Algebra: Equations

Algebra: Equations

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

Math problems require you to read carefully. Identify the bottom line, what the question is asking, and assess your options for solving the problem. There are often multiple ways to solve a problem, but you should be as efficient as possible. Choose a method to attack the problem, then loop back to make sure that you solved for the bottom line.

If  , what is the value of 4n?

Bottom Line: 4n =

Assess your Options You could solve for n and multiply by 4, or you could find a way to solve the equation for 4n. The latter method will save you time because you do not have to do the extra step of multiplying by 4. Both ways will work, but on a timed test you should choose the faster method.

Attack the Problem: Go ahead and cross multiply so that you have the equation 24n = 15 × 4. In other words, 24n = 60. Now, if you were solving for n you would have to deal with a decimal or fraction. Instead, look at the equation that you have. You know that 24 and 60 are both multiples of 4, so it would be easier to solve for 4n than to solve for n. What do you have to multiply by 4 in order to get 24? 6. All you have to do is divide each side of your equation by 6, and you will get 4n = 10.

Loop Back: You solved for your bottom line, so you are ready to look at the answer choices.

(A) 6

(B) 10

(C) 12

(D) 30

(E) 60

The correct answer is (B)


Note: If you went to collegeboard.org on November 23^rd, a programming error would have given you an incorrect answer. Collegeboard.org corrected this error the following day.


On sat.collegeboard.org, 31% of the responses were correct.

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

Pronouns

Writing: Improving Sentences

Happy Thanksgiving!

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 sentence to yourself, listening for errors.  Evaluate the underlined portion of the sentence using the Big 8 Grammar Rules.  Focus on the first error that you find in order to eliminate wrong answer choices.

They determine the color and flavor of honey by the flowers from which the nectar is taken.

Did you find a problem in the original sentence?  Who is “they?”  Not only is there not an antecedent for this pronoun, but there is also not a reason to have a pronoun or noun here.  There is no group of people (or even animals) who determines the color and flavor of honey; the sentence already tells us that these things are determined by the flowers.  Mark the problem that you found.  Realize that you cannot just take out the word “they” without creating new problems in the sentence.  You will have to rearrange the sentence in order to continue using that verb “determine.”  If you cross out “They determine,” where could you add that verb back into the sentence so that it remains a complete sentence?  Take a look at your answer choices to see whether they address the pronoun problem that you found in the sentence.

(A) They determine the color and flavor of honey by the

(B) Honey’s color and flavor is determined by which

(C) Honey’s color and flavor determined by those

(D) The color of honey, and its flavor, determined by the

(E) The color and flavor of honey are determined by the

(A)  You don’t need to reread this option because it matches the original.  Eliminate it.

(B)  There is an extra and unnecessary “which” in this answer choice.  If you select this answer choice, you will have the word “which” twice in the sentence, confusing the meaning of the sentence.  You did not find an error in the phrase “by the flowers,” so there is no need to change it.  Eliminate it. 

(C)  The word “those” is unnecessary.  Also, remember that the Knowsys Writing Strategies tell you to avoid words that end in –‘s.  It is awkward to say that honey possesses its own color and flavor.  Can honey really possess anything when it is only a product of a process?  Eliminate it. Note: this logic would also have worked to eliminate answer choice (B).

(D)  This choice lacks a verb, and it also makes the flavor of the honey less important than the color of the honey.  That is not true in the original sentence where both are given equal weight.  Eliminate it.

(E)  There are no problems with this answer choice.  It is clear and precise.

The correct answer is (E).

On sat.collegeboard.org, 76% of the responses were correct.

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

Sentence Completions

Critical Reading: 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. 

Always cover up your answer choices so that you can approach the sentence with an open mind.  Read the sentence carefully, using context clues to predict what belongs in the blank.  Then you can match your prediction to the correct answer and eliminate any answer choices that do not match.  Be sure to look at all 5 answer choices even if the first one seems to match.

Paradoxically, this successful entrepreneur is sometimes ------- and at other times reclusive.

The word “paradoxically” lets you know that this sentence will contain a contrast.  You don’t really have to know what the word entrepreneur means.  Sometimes this person is one thing and sometimes he is reclusive.  What is the opposite of reclusive?  I like to connect the word “recluse” with the word “hermit;” a recluse shuts himself or herself out of society.  Predict a word such as “outgoing” to fill the blank and look down at your answer choices. 

(A) autonomous

(B) dispassionate

(C) solitary

(D) unthinking

(E) gregarious

(A) The Latin root “auto” means self, so this answer choice will not have anything to do with interacting with others.  Eliminate it.  (B)  Passion does not really have anything to do with being alone or being with others.  You can be passionate in either situation.  Eliminate it.  (C) The Latin root “sol” can mean sun, but it can also mean alone, as in solitary.  This is the opposite of your prediction.  Eliminate it.  (D)  This answer choice is just in case you read the sentence too quickly and thought that you were looking for the opposite of a successful entrepreneur.  Entrepreneurs think a lot to be successful, right?  So maybe the opposite would be unthinking.  Before you start trying to justify the answer, notice that it does not match your prediction and eliminate it.  (E)  The Latin root “greg” means group.  Participating in a group would be evidence of outgoing behavior, and it is the opposite of being reclusive.  This perfectly matches your prediction.

Words used in this SC:

Paradoxically: in a self-contradicting manner

Entrepreneur: someone who organizes or starts a business

Reclusive: withdrawn, preferring isolation

Autonomous: self-governing

Dispassionate: not showing or feeling emotion

Solitary: living alone

Unthinking: careless

Gregarious: outgoing and sociable

On sat.collegeboard.org, 65% of the responses were correct.

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

Functions

Link of the Day

How do medical breakthroughs happen?  How do you feel about animal testing?  A few dogs that were once paralyzed are now walking again with the aid of some cells from healthy dogs.  Scientists recognize that they are not ready to apply their findings to humans with spinal cord injuries, but they are hopeful about the future.  This development would make a great current event example for your SAT essay.  If you choose to use it as a current event, take notes detailing the facts in this article and how you could apply them to a wide variety of topics.

Algebra: Functions

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

Take the time to read math questions carefully because this will save you from wasting time as you solve the problem.  Start by identifying the bottom line and assessing your options for reaching it.  Choose the most efficient method to attack the problem.  When you think you have an answer, loop back to make sure that it matches the bottom line.

• f(2n) = 2f(n) for all integers n
• f(4) = 4

If f is a function defined for all positive integers n, and f satisfies the two conditions above, which of the following could be the definition of f?

Bottom line: f(n) = ?

Assess your options:  You could start by plugging in f(4) to see which of your answer choices results in the number 4, then check any that do against the first condition.  This is the method recommended by collegeboard.com.  However, this requires multiple steps as you compare each answer choice to both conditions.  Instead, take a moment to think logically about the two conditions that you are given.

Attack the problem:  Start with the second condition because it is already in a format that is easy to use.  If f(4) = 4 and you can use the same variable to represent numbers that are the same,  that is the same as saying that f(n) = n.  Now look at the first condition and think about it logically.  If you multiply the variable within the function by 2, that gives you the same number as multiplying the result of the function by 2.  In order for those two numbers to be the same, the final result of the function has to match the number that is plugged into the function.  In other words, f(n) = n.  Both conditions give you the same definition of the function.

Loop Back:  You found a simple way to define both of the conditions for f(n), so look down at your answer choices.

(A)  f(n) = n - 2

(B)  f(n) = n

(C)  f(n) = 2n

(D)  f(n) = 4

(E)  f(n) = 2n – 4

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

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

Sentence Structure

Writing: 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. 

Read the whole sentence to yourself, listening for errors.  Then quickly check each underlined portion of the sentence against the Big 8 Grammar Rules.

Every day, millions of tiny stony bits Afalling quietly into the atmosphere, burning briefly Basmeteors, and leaving Cbehind a vaporized residue that filters Dslowly to Earth. ENo error

(A)   When you read the sentence, you should notice that it is a fragment.  Fragments sound as if they should continue because they lack either a subject or a verb.  Start by identifying the main subject in the sentence.  The main subject is “millions of tiny stony bits.”  What is the main verb?  Right now it is in the wrong form: “falling.”   You cannot put the subject and verb together to create a whole sentence: millions of tiny stony bits falling.  Remember that the Knowsys handbook reminds you to check any word ending in –ing to be sure that the –ing is necessary.  Instead of “falling” you need the words “are falling” or simply “fall.”  Mark this error and quickly look at the rest of the underlined portions.

(B)  Anytime the word “as” is underlined, think about whether the word “like” belongs there instead.  These stone bits are not “like” meteors, they are meteors!  The word “as” can be used to show that something is acting in a particular way.  For example, I can act as a guide and my friend can be hired as a tutor.  There is no error here.

(C)  This answer choice requires you to think about idioms.  The phrase “leave behind” is a common English idiom that can mean anything from forgetting something, to passing something on a journey.  In this sentence “leaving behind” actually means leaving a trace of something, which is exactly what those burning rocks do: they leave a trace of vaporized residue.  This idiom is used correctly, so there is no error here.

(D)  This underlined portion contains an adverb; remember those things ending in –ly?  This particular adverb modifies how a vaporized residue filters.  It is also placed as close as possible to the word that it modifying.  There is no error here.

(E)  This choice cannot be the answer because you have already identified an error.

The correct answer is (A).

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

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

Sentence Completions

Link of the Day

Although most people agree that taking care of the planet is important, individuals and businesses often disagree about the best use of resources.  Clean air also costs money as companies work to find ways to keep their carbon emissions low.  It has been incredibly difficult to motivate care for the environment through legislation, which is one reason that many states and nations are watching California's new plan.  Read about the plan here, and be sure to think about how the broad themes involved in this current event could relate to an SAT essay prompt.

Critical Reading: 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. 

Cover your answer choices so that you can carefully read the sentence without any kind of bias.  When you see two blanks, select the blank that looks easier and focus on it first.  Make a prediction for that blank and eliminate any answer choices that do not match your prediction.  Then use the same process with the other blank.

While a “rock” is usually defined as ------- , or a combination, of one or more minerals, geologists often ------- the definition to include such materials as clay, loose sand, and certain limestones.

The first blank is a simple vocabulary question, so look at it first.  The word “or” signifies that an alternative is coming, so the words “a combination” is an alternative to whatever goes in the blank.  You can just use the words “a combination” as your prediction and look down at the options for the first blank.

(A) a conglomeration . . limit

(B) an aggregate . . extend

(C) an element . . eliminate

(D) a blend . . restrict

(E) a product . . provide

(A) If you know your Latin roots, you know that the prefix “con” often means “with” or “together.”  This word could be a combination of things, so keep this answer choice.  (B) Here are more Latin roots: “ag” and “greg.” The first prefix, “ag,” can mean “toward” or “in addition to,” while the second, “greg,” means “group.”  This word probably has to do with combining things.  Keep it.  (C) Think about elements in science.  They are as basic as they can be – they are not a combination of other things.  Eliminate this answer choice.  (D)  The word “blend” requires two things to come together, so it could work too.  (E)  The word product does not immediately bring to mind a combination of things, but in math the product of two numbers is a third.  Keep this answer choice if you are not sure that you can eliminate it.

Now look at the second blank.  This blank requires you to use some logic.  The first definition for the word “rock” was very specific, but now geologists are saying that other things can also be considered rocks.  They are stretching that definition or increasing it.  You might predict the word “broaden” before looking down at your remaining answer choices.

(A) a conglomeration . . limit

(B) an aggregate . . extend

(D) a blend . . restrict

(E) a product . . provide

(A) is the opposite of your prediction.  Eliminate it.  (B) matches your prediction.  (D) is also the opposite of your prediction.  Eliminate it.  (E) is not specific enough.  An original definition was already provided, what was important in the second blank is that it was changed to add more stuff.  This answer shows no evidence of any kind of change.  Eliminate it.

The correct answer is (B).

Words used in this SC:

Conglomeration: a cohering mass of different materials

Aggregate: a collection of things into one whole or total

Element: an essential part

Blend: a mixture of two or more things

Product: something created by a process

Extend: to expand or increase

Restrict: to limit or confine

On sat.collegeboard.org, 57% of the responses were correct.

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

Triangles

Geometry: Triangles

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

You should start by reading the problem carefully and identifying your bottom line.  Then assess your options and choose the most efficient method to attack the problem.  Finally, loop back to make sure that the answer you found matches the bottom line that you set out to find.

Which of the following CANNOT be the lengths of the sides of a triangle?

Bottom Line: The word “CANNOT” tells you that you are looking for something that is not true.

Assess your Options:  Problems that ask you to find what cannot be true are often impossible to predict; you will have to look at the answer choices to determine whether they work.  Go ahead and look at the answers.

(A) 1, 1, 1

(B) 1, 2, 4

(C) 1, 75, 75

(D) 2, 3, 4

(E) 5, 6, 8

You could try to use logic and your experience with triangles to eliminate some choices.  For example, answer choice A is not the answer because you know there is such a thing as an equilateral triangle and the sides 1, 1, and 1 would create that kind of triangle.  However, after that point you would probably just be guessing.  If you know the Triangle Inequality Theorem, you can systematically check each answer choice.

Attack the problem:  When you are dealing with three sides of a triangle and you do not know that the triangle is a right triangle, you should always think of the Triangle Inequality Theorem.  This theorem states that for any triangle, side x is less than the sum and greater than the difference of the other two sides.  In other words, each side of the triangle must be less than the other two sides added together and greater than the difference of the other two sides.  If you have a triangle with sides x, y and z, you would write the theorem this way:
|y – z| < x < y + z. For the subtraction part you can use absolute value or just always do the bigger side minus the smaller side; the result will be the same. The easiest way to think about the theorem is this: for any triangle, other sides subtracted < one side < other sides added.

Start by checking (E).  It has sides 5, 6, and 8.  Plug these sides into your formula by using the first side, 5, as your x.  Make sure that 5 is greater than the difference of the other two sides, but smaller than the other two sides added together.

|y – z| < x < y + z

8 - 6 < 5 < 6 + 8 

2 < 5 < 14

This is true!  (E) works as a triangle.

Now check (D). It has sides 2, 3, and 4.  Plug those numbers into the Inequality Theorem.

 |y – z| < x < y + z

4 – 3 < 2 < 3 + 4

1 < 2 < 7

This is true!  (D) works as a triangle.

Now check (C).  It has sides 1, 75, and 75.  Plug those numbers into the Inequality Theorem.

|y – z| < x < y + z

75 – 75 < 1 < 75 + 75

0 < 1  < 150

This is true!  (C) works as a triangle.

Now check (B).  It has sides 1, 2, and 4.  Plug those numbers into the Inequality Theorem.

|y – z| < x < y + z

4 – 2 < 1 < 2 + 4

2 < 1 < 6.

Is 2 less than 1?  No!  This is false.  You cannot have a triangle with these three side lengths.

Loop Back:  Your bottom line was to find an answer choice that cannot be a triangle, so you are finished!

The correct answer is (B).

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

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