has an unnecessary  "they". (C) still has the verb "was", since you would not say "the ingredients that was" it won't work either. Answer choice (D) uses the word "being". On the SAT you will almost never want to use an answer choice with the word "being". In this case, only choice (E) matches our prediction and that is the correct answer.

On sat.collegeboard.org, 75% of the responses were correct.

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

# Translating English to Math

If you have trouble studying, bookmark this handy list of study tips that can make your study time more effective.

## Mathematics: Standard Multiple Choice

A florist buys roses at $0.50 apiece and sells them for$1.00 apiece. If there are no other expenses, how many roses must be sold in order to make a profit of $300? First, note the bottom line. p =$300, r = ?

Assess your options. You know how much the roses cost and how much they sell for, so you could calculate the amount of profit earned with each flower. You could also translate the problem into a formula to determine how many flowers will yield $300 in profit. The first solution is more efficient, so begin by finding the amount of profit each flower brings in. Since they cost fifty cents and sell for a dollar, each flower earns fifty cents in profit.$1.00 - $0.50 =$0.50

Next, how many flowers are needed to add up to $300 total? .5r = 300 $r=\frac{300}{.5}$ r = 600 Loop back to the bottom line to check whether you answered the question correctly. You were looking for the number of roses that would yield$300 in profit. Since that is what you found, look at the answer choices to see which one matches.

A) 100

B) 150

C) 200

D) 300

E) 600

On sat.collegeboard.org, 74% of responses were correct.

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

# Sets

If you're feeling nervous about college, check out this page full of advice from real college seniors. They mention many wise "do's" and "don't's" about the freshman year of college.

## Mathematics: Standard Multiple Choice

If S is the set of positive integers that are multiples of 7, and if T is the set of positive integers that are multiples of 13, how many integers are in the intersection of S and T?

First, read carefully. Notice that S includes all multiples of 7 and T includes all multiples of 13. Next, identify the bottom line and note it at the top of your scratch work.

Intersection of S and T = ?

You are looking for the intersection, so you need only the numbers that are in both set S and set T. Since neither set has an upper limit, they both have an infinite number of members; therefore, their intersection also has an infinite number of members. Look at the answer choices.

A) None

B) One

C) Seven

D) Thirteen

E) More than thirteen

# Sentence Completions

Getting a job might not be your top priority in high school--or maybe it is! This link has some good advice. As a student, it is easy to think that you don't have any qualifications, but that isn't true. Think about your personality, your favorite classes in school, any leadership roles you've held, and so on. Some teachers set up Classroom Managers to help with daily tasks, and students often put these positions on real-life job applications as well. Are you exceptionally friendly? Outrageously smart? Good with computers? Do you speak (or even take classes in) another language? Any of these can help you stand out in an application or interview. You have a lot to offer!

## 5/16 Sentence Completions

Choose the word or set of words that, when inserted into the sentence, best fits the meaning of the sentence as a whole.

I wonder sometimes whether you get tired of reading these explanations of the instructions or skip them entirely. I hope you don't. Reading them repeatedly, day after day, is a great way to internalize them so you don't need to waste time on the exam reading the directions--you'll already know what they say. For sentence completion questions, the first thing you should do is cover the answer choices. Next, read the sentence and predict the correct answer. Then look at the answer choices and determine which one best matches your prediction.

His ------ experience notwithstanding, David was judged by the hiring manager to be ------ the job.

This one is a little trickier than most. The sentence does not give enough information to predict either blank with certainty, but you can draw come conclusions about the relationship between the two omitted words. The word "notwithstanding" is your clue here, and it indicates that the two blanks are opposites. It means nevertheless, although, or in spite of, so either David had abundant experience and was still found inadequate, or he had meager experience and was found suitable anyway. Look at the answer choices:

A) illustrious . . entitled to

B) limited . . qualified for

C) applicable . . assured of

D) useful . . overqualified for

E) irrelevant . . perplexed by

Only B demonstrates the opposite relationship needed for this question.

His limited experience not withstanding, David was judged by the hiring manager to be qualified for the job.

Words tested in this SC:
illustrious: dignified
entitled: having the right to own, demand, or do something
applicable: suitable, relevant, appropriate
assured: guaranteed, confident
useful: having a practical or beneficial use
overqualified: having too many qualifications to be deemed appropriate for a job
irrelevant: not related, not applicable, or unimportant
perplexed: confused or puzzled

Need to build your SAT Vocab?  Visit www.myknowsys.com to check out the Knowsys Vocabulary Builder Program.

# Slope

Write it Down! This infographic linked today from www.coolsiteoftheday.com discusses the importance of taking notes, a few different methods, and the potential benefits and drawbacks of taking notes digitally or the old--fashioned way. Did you know that your brain actually processes information differently while you're taking notes? This is a good resource to bookmark and revisit when you notice that your class notes are less than helpful--it might be time to try out a different method.

## 5/15 Slope

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

Remember to read carefully, identify the bottom line, assess your options, attack the problem, and loop back. When you use this method, you will get more problems right and you will move faster through the test.

In the xy-plane, line l passes through the points (a, 0) and (0, 2a), where a > 1. What is the slope of line l?

First, read carefully. You have two points on a line, which means you can visualize that line if you wish. Picking a number for a might make that easier if the variable trips you up. Next, identify the bottom line. The question asks for the slope of line l, so at the top of your scratch work write "slope = ?"

Now assess your options. Since you need to find the slope of the line, a good place to start is with the formula for slope: rise over run. There are two choices here; you can use a as a variable or you can pick a number for a. Using a directly involves fewer steps because you don't need to plug in the value, but manipulating the variable can be confusing for some and can cost time. Which tool you choose to solve the problem is up to your personal preference.

Either way, the first step in the problem is to set up your formula. Since a must by greater than 1, I'll use 2.

$\frac{rise}{run}=\frac{2a-0}{0-a}$                                                             $\frac{rise}{run}=\frac{2(2)-0}{0-(2)}$

$\frac{rise}{run}=\frac{2a}{-a}$                                                                    $\frac{rise}{run}=\frac{4}{-2}$

$\frac{rise}{run}=-2$                                                                    $\frac{rise}{run}=-2$

Now loop back to make sure that you answered the right question. Your bottom line asks for the slope, so you found the change in y-coordinates (rise) and the change in x-coordinates (run), divided one by the other and reduced. That is the slope, so -2 is the answer you need.

A) -2

B) $\frac{-1}{2}$

C) 2

D) -2a

E) 2a

On sat.collegeboard.org, 42% of responses were correct.

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

# Remainders

"Many students discover the need to develop or hone their time management skills when they arrive at college. Unlike high school where teachers frequently structured your assignments and classes filled your day, in college, you will have less in-class time, more outside of class work, and a great deal of freedom and flexibility." Check out these pages for advice on how to manage your time in college.

## 5/3 Remainders

First, always remember to read the question carefully and identify the bottom line. If you don't know what the question is asking, you cannot answer it correctly! Next, assess your options. Based on what you need to find and on what the problem gives you, what COULD you do? What SHOULD you do? Select the most efficient method and attack the problem. Solve it as quickly as you can without making mistakes, then loop back up to the bottom line to check you answer against the question. Finally, once you have found the correct answer, select it from among the answer choices.

If it is now 4:00 p.m. Saturday, in 253 hours from now, what time and day will it be? (Assume no daylight saving time changes in the period.)

First, read carefully. The fact that daylight saving time (commonly mispronounced "daylight savings time") doesn't come or go is helpful to know; that will make the problem simpler. We start at 4 p.m. on Saturday and move forward 253 hours. 253 is a large number, and not one that is easy to work with, so trying to count the days and hours would be a huge waste of time. Instead, divide 253 by 24 to get the number of days that passed during that time.

$253\div 24=??$

Since 253 doesn't divide evenly by 24, what should you do with the left-overs? Keep them in the form of a remainder.

$253\div 24=10r13$

10 days and a remainder of 13 hours passed. If this time started at 4p.m. on Saturday, what day and time did it end? Exactly ten days from 4 p.m. on Saturday would be 4 p.m. on Tuesday. The remaining 13 hours move you forward into Wednesday, landing at 5 a.m. Now look at the answer choices.

A) 5:00 a.m. Saturday

B) 1:00 a.m. Sunday

C) 5:00 p.m. Tuesday

D) 1:00 a.m. Wednesday

E) 5:00 a.m. Wednesday

On sat.collegeboard.org, 50% of responses were correct.

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

# Functions

How to manage time and adapt to college life is a major concern for many students. A site full of College Study Tips could be a great help! This website has "college study tips, college study skills, study guides and tricks to help you manage your time, take better notes, study more effectively, improve memory, take tests, and handle the stresses of college life."

## 4/30 Functions

Always try to solve the problem as if the answer choices weren't there. This has two benefits: it makes you more efficient at multiple-choice questions, and it makes you more confident on grid-in questions. First, read carefully and look for important information. Identify the bottom line, then assess your options and select the most efficient way to solve the problem. Attack the problem, solve it quickly, and loop back to ensure that the answer you found matches the bottom line. Finally, check the answer choices and select the correct answer.

If the function f is defined by $f(x)=\frac{(x-a)(x-b)}{(x-c)}$, where 0 < a < b < c, for which of the following values of x is f undefined?

I. a
II. b
III. c

Read carefully and focus on the bottom line.

f undefined when x = ?

The most important clue here is the word "undefined." What does "undefined" mean in a math problem? It means that you have attempted to do something impossible, such as dividing by zero. In order for this function to divide by zero, x - c must equal zero because it is the denominator. Given that, there is only one possible value for x. Any number minus itself equals zero, so if x = c, then x - c = 0. Any other answer choice would yield a positive or negative number for the denominator, and the function would not be undefined. Only choice III makes the function undefined.

Now look at the answer choices:

A) None

B) I only

C) III only

D) I and II only

E) I, II, and III

On sat.collegeboard.org, 53% of responses were correct.

Want 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