# Inequalities

## Algebra: Inequalities

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

Always read the question carefully, identifying the bottom line.  Assess your options for reaching the bottom line and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.

On the line above, if AB < BC < CD < DE, which of the following must be true?

Bottom Line: wotf must be true = ? (which of the following)

Assess your Options:  For a “wotf” question, you will have to look at the answer choices.  Most students will start with “A,” so Knowsys recommends that you start with “E.”  You may also find that this is a good problem to use the strategy of plugging in numbers.

Attack the problem:  Take a look at your answer choices:
(A) AC < CD
(B) AC < CE
(C) AD < CE
(D) AD < DE
(E) BD < DE

(E) BD < DE  Look up at the figure.  On the figure, does BD look smaller than DE?  No!  It looks slightly larger.  You know that the figure is not drawn to scale, but the figure does give you one possible depiction of the rule.  Use the figure!  If it is possible for BD to be bigger than DE, then this answer is incorrect because you are looking for something that must be true.  Eliminate this choice.

(D) AD < DE  Look up at the figure.  The figure shows you that it is possible for AD to be larger than DE.  Eliminate this choice.

(C) AD < CE  These lengths are very similar on the line.  Break each length down into the parts that compose it so that you can make a precise comparison.  For example, AD contains AB + BC + CD.  CE contains CD + CE.  You now have: AB + BC + CD < CD + DE.  When you have the same thing on both sides of an equation, it cancels.  Eliminate the CD.  You now have AB + BC < DE.
You cannot come to a conclusion about these lengths.  If you want to prove this, try plugging in numbers.  Suppose AB starts at 10 and each portion along this line gets larger by 1.  AB = 10, BC = 11, CD = 12, DE = 13.  Is 10 + 11 < 13?  No.  Eliminate this choice.
(B) AC < CE  This one looks like it could be true, based on the figure.  See if you can prove it.  Break it down into its parts just as you broke down the last answer choice.  AC contains AB + BC.  CE contains CD + DE.  At first it seems as if you cannot compare these either because all of the numbers are different.  Try plugging in the same values as you used before: AB = 10, BC = 11, CD = 12, DE = 13.  Is 10 + 11 < 12 + 13?  Yes!  Will this work for all numbers?  Yes!  You are adding a small number plus a medium number and comparing it to a big number plus an even bigger number.  The former will always be smaller than the latter.  Once you know this, you do not even need to check (A).

(A) AC < CD You can tell from the figure that this does not have to be true.

Loop back:  You solved for what must be true, so you should select the answer you found.

The correct answer is (B).

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

To get help preparing for the SAT exam, visit www.myknowsys.com!

## ACT Question of the Day:

If you have gone 4.8 miles in 24 minutes, what was your average speed, in miles per hour?

Your bottom line here is in miles per hour.  That would be miles over hours.  Your distance (miles) is in the correct unit, but your time (minutes) is not.  You know that there are 60 minutes in an hour.   Find the fraction of an hour that was spent traveling. Take your minutes and put them over the total minutes in an hour:

$\frac{24\: min}{60\: min}=\frac{2}{5}\: of\: an\: hour =.4\: hr$

Now you know that you went 4.8 miles in .4 hours.  How many miles per hour was that?  Divide 4.8 by .4 and you will see that the answer is 12.

Note: You can do this in your head if you realize that this is the same thing as dividing 48 by 4.  This whole problem can be done in seconds if you know your times table all the way up to 12.

(A)  5.0
(B) 10.0
(C) 12.0
(D) 19.2
(E) 50.0

The correct answer is (C).

For the ACT Question of the Day, visit http://www.act.org/qotd/.

To get help preparing for the ACT exam, visit
www.myknowsys.com!

# Roots and Radicals

## Algebra: Roots and Radicals

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

Always read the question carefully and identify the bottom line.&nbsp. Assess your options for reaching the bottom line – what is the easiest and most time-efficient method to reach the answer? Use that method to attack the problem. When you have an answer, loop back to make sure that you reached the bottom line and did not just solve a portion of the problem.

If $\sqrt{x}=16$, what is the value of $\sqrt{4x}$?

Bottom Line: $\sqrt{4x}=?$

Assess your Options: You might be tempted to find the value of x first, but look at your bottom line. Do you need to know the value of x? No! Don’t waste your time! You just need to know the value of the square root of x multiplied by another number. Use your knowledge of radicals to rearrange your bottom line so that you have fewer steps to solve the problem.

Attack the Problem: Focus on the 4 under the radical. If this question simply asked for the square root of 4, you could easily answer. What is the square root of 4? 2! That value now goes in front of the radical. This could also be written as 2 multiplied by the square root of x. Plug in the value of 16 that you were given for the square root of x. All you have to do to reach a single number is multiply 2 by 16. Here are the steps you just completed:

$\sqrt{4x}= 2\sqrt{x}=(2)(\sqrt{x})=(2)(16)=32$

This method is much easier and faster than finding that x = 256, multiplying 256 by 4, and then taking the square root of 1024. You should not need to waste time typing numbers into a calculator in order to solve this problem.

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

$\sqrt{4x}= 2\sqrt{x}=(2)(\sqrt{x})=(2)(16)=32$

$\sqrt{4x}=?$

(A) 16

(B) 32

(C) 64

(D) 128

(E) 256

The correct answer is (B). On sat.collegeboard.org, 52% of the responses were correct.

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

# Rates

## Arithmetic: Rates

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

Use the same method with each math question to avoid making mistakes.  Start by reading carefully and identifying the bottom line.  What question must you answer?  Then assess your options for answering the question, choosing the most time efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.

Machine X, working at a constant rate, can produce x bolts per hour. Machine Y, working at a constant rate, can produce x + 6 bolts per hour. In terms of x, how many bolts can both machines working together at their respective rates produce in 4 hours?

Bottom line: #bolts in 4 hr = ?

Assess your Options:  You could choose numbers for x and y and then see which of your answer choices matches the answer that you get, but you will still have to write an equation.  It will be much faster to leave the variable in the problem and write an equation to find the answer.

Attack the Problem:  You know that you have two machines, X and Y.  You know how much each of these machines produces in an hour.  Find out the total that they can produce in one hour.

X + Y (both machines)= x + x + 6          Combine like terms.
X + Y (both machines)= 2x + 6

In one hour you can produce 2x + 6 bolts.  However, your bottom line requires you to find the number of bolts that can be produced in 4 hours.  Multiply 2x + 6 by 4.

4(2x + 6)          Distribute the 4.
8x + 24

Loop Back:  You solved for 4 hours rather than just 1 hr, so you are ready to look at the answer choices.

(A) 4x + 12
(B) 4x + 24
(C) 6x + 30
(D) 8x + 24
(E) 8x + 36

The correct answer is (D).

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

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

# Equation of a Line

## Link of the Day

How do you make sure that you have the best doctors and the best conditions for patients?  First there was a push for doctors to get more sleep.  Now there is a push to make sure that doctors are getting more hours to finish their work.  Take a look at the debate in this current event.  Write down the broad themes in this article, and the specific details that will make you sound informed.  Then try linking this current event to the following previous SAT essay prompts:  Is there always another explanation or another point of view?  Can success be disastrous?  Should people let their feelings guide them when they make important decisions?  Should people change their decisions when circumstances change, or is it best for them to stick with their original decisions?

## Geometry: Coordinate Geometry

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

Always read the question carefully and identify the bottom line.  Assess your options for reaching the bottom line, and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.

If the graph of the function f is a line with slope 2, which of the following could be the equation of f?

Bottom Line: WOTF (which of the following)

Assess your Options:  For a “which of the following” question you should look at the answers choices, but not until you have used what you know about the equation of a line to decide what kind of equation you need to find.  Start with the information that you are given.

Attack the Problem:  Remember the generic equation for a line is y = mx + b.  In any equation, f(x) and y can mean the same thing.  The variable m is the slope of the line.  You know that your slope must be 2.  Plug that 2 into the equation.  You now have:

f(x) = 2x + b

(The variable b is the y-intercept.  You were not told anything about the y-intercept, so that could be any number.  All you need to do is match the part that you do know, the 2x.)

Loop Back:  You used all the information that you were given, so look down at your answer choices.

(A) f(x) = 4x - 2
(B) f(x) = 2x + 4
(C) f(x) = -2x – 2
(D) $f(x)=\frac{1}{2}x+2$
(E) $f(x)=-\frac{1}{2}x+\frac{1}{2}$

The correct answer is (B).

On sat.collegeboard.org, 64% of the responses were correct.

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

# Fractions

## Arithmetic: Fractions

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

Use the same process with every SAT question.  Read carefully and identify the bottom line.  Then assess your options for reaching the bottom line and choose the most time efficient method to attack the problem.  When you have an answer, loop back to check that you solved for the bottom line.

$\frac{1}{2}\cdot \frac{2}{3}\cdot \frac{3}{4}\cdot \frac{4}{5}\cdot \frac{5}{6}\cdot \frac{6}{7}=$

Bottom Line: just solve

Assess your options:  When you see a problem like this, get excited!  Some people will multiply all of the numbers, or change the fractions into decimals, but you should recognize a pattern!  Use what you know about fractions to solve this problem in less than 5 seconds.

Attack the problem:   The way you would normally solve the problem is to multiply all of the top numbers and multiply all of the bottom, then simplify the resulting fraction.  There is a faster way!  Although this problem starts out with separate fractions, you can think of the numbers that you are given as factors of the product you would get.  Remember that a number on top of a fraction will cancel if the same number is on the bottom of a fraction. Envision the problem this way:

$\frac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6}{2\cdot3\cdot 4\cdot 5\cdot 6\cdot 7 }=$

Then simply eliminate any numbers that are both on top and bottom!  The 2s cancel.  So do the 3s.  Keep going, and what do you have left?

$\frac{1}{7}$

Loop back:  You solved the original equation, so you are ready to look down at the answer choices.

(A) $\frac{1}{7}$
(B) $\frac{3}{7}$
(C) $\frac{21}{27}$
(D) $\frac{6}{7}$
(E) $\frac{7}{8}$

The correct answer is (A).

On sat.collegeboard.org, 60% of the responses were correct.

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

# Multiples

## Arithmetic: Multiples

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

Approach each problem the same way so that you feel confident about your ability to solve it.  Start by reading the question carefully and identifying your bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer addresses the bottom line.

Add 8x to 2x and then subtract 5 from the sum. If x is a positive integer, the result must be an integer multiple of

Bottom Line:  multiple of = ?

Assess your Options:  You have to write an equation for this problem, but after doing so you can use logic or the strategy of plugging in numbers to find possible answers to the equation.  Both methods are quick and will result in the correct answer.

Attack the Problem:  Your first step is to translate all the words you are given into an equation. If you add 8x to 2x, you get 8x + 2x.  Then subtract 5.  You should have:

8x + 2x – 5

Always simplify as much as possible before moving to the next step.  Here, you can combine like terms.

10x – 5

Now go back to the other information that you are given.  The variable x must be a positive integer.  Plug in the smallest possible value for x, and you will get the smallest possible result of this equation.  Plug in x = 1.

10(1) – 5 = 5

Now, multiples will always get larger, so there are other possible answers to this equation.  However, this is the smallest answer and you are looking for what the result “must” be an integer multiple of.  Multiples are simply the product of a number and an integer.  5 is a prime number, so the only thing that the answer must be a multiple of is 5.

(If you want to make sure you are on the right track, plug in x = 2.  The answer is 15.  15 is still a multiple of 5.  Any positive number that you plug in will still be a multiple of 5 because when you subtract 5 from a multiple of 10, you will always get a number ending in a 5.)

Loop Back:  You found that the answer must be a multiple of 5.  Look down at your answer choices.

(A) 2
(B) 5
(C) 8
(D) 10
(E) 15

The correct answer is (B).

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

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

# Coordinate Geometry

## Geometry: Coordinate Geometry

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

Read the question carefully and identify the bottom line.  Assess your options for solving the problem and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

What is the area of the triangle in the figure above?

Bottom Line: a =?  (What is the area?)

Assess your Options:  The best way to solve this problem is to use the formula for the area of a triangle.  You have already been given all the information that you need to solve the problem.

Attack the Problem:  Start with the formula for the area of a triangle.

$area =\frac{1}{2}(base)(height)$

The base of the triangle extends to the right of the origin (5 units).  The height of the triangle extends upwards from the origin (3 units).

$area =\frac{1}{2}(5)(3)$

Work with the easy numbers first: 5 times 3 is 15.  If you divide 15 by 2 you get 7.5.

Loop Back:  You solved for area, so you are ready to look down at the answer choices.

(A) 4.0
(B) 7.5
(C) 8.0
(D) 8.5
(E) 15.0

The correct answer is (B).

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

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

# Coordinate Geometry

## Link of the Day

Isn't it fascinating that no matter how long people study people, there is still more to learn?  Take a look at this current event article that endeavors to explain why women talk more than men.  Pick out the broad topics in this article.  How could you use the facts from this article to support a position on the following SAT essay prompts?

1. Do we need other people in order to understand ourselves?
2. Should heroes be defined as people who say what they think when we ourselves lack the courage to say it?
3. Are people best defined by what they do?

## Geometry: Coordinate Geometry

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

Always read the question carefully and identify the bottom line.  Assess your options for reaching the bottom line and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that the answer matches the bottom line.

What is the equation of the line parallel to the x-axis and four units above the x-axis?

Bottom Line: equation of a line

Assess your Options:  You could look down at the answer choices, but if you look down without thinking first you will often confuse yourself.  Instead, use the information that you are given to write an equation.

Attack the Problem:  You know that you are dealing with an x-axis, which means you must use a normal xy-graph with a vertical y-axis and a horizontal x-axis.  Draw this on your paper.  Next, imagine 4 ticks on the y-axis and put a little dot four units above the x-axis.  Draw a horizontal line that is parallel to the x-axis.  Does that line ever leave y = 4?  No!  That is the equation of the line.

Note:  If you write x = 4, you create a vertical line.  Think about it this way: the x values change from negative infinity to positive infinity.  If you choose a single x value, the line along this value cannot be parallel to the x-axis because it is limited to a single value.

Loop Back:  You needed an equation of a line, and not necessarily one that mentioned x at all.  You found one.  Look down at your answer choices.

(A) x = -4
(B) x = 4
(C) y = -4
(D) y = 0
(E) y = 4

The correct answer is (E).

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

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

# Writing Equations

## Link of the Day

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!

# Writing Equations

## Algebra: Writing Equations

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

Approach all math questions the same way.  Read the question carefully to avoid making careless mistakes.  Identify the bottom line, the question you must solve, and note it on your test.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer addresses the bottom line.

First, 3 is subtracted from x and the square root of the difference is taken. Then, 5 is added to the result, giving a final result of 9. What is the value of x?

Bottom line: x = ?

Assess your options: You could try to plug in answer choices and see which one equals 9, but you may have to write and solve the equation multiple times.  Instead, translate the two sentences into “math” and use algebra to find x.

Attack the problem: Work through the words step by step.  First, 3 is subtracted from x.  Write:

x – 3

The square root of the difference is taken.  That means both numbers involved in the difference are under the radical.

$\sqrt{x-3}$

Then 5 is added and the final result is 9.

$\sqrt{x-3}\, +5=9$

Now that you have your equation written, all you have to do is solve for x:

$\sqrt{x-3}\, +5=9$           (subtract 5 from each side)
$\sqrt{x-3}\, =4$                 (square each side to remove the radical)
$x - 3= 16$
$x = 19$

Loop Back: You solved for your bottom line, so look down at the answer choices.

(A) 3
(B) 4
(C) 5
(D) 16
(E) 19

The correct answer is (E).

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

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

# Functions

## Algebra: Functions

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

Read each question carefully and identify the bottom line to avoid making careless mistakes.  Then assess your options 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.

If f(x) = x + ax, and $a =\frac{7}{2}$ what is $f(\frac{3}{2})$?

Bottom Line$f(\frac{3}{2})=?$

Assess your Options:  You could use your graphing calculator to solve this problem, but it would probably take you more time to type in the fractions than to just solve the problem.  You are given a value for each variable in the problem so all you need to do is plug them in.

Attack the Problem:  Start by plugging in the value of a to the function that you were given.

$f(x)=x+ax$
$f(x)=x+\frac{7}{2}x$

Simplify the problem by adding.  Remember that the first x is a whole 1x, but that you must have like terms before you can add fractions.

$f(x)=\frac{2}{2}x+\frac{7}{2}x$
$f(x)=\frac{9}{2}x$

Now solve your function by plugging in the value for x that you were given.

$f(\frac{3}{2})=(\frac{9}{2}\)(\frac{3}{2})$
$f(\frac{3}{2})=\frac{27}{4}$

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

(A) $\frac{1}{3}$
(B) $\frac{3}{2}$
(C) $\frac{7}{2}$
(D)$\frac{21}{4}$
(E) $\frac{27}{4}$

The correct answer is (E).

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

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

# Scatterplots

## Data Analysis: Scatterplots

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

Always read the question carefully so that you can glean as much information from it as possible.  Identify the bottom line – what is it asking?  Then assess your options and choose the most efficient method to attack the problem.  When you have a solution, loop back to make sure that it matches your bottom line.

The scatterplot above shows the number of items purchased at a grocery store by 28 customers and the total cost of each purchase. How many of these 28 customers bought more than 10 items and spent less than $20? Bottom Line: # of people. Notice that this number must reflect those that meet 2 requirements: buying more than 10 things and spending less than$20.

Assess your Options:  When you have a graph, use the graph!  You can draw directly on it to help you visualize what you need.

Attack the Problem:  The dots represent each person.  Start with the first restriction.  If people must buy more than 10 items, then only the dots to the right of the 10 on the horizontal axis will be counted; those on the line do not count because they are equal to 10 rather than more than 10.  Draw a vertical line on the 10.  Now look at the second restriction.  If the people must spend less than $20, then they must be under the$20 hash mark on the vertical axis.  Draw a line at \$20.  Your graph should look like this:

Count the number of dots in the lower right hand region that you created.  Your answer is 4.

Loop Back:  Each dot represents a customer, a person, so you reached your bottom line.  Look down at your answer choices.

(A) Four
(B) Five
(C) Six
(D) Seven
(E) Eight

The correct answer is (A).

On sat.collegeboard.org, 66% of the responses were correct.

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

# Rates

## Arithmetic: Rates

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

Use the same process for every math problem so that you are not intimidated by any question.  (1) Read the question carefully.  (2) Identify the bottom line – what is the question asking?  (3) Take the time to assess your options – which methods can you use to solve this problem most efficiently?  (4) Attack the problem and work though it logically.  (5) Loop back to make sure that your answer matches the bottom line – did you complete every step of the problem?

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?

Bottom line: Make a quick note that you are solving for hours: hrs = ?

Assess Your Options:  You could try to use logic for this problem by thinking that if a train goes more slowly, it must take longer to go the same distance as it did at a faster speed.  Unfortunately, logic will not eliminate all of your answer choices.  Use the distance formula to solve this problem.

Attack the problem:  The distance formula is distance is equal to rate(speed) times time:  D = R × T.  Start with the first train and multiply the rate (60 m/hr) by the time (1 hr) to get the distance:

60 × 1 = 60

The first train traveled 60 miles.  You know that both trains traveled the same distance, so plug in 60 as the distance for the second train. You also know that the rate is 30 and the time is unknown.  That should look like:

30 × T = 60
30T = 60
T = 2

Note:  If you are good at balancing equations, there is an even faster way to do this problem.  Look at the distance equation:  D = R × T.  If the distance for a problem stays the same, but you increase the speed (rate), then you must decrease the time by the reciprocal of the speed increase.  That keeps the equation balanced.  Ex:  If you double the speed, you must halve the time.  In this particular problem you halve the speed (from 60 to 30), so you must double the time.  2 × 1 hour = 2 hours.  This reciprocal rule will always work!

Loop Back:  You solved for the time of the second train, which is already in hours, so you are ready to look at your answer choices!

(A) 3
(B) 2
(C) 1
(D)
(E)

The correct answer is (B).

On sat.collegeboard.org, 78% of the responses were correct.

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

# Percents

## Arithmetic: Percents

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

Always read math questions carefully so that you can absorb all the information and avoid mistakes.  Identify the bottom line, what the question is asking you to find, and assess your options for reaching that bottom line.  Choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer matches the bottom line and you have finished all the steps in the problem.

If p percent of 75 is greater than 75, which of the following must be true?

Bottom Line: p = ?

Assess your Options:  It is often tempting to look down at the answer choices before you need them, but they could mislead you since most of them are wrong!.  You could take numbers that fit each answer choice and see if they give you a number greater than 75.  However, by applying what you know about percents, you can solve the problem much faster than you can by trying out 5 different numbers.

Attack the Problem:  There are a number of ways to think about percentages: as percents, decimals, numbers out of a hundred, parts of wholes….  The list continues.  Here is one of the fastest ways to think about the problem:

If you have one hundred percent of something, you have all of it.  So 100% of 75 is going to be 75.  If you want a result that is greater than 75, you are going to need more than 100% of 75.  Therefore, p must be bigger than 100.

Or, if you normally think about percents in terms of decimals, you know that 50% of something is .5.  In order to get a decimal from a percent, you had to move the decimal twice to the left.  So with 100%: 75 × 1.00 = 75.  Try writing an inequality to find the decimal that you would need in order to get a number bigger than 75: 75p > 75.  The p represents the unknown percent of 75 (remember, "of" means multiplication in math).  If you solve the inequality, you get p > 1.  Then you have to move the decimal back in order to get a percent: p > 100.  Your percent must be bigger than 100%.  This method takes much longer than the first one, but it proves that the first method is correct.  The testers realize that students are not used to working with percentages greater than 100, so it is a good idea to review how these work before the test!

Loop back:  You know what p must be greater than, so look down at your answer choices.

(A) p > 100
(B) p < 75
(C) p = 75
(D) p < 25
(E) p = 25

The correct answer is (A).

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

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

# Rates

## Arithmetic: Rates

Read the following SAT test question and then click on a button to select your answer.

Use the same method for every math problem on the SAT.  Read the problem carefully, identify the bottom line, and assess your options for solving the problem.  Choose the most efficient method to attack the problem.  Often there will be multiple steps to a single problem, so when you have an answer, be sure to loop back and verify that it matches the bottom line.

A machine can insert letters in envelopes at the rate of 120 per minute. Another machine can stamp the envelopes at the rate of 3 per second. How many such stamping machines are needed to keep up with 18 inserting machines of this kind?

Bottom Line:  # stamping machines = ?

Assess your Options:  You could try to work backwards from the answers, but there is no need.  It will be faster just to solve the problem.

Attack the Problem:  You have been given two different units of time: minutes and seconds.  There are 60 seconds in a minute.  Changing the minutes to seconds will be easiest, so the letter inserting machine works at a rate of 120 letters per 60 seconds.  120 divided by 60 is 2 letters in envelopes per second.  If there are 18 letter inserting machines, then together they will insert 36 letters in envelopes per second (2 × 18 = 36).

You don’t know how many stamping machines you need, so use x to represent that number.  Stamping machines have a rate of 3 envelopes per second, so each machine will finish 3 envelopes in a second.  You know that the stamping machines must keep pace with the 18 letter inserting machines that finish 36 envelopes per second, so the outcome must be 36.  Write 3x = 36.  When you solve for x,  x = 12.

Loop Back:  You used x to represent the number of stamping machines, your bottom line, so you are ready to look at the answer choices.

(A) 12
(B) 16
(C) 20
(D) 22
(E) 24

The correct answer is (A).

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

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

# Coordinate Geometry

## Link of the Day:

A new year symbolizes a new start for many.  Although the world is essentially the same as it was before the clock struck midnight, there is a new optimism about the future.  People want to focus on goals such as peace and prosperity.  Read this current event about an unexpected gesture from North Korea, and then ask yourself what you can expect from 2013.  What themes can you identify in this article that are likely to be part of an SAT essay question?

## Geometry: Coordinate Geometry

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

Always read the question carefully and identify the bottom line.  Assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it addresses the bottom line.

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

Bottom Line: k = ?

Assess Your Options:  You could start from the point (1, 1) and use the slope to find new points, hoping that by adding 3 to the y value and 2 to the x value you will reach a point that contains a 7 y value.  Unfortunately, it is very easy to make a mistake using this method, such as adding the y change to the x value or vice versa.  Instead, use the information that you are given, the slope, to write an equation.

Attack the problem:  Although you are given the slope, you also know how the slope was obtained.  Think about it:  The slope is rise over run or the change in y over the change in x
$slope=\frac{rise}{run}=\frac{\bigtriangleup y}{\bigtriangleup x}=\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}$
You know two different y values, and two different x values, so you can plug in all the information that you know for the slope.
$slope =\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}=\frac{7-1}{k-1}$
Now you need to set this formula for slope equal to the value for slope that you were given in the problem, isolate the variable k, and solve for it.
$\frac{7-1}{k-1}=\frac{3}{2}$
$\frac{6}{k-1}=\frac{3}{2}$
3(k – 1) = 6 × 2
3k – 3 = 12
3k = 15
k = 5

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

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

The correct answer is (B).

On sat.collegeboard.org, 51% of the responses were correct.

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

# Equations

## Algebra: Equations

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

You should read every math problem on the SAT carefully.  Identify your bottom line, assess your options for reaching it, and then select the most efficient method to attack the problem.  Once you have an answer, loop back to make sure that it matches your bottom line.

y = x² - 4x + c

In the quadratic equation above, c is a constant. The graph of the equation in the xy-plane contains the points (-2, 0) and (6,0). What is the value of c?

Bottom Line: c = ?

Assess your Options:  For this problem, you are given two points (x, y).  That means that you could plug in the x and y values for either point and solve for c (this is the method that most students will use):

 (-2, 0) (6, 0) 0 = (-2)² - 4(-2) + c 0 = (6)² - 4(6) + c 0 = 4 – (-8) + c 0 = 36 – 24 + c 0 = 12 + c 0 = 12 + c -12 = c -12 = c
Isolating the variable in either of these two equations will get you the correct value of c.  However, notice how many steps there are.  Can you just look at the two points and know the answer?  Yes!  Think about you find the roots of an equation.

Attack the Problem:  Your original equation is already set equal to zero.  You know this because both of the points have a y value of 0.  In order to factor a polynomial, you need two binomials.  Here you already know that x = -2 or 6.  That means that your two binomials are (x + 2) and (x – 6).  Now c is the last number that you would get in your polynomial if you multiplied (x + 2) by (x – 6).  What number is that? -12!

All you had to do was multiply the last two numbers (2 × -6) because every other combination would have an x.  If you don’t see how that works, multiply out (x + 2)(x – 6):

Use FOIL (First, Inner, Outer, Last)
x² + 2x – 6x – 12 (combine like terms)
x² - 4x – 12

When you compare this equation to the original equation, you will see that in place of the c you now have a -12.

Loop Back:  During a test, you would never work through a problem three times (time waster!), so this is where you would check to make sure that you solved for the correct variable.  Look down at your answer choices.

(A) -12
(B) -6
(C) 4
(D) 6
(E) 12

The correct answer is (A).

On sat.collegeboard.org, 51% of the responses were correct.

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

# Multiple Figures

## Geometry: Multiple Figures

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

Use the same steps for every math problem.  First, read the question carefully and identify the bottom line.  Next, assess your options and choose the most efficient method to attack the problem.  Finally, loop back to verify that your answer addresses the bottom line.

In the figure above, if PQRS is a quadrilateral and TUV is a triangle, what is the sum of the degree measures of the marked angles?

Bottom Line:  Sum of degrees of the marked angles = ? (Write Sd = ?)

Assess your Options:  You could try to find the individual angles, but you don’t have enough information to do this.  Instead, use the rules you have memorized about each shape.

Attack the Problem:  You know that TUV is a triangle.  All the angles of a triangle add up to 180 degrees.  You know that PQRS is a quadrilateral.  All the angles of a quadrilateral add up to 360 degrees.  In the image, you can see that all of these angles in each of these two shapes are marked, and you know that you are looking for a sum, so add them together.  180 + 360 = 540.

Loop back: Your answer is in degrees and you have found the total of all the marked angles.  Look down at your answer choices.

(A) 420
(B) 490
(C) 540
(D) 560
(E) 580

The correct answer is (C).

On sat.collegeboard.org, 77% of the responses were correct.

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

# 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!

# 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 DD 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!