# 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!

# 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.

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:

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?

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

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

On sat.collegeboard.org, 60% 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!

# 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!

# 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!

# 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

On sat.collegeboard.org, 71% 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

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

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

# Group Problems

## Arithmetic: Group Problems

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

Work all math problems the same way so that you can approach even the most difficult problems with confidence.  Start by reading the question carefully.  Many problems have several steps, so you must identify the bottom line: what is the question asking?  Assess your options and choose the most efficient method to attack the problem.  Finally, loop back to make sure that your answer matches the bottom line.

In a community of 416 people, each person owns a dog or a cat or both. If there are 316 dog owners and 280 cat owners, how many of the dog owners own no cat?

Bottom Line: just dogs = ?
When you get to the step where you look at the answers, notice that (E) comes from not reading carefully.  Yes, there are 316 total dog owners, but some of them also own cats.  You must find how many own only dogs.

Assess your Options:  You could try to work backwards using the answer choices, but trying to think about the steps of a problem backwards often leads to mistakes.  You could also realize that this is a problem involving two overlapping groups and draw a Venn Diagram.  Forget those methods because the fastest method is to use the Group Formula.  Take a moment now to memorize this formula if you have not already done so: Total = Group 1 + Group 2 + Neither – Both.

Attack the Problem:  Plug all the information that you know into the formula.  How many total people are there? 416.  Then there are your two groups: Those who own dogs and those who own cats.  Plug in the numbers 316 and 280 to represent these groups.   Now, the problem tells you that “each person owns a dog or a cat or both,” so how many people own neither animal?  Zero.  The only thing that you are not given in the problem is how many people own both a dog and a cat.  Your formula should now look like this:

Total = Group 1 + Group 2 + Neither – Both
416 = 316 + 280 + 0 – B

That B represents the unknown Both, but you can now solve for it because it is the only variable left in your equation.  Start by simplifying the problem.

416 = 316 + 280 + 0 – B
416 = 596 – B   (add B to each side to make it positive)
416 + B = 596   (subtract 416 from each side)
B = 180

You have finished one step, but you have not yet reached your bottom line!  Do not look at the answer choices yet or you will be tempted to pick a wrong answer!

You just solved for the number of people who own both a cat and a dog.  How do you find the number of people who own just a dog?  Take the number who own both and subtract it from the total number of dog owners.  Remember that the total number of dog owners was given in the problem as 316.

316 – 180 = 136

(A) 36
(B) 100
(C) 136
(D) 180
(E) 316

On sat.collegeboard.org, 46% 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.

Read each question carefully.  Then identify the bottom line and assess your options for finding it.  Choose the most efficient method to attack the problem.  Before selecting your answer, loop back to make sure that you solved for the bottom line.

If  then N =

Bottom Line:  N = ?

Assess your options: Normally, you would begin working this problem by multiplying the two fractions on the right and then multiplying them by the reciprocal of the fraction on the left in order to find N.  Before you jump into the problem, think about the properties of multiplication and you will see that there is a much faster way to solve the problem.

Attack the problem:  The commutative property of multiplication tells you that order is not important when you are multiplying; 3 × 5 = 5 × 3.  If you rearrange your equation, you will see that  .   When you see the same thing, such as a fraction with 5 over 14, on both sides of the equation, you know that you can ignore that information.  No matter what number or variable you have, if it is the same on both sides of the equation, you will eliminate one side when you eliminate the other.  You can check this fact by multiplying both sides by the reciprocal of .  If you multiply both sides by , the  will cancel on each side and you are left with N = .

Loop back:  You solved for your bottom line, N, so you should look down at your answer choices.

(A)
(B)

(C) 5

(D) 7`

(E) 14

On sat.collegeboard.org, 82% of the responses were correct.

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

# Number Line

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

For all math problems, read the problem carefully.  Identify your bottom line, and assess your options for reaching that bottom line.  Select the most efficient method to work the problem, and attack the problem.  Your last step is to loop back and make sure that the answer addresses the bottom line. On the line above, if AB < BC < CD < DE, which of the following must be true?

(A) AC < CD
(B) AC < CE
(E) BD < DE

You must decide which of the answer choices is true.  This is a “which of the following” question, and it would be really difficult to predict an answer choice, so start with answer choice (E).

(E) BD < DE

Normally you cannot depend on a drawing that has the words "Note: Figure not drawn to scale" underneath it, but this particular line follows the rule that each line segment is longer than the last.  Once you have ascertained that the image matches the information that you have been given, you can use the image to draw conclusions.  You can look at the line provided and see that this does not have to be true:  BD actually looks longer than DE.  You can also think of BD as BC + CD.  You don’t have any information to compare BC + CD and find out whether it is less than DE.  One way of proving this is to imagine that AB is 1, BC is 2, CD is 3 and DE is 4.  That fits AB < BC < CD < DE.  However, 2 + 3 is not less than 4.  Eliminate (E).

Look back at the line.  You can clearly see that AD is longer than DE.  Eliminate (D).

This one is not obvious from a glance at the line.  Think of AD as AB + BC + CD and CE as CD + DE.  Both equations share CD, but do we know that AB + BC is smaller than DE?  No.  Think about what would happen if AB had a high value.  Suppose that AB = 10, BC = 11, CD = 12 and DE = 13.  In that case AD would be 33 while CE would only be 25, and this answer choice would be false.  Eliminate (C).

(B) AC < CE

Think of AC as AB + BC and CE as CD + DE.  Compare the two.  You know that AB must be smaller than CE and BC must be smaller than DE.  What you really have is: small number + small number < big number + big number.  Is that always true?  Yes.  You do not have to check the last answer choice.

On sat.collegeboard.org, 70% of the responses were correct.

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

# Number Properties

Arithmetic: Number Properties

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

Every time you work a math problem, read the problem carefully.  Identify the bottom line and think about the most efficient method to solve for the bottom line.  Choose a method to solve the problem and attack the problem without hesitation.  When you think you have the answer, loop back to make sure that the answer addresses the bottom line because questions often require multiple steps to get to the answer.

When the positive integer n is divided by 5, the remainder is 0. What is the remainder when 3n is divided by 5?

Make a note that your bottom line is the remainder of 3n.  Then think carefully about the first portion of information that you are given.  Some number, n, is divided by 5 and there is no remainder.  That means that n must be a multiple of 5.  If you do not immediately see this, think about a concrete number that will not result in a remainder:

or    or
All of these values for n result in a whole number with no remainder.

If n is a multiple of 5, what will 3n be?  It will still be a multiple of 5!  It will still result in a remainder of 0.  If you cannot see this, look back at the examples above using 5 and 15.  If 5 is your n value, 3 times 5 is 15, and when you divide 15 by 5 the answer is 3 with a remainder of 0.  Now that you have found your bottom line, look down at your answer choices.

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

On sat.collegeboard.org, 70% of the responses were correct.

For more help with math, visit

# Logic

Today’s question is a numbers game, and numbers are very important to species on the edge of extinction.  Some politicians support measures to save endangered species, but there is one Russian politician who is going above and beyond to show his devotion to restoring healthy populations of Siberian white cranes.  Read about Vladimir Putin here and think about the impact that stunts such as this can have on politics.  What would you think if the President of the United States did this?  Is this a current event that you could remember for your essay?

## 9/6 Logic

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

It is just as essential to read math questions carefully as it is to read reading questions carefully.  If you miss any information, you could solve for a question that is not being asked.  To avoid errors, make a note of the bottom line.  Then assess your options and choose the most efficient method of working the problem.  Attack the problem, clearly writing any scratch work, and loop back to make sure that your answer addresses the bottom line.

The sum of the positive odd integers less than 50 is subtracted from the sum of the positive even integers less than or equal to 50. What is the resulting difference?

Your bottom line is the difference between two sums.  This question is very simple mathematically because it only asks you to add and subtract integers, but it would take a long time to add and subtract all of the integers involved.  The people who created this test know that you do not have a lot of time, so there must be an easier way to solve this problem than writing every single number out.  Think about the question logically.

Start with the two sums.  One is the sum of all the even integers less than or equal to 50.  Write out a few of these numbers so that you can check for patterns.  Remember that zero is an integer, but not a positive number!  The problem calls for positive even and odd integers.  Did you read carefully?

2 + 4 + 6 + 8… 50

The sum of the positive odd integers is subtracted from the first sum, so write a few of these numbers underneath the first set.

2 + 4 + 6 + 8… 50
- ( 1 + 3 + 5 + 7… 49)

Notice that if you subtract each number individually, you will always get 1. Two minus one is one.  Four minus three is one.  You get the idea.  This happens all the way up to fifty minus forty-nine is one.

Your problem really looks like this:

2 + 4 + 6 + 8… 50
- ( 1 + 3 + 5 + 7… 49)
1 + 1 + 1 + 1…+1

Now you just need to figure out how many ones to add together.  Think about it this way: you need an even integer and odd integer for each additional one.  How many positive even integers are less than or equal to 50?  Well, half of the integers are even, so the answer is 25.  If you add the number one twenty-five times, what will you get?  Look down at your answer choices.

(A) 0
(B) 25
(C) 50
(D) 100
(E)200