# 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.

(A) AC < CD
(B) AC < CE
(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.

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

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!

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

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

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

# Equation of a Line

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) $f(x)=-\frac{1}{2}x+\frac{1}{2}$

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) $\frac{7}{8}$

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

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

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

Always read the problem carefully and determine the bottom line, the question that you must answer.  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 you completed all the necessary steps and solved for the bottom line.

If $\sqrt{x-a}=\sqrt{x+b}$ , which of the following must be true?

Bottom Line: Which of the following . . . ?

Assess your Options:  Many "Which of the following . . . " questions require you to look at the answer choices to solve the problem, but you should always check to see whether you can simplify the equation that you have been given.  Instead of jumping to the answer choices, work the equation into a form that is not as intimidating.

Attack the Problem:  The original equation has a square root on each side.  How do you get rid of these square root signs?  Square both sides of the equation, and the roots will cancel out.  You are left with:

xa = x + b

You just showed that when something is on both sides of the equation, you can cancel it out.  There is a positive x on both sides of the equation.  If you subtract it from one side, you must subtract it from the other, and the x is eliminated.  You are left with:

-a = b

This looks fairly simple, so glance down at your answer choices.  All of them are set equal to 0.  Set your equation equal to zero by adding an a to each side.

0 = b + a

Remember, it doesn’t matter what order you use when adding two variables.

Loop Back:  You put your answer in the same form as the answers on the test, so now all you have to do is match your answer to the correct one!

(A) a = 0
(B) b = 0
(C) a + b = 0
(D) a b = 0
(E) a² + b² = 0

On sat.collegeboard.org, 54% of the responses were correct.

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

# Circles

Anytime something new happens to something very old, the result is a rich current event that could be interpreted in many different ways.  The Catholic Church has chosen a new pope, and for the first time ever, the pope is from the Americas.  Look for broad themes in this article that would make it easy to relate this current event to an SAT essay topic.

## Geometry: Circles

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

Read the question carefully and identify the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that your answer matches the bottom line.
In the figure above, a shaded circle with diameter  is tangent to a large semicircle with diameter  at points C and D. Point is the center of the semicircle, and  is perpendicular to . If the area of the large semicircle is 24, what is the area of the shaded circle?

Bottom Line: A sm =? (What is the area of the small, shaded circle?)

Assess your Options:  There are two good ways to approach this problem.  Both ways require you to know the formula for the area of a circle. On collegeboard.org you will find a method that is especially efficient for students who are good at writing equations.  The method used here will focus on geometry skills and estimation in order to avoid the mistakes that often come with working more abstract formulas.

Attack the Problem:  You know the most about the large circle, so start there.  A semicircle is just half of a whole circle.  Therefore, to find the area of the whole circle, you would simply double the 24.

24 × 2 = 48

If you know the area of the large circle, you can use the area formula to find out more information.  The area of a circle is $A=\Pi(r)^{2}$  Plug in the area you just found to find the radius.

Note: working backwards using the area formula for a circle is difficult, because using pi will always result in icky decimals.  If you glance at your answer choices, all of them are whole numbers.  You can estimate pi as 3 instead of 3.14 in order to keep this problem as easy as possible.

48 = 3r²
16 =
4 = r

You now have the radius for the big circle.  Now look back up at the diagram.  The radius for the big circle is also the diameter for the little circle!  If the diameter of the little circle is 4, the radius will be half of that.  Once you know that the radius of the little circle is 2, you are ready to find the area!

$A=\Pi(r)^{2}$
A = 3 × 2²

A = 3 × 4
A = 12

Loop Back:  You found the area of the small circle, so you are ready to look at your answer choices.

(A) 8
(B) 10
(C) 12
(D) 14
(E) It cannot be determined from the information given.

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

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

# Sequence Problems

## Arithmetic: Sequence Problems

Always read the question carefully and identify the bottom line.  Then assess your options and use the most efficient method to attack the problem.  When you have an answer, loop back to make sure that you solved for the bottom line.

8, a, 14, b, 20, …
The first term of the sequence above is 8. Which of the following could be the formula for finding the nth term of this sequence for any positive integer n?

Bottom Line: You want a formula to describe this number sequence.

Assess your Options:  You could try to write a formula, but you will have a hard time doing that because you do not know the second and fourth terms in your pattern.  You also do not need to find numbers for the variables a and b in order to solve this problem.  Instead, use the answer choices to help you find an answer.

Attack the Problem: The first thing to do is realize that n is not a variable that you have to find algebraically; the nth term just describes the number of that term in the sequence, like the first, second, third, fourth, or fifth.  Therefore:
8,   a,  14,  b,   20, …
1,   2,   3,   4,     5

That means that when you plug in 1 to the formula, you should always get 8, when you plug in 3, you should always get 14, and when you plug in 5, you should always get 20.

(A) 2n + 6
(B) 3n + 5
(C) 5n + 3
(D) 6n + 2
(E) 6n + 5

You could start by plugging in 1 and finding out which of these equals 8, eliminate any that do not, and then try plugging in 3 and then 5 (this method is used on collegeboard.org).  However, just by looking at the numbers (a lot of 2s and 6s and a lot of 3s and 5s) you should be able to tell that a lot of these will equal 8.  To save time, start by plugging in the biggest term you know, the fifth, and see which answer choices equal 20.

(A) 2(5) + 6 = 16
(B) 3(5) + 5 = 20
(C) 5(5) + 3 = 28
(D) 6(5) + 2 = 32
(E) 6(5) + 5 = 35

Note: if you use logic, you do not even have to work out (C), (D), and (E) because the product of the first two numbers is larger than 20 before you even add to them.

Only one answer choice results in the correct 5th term of 20.  You don’t need to check any other numbers!

Loop back:  You found the only formula that will work for every number in the sequence, so select that answer.

(A) 2n + 6
(B) 3n + 5
(C) 5n + 3
(D) 6n + 2
(E) 6n + 5

On sat.collegeboard.org, 43% of the responses were correct.

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

# Coordinate Geometry

## Geometry: Coordinate Geometry

Approach every question the same way to minimize mistakes.  Start by reading the question carefully and identifying the 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 it matches your bottom line.

In the xy-plane, line l passes through the points (0, 0) and (2, 5). Line m is perpendicular to line l. What is the slope of line m?

Bottom line: slope m = ?

Assess your Options:  You could draw out a graph and solve this visually, but that is a waste of time if you know the formula to find the slope of a line.

Attack the Problem:  You are given the most information about line l, so start with that line.  You should have the formula for slope memorized:

$slope=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

It is easiest just to think about slope as the change in y-values over the change in x-values.  If you look up at the original points that you have been given, from zero the y-values go up to 5 and the x-values go up to 2.  You now have 5 over 2.

The slope of line l is $\frac{5}{2}$.

At this point, some students will think they are finished and select answer (D).  However, your bottom line was the slope of line m!  The problem tells you that line m is perpendicular to line l.  In order to find a perpendicular line, you must take the opposite reciprocal of the first line; in essence you must flip the sign (negative or positive) and the numbers (a fraction or whole number).

The slope of line m is $-\frac{2}{5}$.

(A) $-\frac{5}{2}$
(B) $-\frac{2}{5}$
(C) $\frac{2}{5}$
(D) $\frac{5}{2}$
(E) 5

On sat.collegeboard.org, 47% 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

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

On sat.collegeboard.org, 75% 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.

Always read the question carefully and identify the bottom line so that you do not waste time finding something unrelated to the question.  Assess your options for solving the problem and choose the most efficient method to attack the problem.  When you have an answer, take a second or two to loop back and make sure that your answer matches the bottom line.

If a, b, and c are numbers such that $\frac{a}{b}=3$ and $\frac{b}{c}=7$, then $\frac{a+b}{b+c}$ is equal to which of the following?

Bottom line:  $\frac{a+b}{b+c}$

Assess your Options:  There are two ways that you can solve this equation, and both will arrive at the correct answer.  You can solve it algebraically by substituting information into the equation, or you can pick your own numbers for the variables.  Choose the method that is easier and faster for you.

Attack the problem:  If you are going to solve a problem algebraically, always look for ways to simplify the problem that you are given.  In this case, you will want to get rid of unnecessary fractions.  Look at the first piece of information that you are given.  If a divided by b is 3, you can get rid of the fraction by multiplying each side of the equation by b.

Now you have a = 3b.

Look at the numerator (the top part of the fraction) of your bottom line.  You can now make sure that there is only one variable in this portion of the equation.   Substitute 3b for a.  Now you have 3b + b, which will simplify to 4b

Here are the steps you just completed:

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

Look at the denominator of your equation.  How can you simplify b + c?  You might be tempted to substitute 7c for b, but remember your goal is to get to a number without a variable.  If you have the same variable in the top and bottom, the two variables cancel. Therefore, you need to find what c is equal to in terms of b

When you are given the information that b divided by c is 7, then you know that c divided by b is 1 over 7.  You flip both equations.  Solve for c by multiplying both sides of the equation by b.

$\frac{b}{c}=7$ so  $\frac{c}{b}=\frac{1}{7}$ so $c =\frac{1}{7}b$

Plug this information into your bottom line equation and combine like terms.

$\frac{4b}{b+c}=\frac{4b}{b+\frac{1}{7}b}=\frac{4b}{\frac{8}{7}b}$

A fraction over a fraction is ugly, but remember that dividing by a fraction is the same thing as multiplying by the reciprocal of that fraction.  In other words:

$\frac{4b}{\frac{8}{7}b}=4b(\frac{7}{8b})=4(\frac{7}{8})=\frac{28}{8}=\frac{7}{2}$

Notice that the variable b moves to the bottom of the second fraction and cancels out.  You solved the equation!

Alternatively:  If you dislike algebra, use the strategy of picking numbers to solve this problem.  You want to get rid of ugly fractions, and the best way to do that is to put a number over 1.  You cannot just put b = 1 because b affects two different equations and you might end up with numbers that are difficult to use in your other equation.   However, c is on the bottom of a fraction in one equation.  Pick c = 1.  Plug 1 into the second piece of information with c and solve for b.

$\frac{b}{c}=7$ so $\frac{b}{1}=7$ so b = 7.

The variable b must equal 7. Now plug that into the first piece of information that you were given.  If b is 7, then a must equal 21.

$\frac{a}{b}=3$ so $\frac{a}{7}=3$ so a = 21.

Now that you have numbers for a, b, and c, plug those into your bottom line equation:

$\frac{a+b}{b+c}=\frac{21+7}{7+1}=\frac{28}{8}=\frac{7}{2}$

Bottom Line:  As soon as you have a value to represent your bottom line, look down at your answer choices.

(A) $\frac{7}{2}$
(B) $\frac{7}{8}$
(C) $\frac{3}{7}$
(D) $\frac{1}{7}$
(E) 21

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

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

# Coordinate Geometry

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

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

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

# 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!

# Sets

## Arithmetic: Sets

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

Approach each math question on the SAT the same way.  Read the question carefully to be sure you take into account all of the information as you solve it, and be sure to identify and note the bottom line.  Assess your options for solving the problem, and then choose the most efficient method to attack the problem.  Never forget to loop back and make sure that your final answer solves for the bottom line, the question that you were asked.

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?

Bottom Line: # of intersections = ?

Assess your Options:  When you have a question that asks about number properties, ignore your answer choices!  If you look down and see a 0, you could think to yourself that both 7 and 13 are prime, so they have nothing in common.  Are you looking for factors?  No!  You are looking for multiples.  Think through all of the information that you are given before looking at the answer choices.

Attack the Problem:  A set is just a collection of data.  You are given two different sets and asked to find the intersections, the data that the two have in common.  The only restriction on both sets is that all of the numbers must be positive.

Now think about what multiples are.  Multiples are the product of a number and an integer.  So Set S contains 7, 14, 21, 28… and continues in this manner into infinity.  Set T contains 13, 26, 39, 52… and continues in this manner into infinity.

If you keep listing numbers in each set, it will take you forever to find the answer to this problem.  Instead, think logically about where you know you must have multiples that match.  For example, if you multiply 7 times 13, you will find a number that belongs in both sets.  If you multiply 14 times 13, you will find another intersection.  Notice that you can keep doing this because you will never reach infinity.  The answer to this problem is that there are an infinite number of intersections between S and T.

Loop Back:  You found your bottom line, so look down and see which answer choice it matches.

(A) None
(B) One
(C) Seven
(D) Thirteen
(E) More than thirteen

On sat.collegeboard.org, 40% 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

On sat.collegeboard.org, 57% 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.

Always be sure to read the question carefully and make a note of the 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 it matches the bottom line.

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?

Bottom Line: slope of l = ?

Assess Your Options: You could select a number larger than 1, plug it in for the variable a, and then work the problem.  However, if you peek down at the answer choices, notice that some have a variable still in the problem.  It will take you longer to plug in a number than to work the problem using the variables.

Attack the Problem: Your bottom line is a slope, so use the formula for the slope of a line. The formula for slope of a line is:

$\frac{rise}{run}\: or\: \frac{\Delta y}{\Delta x}$

To find the change in y coordinates, subtract the first y-value from the second y-value.  Do the same with the x values:

$\frac{2a-0}{0-a}=\frac{2a}{-a}=-2$

The variable will cancel when you simplify the problem.  Your answer is -2.

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

(A) -2
(B)$-\frac{1}{2}$
(C) 2
(D) -2a
(E) 2a

On sat.collegeboard.org, 42% 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.

Always read the question carefully and identify the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that your answer matches the bottom line; the specific question the problem asked you to solve.

The c cars in a car service use a total of g gallons of gasoline per week. If each of the cars uses the same amount of gasoline, then, at this rate, which of the following represents the number of gallons used by 5 of the cars in 2 weeks?

Bottom line: gal in 2 wks = ?

Assess your Options:  You could try to work backwards from the answer choices by plugging in a number for each variable, but you want to avoid working from the answer choices when you do not have to.  Instead, write an equation using the information that you are given in the problem.

Attack the Problem:  Start with the most basic information that you are given and logically translate the words into a math problem.  You know that c stands for cars and g stands for gallons of gasoline.  If all of the cars use the same amount of gasoline, then the total number of gallons must be divided evenly among each of the cars:

$1\: week = \frac{g}{c}$

Now you know that there are 5 cars.  You might be tempted to put the 5 with the c, but think about it this way: that would mean that the same number of gallons was divided among more cars, so each car was using less gasoline, which is impossible!   If there are more cars, the total amount of gasoline must increase:

$1\: week = \frac{5g}{c}$

Now all you have to do is turn 1 week into 2 weeks by multiplying both sides of your equation by 2:

$2\: week = \frac{10g}{c}$

Loop Back: You found the gallons for 2 weeks, so look down at your answer choices.

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

Alternative method using Knowsys strategies:  If you struggle with writing equations, choose a number to represent the variable you are given in the problem.  You know you have 5 cars, but pick a number to represent the gallons that these cars use.  Any number that is not already in the problem will work; avoid  0 or 1 because multiple equations may work with these choices. Let’s say that g = 10.  In one week, those 5 cars will use 10 gallons.  How many gallons will they use in 2 weeks?  20 gallons!

Plug in the 10 for g and the 5 for c.  10 times 10 is 100, and then if you divide 100 by 5, you get 20.  That matches the answer that you found, so E must be correct.  None of the other answer choices will equal 20.  Strategies are tools to help you – remember that you get the same number of points for the correct answer no matter how you work the problem!

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

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

# Triangles

## Geometry: Triangles

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

Always read the question carefully so that you don’t misapply any information.  Identify the bottom line and assess your options for solving the problem.  Choose the most efficient method to attack the problem.  When you think you have the answer, loop back to make sure that it matches the bottom line.

If triangle ABC above is congruent to triangle DEF (not shown), which of the following must be the length of one side of triangle DEF?

Bottom Line: side of DEF = ?

Assess your Options: Many students go straight to the Pythagorean Theorem whenever they see a right triangle.  This formula, a² + b² = c², will not help you in this case because you do not know a or b.  Instead, use your knowledge of special triangles to solve this problem.

Attack the problem:  As soon as you see that this is a 30° – 60° – 90° triangle, you should think about the sides that relate to this special triangle.  Those sides, which you should have memorized, are x - x√3 – 2x.  Remember that the longest side has to be across from the biggest angle, the 90° angle.  That is your 2x.  This triangle has a 12 in that position.  Solve for x.

2x = 12
x = 6

Now you know that the side across from the 30° angle, AB, must be 6.  Label it.  Look at the side across from the 60° angle.  AC must be x√3.  You know that x = 6, so this side must be 6√3.  Label it.  You now know all the sides of triangle ABC:

x - x√3 – 2x
6 -6√3 – 12

Your bottom line is a side on triangle DEF, not on triangle ABC.  However, the problem tells you that ABC is congruent to DEF.  Congruent triangles have the same shape and size; they are basically the same triangle with different labels.  That means that the side lengths from triangle DEF will match the lengths you already found for ABC.

Loop back: You took into account all of the information that you were given and solved for your bottom line.  Look down at your answer choices.  One of the three side lengths you found will be there.

(A) 18
(B) 24
(C) 3√6
(D) 6√3
(E) It cannot be determined from the information given.

On sat.collegeboard.org, 38% of the responses were correct.

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

# Functions

Is there always another explanation or point of view?  Before you answer this released SAT essay prompt, check out this article that is part current event and part historical example with a literary connection thrown in just for fun.  Richard the III was a real king who is best known as a villain in Shakespeare’s work.  Read about what happened to him and why he is appearing in the news now.  There are far too many themes in this article to name them all, so come up with about a dozen ways you could connect this example to an essay prompt.  Then memorize a few of the most interesting facts so that you can use them to support your opinion on any of the themes that show up in your SAT essay prompt.

Note: The identity of King Richard the III has been confirmed.  Read here for details.

## Algebra: Functions

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

Always read the problem carefully, identify the bottom line, and assess your options for solving the problem before you attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

The function y = f(x), defined for -1.5 ≤ x ≤ 1.5, is graphed above. For how many different values of is f(x) = 0.2?

Bottom Line: # times f(x) = .2

Assess your Options:  Some students will skip this problem, thinking that it requires a lot of time to somehow write a formula for the function from the graph.  However, once you know what you are looking at, this is one of the easiest and fastest problems on the test!  All that you have to do is read the graph!

Attack the Problem:  You know that f(x) = .2 is the same thing as y = .2.  Anytime you see f(x), you can just substitute a y for f(x) if that clarifies the problem in your head.  If y is constant, you know that it will be a horizontal line at .2.  Draw that line on your graph.

Anywhere that the line crosses the function f(x), that function is equal to .2.  Count up the number of intersections between the line that you drew and the original function.  There are four.  That means that f(x) = .2 four times.