Writing Equations

Algebra: Writing Equations

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

Use the same method for all SAT math questions.  Read the question carefully, identify the bottom line, assess your options for reaching the bottom line, and choose the most efficient option to attack the problem.  When you have an answer, loop back to make sure that it matches your bottom line.

A geologist has 10 rocks of equal weight. If 6 rocks and a 10-ounce weight balance on a scale with 4 rocks and a 22-ounce weight, what is the weight, in ounces, of one of these rocks?

Bottom line: Remember to write your bottom line in easy-to-understand shorthand. You could write "weight of 1 rock = ?" but "w = ?" is much shorter. 

Assess your options: You could try each of your answer choices in this scenario, but that will waste time because you will most likely need to try multiple answers.  Start by writing an equation so that you only have to solve one problem.

Attack the problem:  On one side of the scale you have 6 rocks and a 10 oz. weight.  You don’t know how much each rock weighs, so you will need to add a variable to represent that number.  There are 6 of that missing weight (w), plus 10 oz.

6w + 10

All of this balances with, is equal to, 4 rocks of the same weight plus 22 oz.

6w + 10 = 4w + 22

Now solve the equation that you wrote by combining like terms and isolating the variable.

2w + 10 = 22

2w = 12

w = 6

Loop Back: You solved for the weight of one rock, so you are ready to look down at your answer choices.

(A) 4

(B) 5

(C) 6

(D) 7

(E) 8

The correct answer is (C).

On sat.collegeboard.org, 67% of the responses were correct.

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

Idioms

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

The jury Aawarded first prize in the architectural competition to a firm Bknown not only for Citsinnovative design Dand also for its environmentally friendly building practices. ENo error.

(A) When a verb is underlined, check to make sure that it matches the subject of the sentence.   The phrase “jury awarded” has no error.

(B) Someone or something can be “known for” a specific trait or traits.  A “for” follows the underlined “known,” so there is no error here.

(C) Check any pronoun to make sure that it matches its antecedent.  “It” refers to the “firm” stated in the sentence.  Now check to make sure that you need “its” rather than “it’s.”  “It’s” means “it is,” which does not make sense in context.  The possessive “its” is needed here, so there is no error.

(D) The “and” links two things, but it does not express the full meaning that is necessary for this sentence.  Anytime you have the phrase “not only” it must be followed by the word “but.”  This fact is listed on your Knowsys Frequently-Tested Idioms chart.  For example, I will order not only this but also that.  Mark this error.

(E) This answer choice cannot be correct because you already marked an error.

The correct answer is (D).

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

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

Sentence Completions

Link of the Day

Teachers and parents are often so caught up in trying to prepare teens for the future that they don’t realize that teens have a lot to contribute in the present.  Your life is happening now; instead of waiting for the future, get out there and make a difference with the talents that you have.  Here is the story of one teen, Jack Andraka, who created a method to detect pancreatic cancer.  (And yes, this would make a great current event example for your SAT essay!)

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 up the answer choices until you have read the sentence and made a prediction to fill the blank.  Then match your prediction to the correct answer and eliminate any words that do not match.  Make sure to examine all of the answer choices as possibilities. 

The salmon's extraordinary ability to smell a single drop of its home river in almost two million gallons of seawater is only one of the fish's many ------- skills.

Ask yourself, what kind of skill does this fish have?  It is extraordinary!  You can use words that are in the sentence to save yourself time.  This extraordinary ability is just one of the many extraordinary skills that the fish has.  Predict the word “extraordinary” and look down at your answer choices.

(A) sinuous

(B) encumbered

(C) colossal

(D) prodigious

(E) furtive

(A) The Latin prefix “sinu-“ has something to do with drawing a line.  Think of the graph of sine from your math classes: it is curvy.  Curvy doesn’t match your prediction of extraordinary, so eliminate this choice.  (B)Encumber is a negative word:  Too many shopping bags encumber a person.  You are looking for a positive word, so eliminate this choice.  (C) Colossal has something to do with big, so this might be a tempting answer.  However, it actually means big as in large, not big as in a big idea.  Eliminate this choice. (D)  The prefix “pro” from prodigious means for or forward, and is often used with positive words.  Think of the related word, prodigy.  Are prodigies extraordinary?  Yes, so keep this choice.  The Latin prefix “furt” means “steal,” so you can identify this as a negative word.  Eliminate this choice.

The correct answer is (D).

Words used in this SC:

Sinuous: curvy and graceful or indirect and devious

Encumbered: weighed down with a burden

Colossal: extremely large, gigantic

Prodigious: extraordinary, wonderful, enormous

Furtive: stealthy, exhibiting guilty or evasive secrecy

On sat.collegeboard.org, 56% of the responses were correct.

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

Number Line

Arithmetic: Number Line

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

Work all math problems by reading the question carefully and identifying the bottom line.  Assess your options for solving the problem and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that it satisfies the bottom line.

A, B, C, and D are points on a line, with D the midpoint of segment . The lengths of segments , , and  are 10, 2, and 12, respectively. What is the length of segment ?

Bottom Line: distance A to D

Assess your Options:  Drawing out the situation will give you a visual to understand the situation. 

Attack the Problem:  Start with what you know.  You have a lot of points named, but the first information that you are given is that D is the midpoint between B and C.

Now you are given three lengths.  You can’t label the ones involving A yet, but you can label the length from B to C.  Remember that D is the midpoint, and you will also know the lengths of B to D and D to C.

Now go back to those other lengths you were given that involved point A.  Point A is 2 units away from C and 10 units away from B.  The only possible location for A is between B and C, but closer to C.

Now that you have all your points labeled, it is time to go back and look for your bottom line.  What is the distance from A to D?  D to C was 6 units, and A to C was 2 units, so what is 6 minus 2?  The answer is 4.  (You could also use B to A is 10 and subtract the length of B to D, 6, and get the same answer of 4.)

Loop Back:  You solved for the distance from A to D, so you are ready to check your answers.

(A) 2

(B) 4

(C) 6

(D) 10

(E) 12

The correct answer is (B).

On sat.collegeboard.org, 61% of the responses were correct.

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

Modifiers

Link of the Day:

Is competition always a good thing?  Read this article, which explains why there is no such thing as a brontosaurus, before you answer.  What themes other than competition do you see in this historical example?  How could you use this story as an interesting addition to your SAT essay?

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

Concerned that people will click to borrow an e-book from a library rather than click to buy it, library access to the e-book forms of most U.S. publishers' titles is blocked.

Think about the structure of this sentence.  The part of the sentence that is not underlined is an introductory phrase, modifying the part that comes next.  You know that because it does not name the subject of the sentence.  Who is concerned?  You don’t know.  When you have an introductory phrase followed by a comma, the very next independent noun must be the subject of the phrase.  However, in your original sentence, it sounds as if “library access” is what is concerned.  This does not make sense.  Look down in your answer choices for an answer that follows the rule.

(A) library access to the e-book forms of most major U.S. publishers' titles is blocked

(B) it is important to block library access to the e-book forms of most major U.S. publishers' titles

(C) the e-book forms of most major U.S. publishers' titles are blocked from library access

(D) most major U.S. publishers block library access to the e-book forms of their titles

(E) most major U.S. publishers blocking library access to the e-book forms of their titles

(A) You don’t need to read this option because you already found a modifier error in the original sentence.  Eliminate it.

(B) “It” is not who is concerned.  This does not fix the modifier error.  Eliminate it.

(C) “The e-book forms” are also not concerned.  Eliminate it.

(D) “Most major U.S. publishers” would definitely be concerned about sales of e-books.  This fixes the error that you found.  Keep it.

(E) This choice also fixes the original error that you have, but it changes “block” to “blocking.”  The Knowsys rule is to avoid words ending in “-ing” unless they are necessary in the sentence.  In this case, the extra “-ing” is not necessary but creates a sentence structure problem: you no longer have a complete sentence.  Eliminate it.

The correct answer is (D).

On sat.collegeboard.org, 55% of the responses were correct.

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

Sentence Completions

Link of the Day:

Are you able to memorize and analyze sports statistics even though other numbers leave your head in an instant?  You can use athletes and important events from the world of sports in your SAT essay as one excellent historical or current event example.  However, this example must be intellectual; be able to explain why the player or event is important and what kind of conclusions you can draw from this example in a wider context than sports.  Make sure you can draw out the themes of the example and then name specific statistics to back up your claims.  One current event that would make an excellent essay example is freshman Johnny Manziel's acceptance of the Heisman.  No freshman has ever won the award before.  Read more about this topic here.

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 they do not prejudice your thoughts as you read the sentence carefully.  Then make a prediction for the easier blank and check it against the answer choices.  Eliminate any word that does not match your prediction.  Do the same for the other blank.  Always make sure to look at all the choices, even if one matches your prediction exactly.

Not wanting to come across as -------, the award-winning biologist ------- her many impressive accomplishments as she addressed the team of scientists.

Think about this sentence logically.  If someone has a lot of awards, how will others perceive that person?  Awards are a good thing unless someone becomes too proud and haughty.  Since the first blank has to do with what the award-winner does not want to be, predict the word “haughty” and look down at your answer choices.

(A) egotistical . . downplayed

(B) experienced . . explored

(C) celebrated . . highlighted

(D) arrogant . . evoked

(E) incompetent . . omitted

(A) Egotistical matches your prediction.  Keep it.  (B)  Experience is a good thing, people want to be seen as experienced.  It does not match your prediction, so eliminate it.  (C) Celebrated is a very positive word, but you are looking for a negative trait.  Eliminate it.  (D) Arrogant and haughty are synonyms.  Keep it.  (E) Incompetent is a negative thing, but you already know that the biologist is award-winning.  Her problem is not looking incompetent, it is looking too proud.

Look now at the second blank.  If this biologist does not want to be seen as too proud, is she going to enumerate or list out all of her accomplishments?  No!  She is going to avoid emphasizing them.  Predict a word such as “minimize” and look down at your remaining answer choices.

(A) egotistical . . downplayed

(D) arrogant . . evoked

(A) Downplayed matches your prediction.  (D) Evoked is the opposite of your prediction.

The correct answer is (A).

Words used in this SC:

Egotistical: vain, selfish, boastful, proud

Downplayed: reduced emphasis on

Arrogant: having excessive pride in oneself, conceited

Evoked: called up

Incompetent: unskilled

Omitted: left out

On sat.collegeboard.org, 67% of the responses were correct.

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

Writing Equations

Algebra: Writing Equations

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

Read the question carefully so that you don’t miss any important information.  Identify the bottom line and assess your options to find it.  Choose the most efficient method to attack the problem.  Always loop back to make sure that your answer addresses the bottom line.

Milk costs x cents per half-gallon and y cents per gallon. If a gallon of milk costs z cents less than 2 half-gallons, which of the following equations must be true?

Bottom Line: equation

Assess your Options:  The question asks you about "the following equations," so your first instinct is going to be to look down at the answer choices.  Don’t do it!  Most of them are wrong and they are there to distract you from the correct answer.  Instead, write your own equation using what you know from the problem.

Attack the Problem: Start with what you know: “Milk costs x cents per half-gallon and y cents per gallon.”  Make a note:

x = half-gallon

y = gallon

Now look at the conditions that you are given “a gallon of milk costs z cents less than 2 half-gallons.”  The word “costs” is just like the word “is;” it shows you where to put the equal sign.  The words “less than” signal that you will need to subtract the z. Use the variables you have been given to write an equation.

y = 2x – z

Once you have an equation, glance down at your answer choices.  Notice that all of them are set equal to zero, and all the x values are positive.  Set your equation equal to zero and keep the x value positive by subtracting the y variable from each side.

0 = 2x – z – y

As you look at your answer choices, realize that when you are adding and subtracting numbers, order does not matter.  In fact, all of the answers have the variables arranged alphabetically.  Do the same to your equation.

0 = 2x – y – z

Loop Back:  You can be confident in your answer because you reached it by writing your own equation.

(A) x – 2y + z = 0

(B) 2x – y + z = 0

(C) x – y – z = 0

(D) 2x – y – z = 0

(E) x + 2y – z = 0

The correct answer is (D).

On sat.collegeboard.org, 56% of the responses were correct.

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