# SAT Math: Data Analysis

## Average (arithmetic mean)

Each of four people were given a blank piece of paper on which they wrote a positive integer.  If the average (arithmetic mean) of these integers is 21, then what is the greatest possible integer that could be written on one of the pieces of paper?

## Knowsys Method

Note: In the math section of the SAT, you will encounter questions that do not have answer choices.  Instead of bubbling in a letter, you will bubble in your answer.  These questions are called grid in questions, and you should always guess an answer for them because there is no penalty for getting the question wrong!

Read the problem carefully.  Make sure you understand the situation described in the question and take note of important details.  The integers written on the pieces of paper are positive, and they are NOT distinct (different), so there could be repeats of the same number.

Identify the bottom line.  Greatest possible integer (out of these 4) = ?

Assess your options.  There is only one possible method for solving this problem, and it is demonstrated below.

Attack the problem.  Whenever a problem asks about average (arithmetic mean), the first thing you should do is write out the average formula.

Now, plug in the information you know from the problem.  You were given the average and the number of people.

Multiply both sides by 4 to find the sum, which is 84.  Now, think about it.  The four integers written on the pieces of paper all add up to 84.  What is the greatest possible number that could be included among those four?  Figure out how to make the other three integers as small as possible to leave the greatest possible number left over.  The integers must be positive, so each one must be at least 1.  Three of them can be 1 because they do not have to be distinct.  So, subtract 3 from 84 and get 81.

Loop back.  Check to verify that you have solved for the bottom line.

This is a medium level problem.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

# SAT Math: Geometry

## Coordinate Geometry

In the xy-coordinate plane, line m is parallel to the x axis, and it passes through the point (4, -7).  Which of the following is an equation for line m?

A. x = -4
B. y = 4
C. y = -7
D. y + 7 = x - 4
E.  y – 7 = x + 4

## Knowsys Method

Read the problem carefully.  Take careful note that the line in the question is parallel to the x axis.

Identify the bottom line.  equation for line m = ?

Assess your options.  We must solve this problem by plugging the line into the point-slope formula.  The point slope formula is:

Attack the problem.  We need to use the point-slope formula, but to do so, we must determine the slope (m) and the y-intercept (b).  It makes it a little easier to explain this problem if we draw out the line on the coordinate grid, but you do not have to do this step.

As the illustration shows, line m is parallel to the x axis, which makes it is a completely horizontal line.  A horizontal line has a slope of 0.  Think about it: if the y value never changes, then the change in y is zero, and no matter what the change in x, 0 divided by anything is 0.  Since you know that the y value never changes, you also know the y-intercept will be (0,-7).  Plug these values into the point-slope formula.

y = mx + b
y = (0)x -7
y = -7

Loop back.  Verify that you solved for the bottom line.

This is a medium level problem.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Arithmetic

## Sequence

3, -6, 12, …

The first term in the sequence shown above is 3, and every term following is equal to -2 times the preceding term.  Out of the first 30 terms in this sequence, how many are less than 50?

A.  5
B.  6
C.  15
D.  18
E.  27

## Knowsys Method

Read the problem carefully.  Make note of how the sequence works and note that you are focusing on the first 30 terms in the sequence only.

Identify the bottom line.  Out of the first 30 terms, how many are less than 50?

Assess your options.  You could follow the sequence, list out the first 30 terms, and count how many terms are less than 50, but you do not need to do that much work.  Create a chart to help you notice the pattern, and use your logical thinking skills to help you figure out the answer.

Attack the problem.  First, use your logical thinking skills.  Every other number in this sequence is going to be negative, and therefore less than 50.  So out of the first 30 terms, at least 15 of them will be less than 50.

Next, create a chart of positions and values to help you figure out what positive values are also less than 50.

As you can see, 3 positive values in the sequence are less than 50.  Starting with the 7th term in the sequence, all positive values will be greater than 50.  So add 3 to 15, and you get 18 terms out of the first 30 that are less than 50.

Loop back.  Did we solve for the bottom line?  Yes.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Data Analysis

## Knowsys Method

Read the question carefully.  Take notice of important information.  You need to determine the probability of rolling two numbers that add up to 8.

Identify the bottom line.  probability that sum will be 8 = ?

Assess your options.  You need an organized way to keep track of all the possible rolls that could add up to 8.  A simple method is to create a table.

Attack the problem.  Imagine that one of these dice is blue and one is red.  Set up a table with two columns, one for the blue die, and one for the red die. These dice have values 1-6 on their faces, so what sets of numbers between 1 and 6 add up to 8?  There’s 2 and 6, 3 and 5, and 4 and 4.

But you cannot stop there.  You could also have a blue 6 and a red 2 or a blue 5 and a red 3.  Fill in those values now.

To find the probability of rolling one of these combinations, you have to understand probability.

You need to find the probability for each of the combinations you have listed on your chart.  You are trying to figure out the probability of rolling two numbers at the same time, so you need to use probability rule 1.

Now, you need to figure out the probability of rolling any of these 5 combinations, so use probability rule 2 and add the probabilities together.

Loop back.  Verify that you solved for the bottom line and select the correct letter.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Geometry

## Polygons

If all angles in the polygon above are congruent, then a =
A.  20
B.  35
C.  60
D.  72
E.  120

## Knowsys Method.

Read the problem carefully.  The problem states that all angles in the polygon are congruent, which means that you can apply rules for regular polygons.

Identify the bottom line.  a = ?

Assess your options.  There are two ways to find the total number of degrees in a regular polygon.  You can:

1) use the polygon rule:
180 (n – 2 )
n is the number of vertices

2) divide the polygon into triangles and multiply the number of triangles by 180 degrees

We will demonstrate both methods below.

Attack the problem.  Either use the polygon rule or divide the polygon into triangles to find the total number of degrees in the polygon.

Now, divide the total number of degrees in this polygon by the number of vertices to find the measure of each individual angle.

The interior angle and a are supplementary, so subtract 120 from 180 to get the value of a.

180 – 120 = 60

Loop back.  Did you find the value of a?  Yes.  a = 60, which is answer choice C.

This is a medium level problem.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Algebra

## Simultaneous Equations

Emmanuel owns a printing business.  He ordered boxes of copy paper for \$10 each and boxes of photo paper for \$12 each.  If Emmanuel ordered 10 boxes of paper and spent \$112 (before tax), how many boxes of photo paper did he order?

A.  2
B.  3
C.  4
D.  6
E.  7

## Knowsys Method

Read the problem carefully.  Be careful to identify the correct bottom line- which type of paper is the problem asking about?

Identify the bottom line.  boxes of photo paper = ?

Assess your options.  At Knowsys we call these problems “How Many? How Much?” problems.  To solve these problems, you can test out each of the answer choices, or you can create two simultaneous equations.  Unless the first answer choice you test out is correct, the latter strategy will save you time.

Attack the problem.  Create two equations.   One should tell you “how many,” as in “how many boxes of paper are there total?”  The other equation should tell you “how much,” as in “how much do the boxes of paper cost?”  We will use c to represent copy paper and p to represent photo paper.

The 10 and 12 in the second equation are the prices of each box.  The next step is to eliminate the variable that you are not focusing on, in this case, c.  To get rid of 10c in the second equation, you need to have -10c in the first equation, so multiply the entire first equation by -10 and then add the equations together to solve for p.

Loop back.  Did you solve for p (photo paper)? Yes.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Arithmetic

## Digits

X and Y are distinct integers less than 6.  Given the correctly worked addition problem below, which of the following could be equal to XY?

A. 26
B. 33
C. 35
D. 42
E. 44

## Knowsys Method

Read the problem carefully.  A very important word to note in this problem is “distinct.”  If x and y are distinct, they cannot be the same number.  That eliminates answers B and E already.  Also, notice that X and Y must be less than 6.  That eliminates A.

Identify the bottom line.  XY = ?

Assess your options.  You can either plug in the answers or think about the problem logically.  The latter method is quicker, so we will demonstrate that below.

Attack the problem.  If you look at the addition problem, you will see that X and Y must add up to 8.  The numbers in choice A add up to 8, but we already eliminated choice A in step 1.  That leaves us with choice C.  3 and 5 add up to 8, and if you sub in 3 as X and 5 as Y, you get this:

Loop back.  Did you solve for the bottom line?  Yes.

This is an easy level problem

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Geometry

## Triangles

In the figure above, what is the sum, in terms of a, of angles b, c, d, and e?

A.  a
B.  2a
C.  180 – a
D.  180 – 2a
E.   360 – a

## Knowsys Method

Read the problem carefully.  Notice that the image is drawn to scale because it does not indicate otherwise.

Identify the bottom line.  b + c + d + e = ?

Assess your options.  If you find yourself stumped on a triangle problem on the SAT, run through the facts and rules you know about triangles and determine whether any of this knowledge applies to the problem at hand.  This problem is categorized as a “hard” level problem, but knowing one simple fact about triangles makes this problem easy to solve.

FACT: The exterior angle d equals the sum of the opposite interior angles a and b.

So, d = a + b.

Attack the problem.
Angle a is an exterior angle to both triangles in the image.  That means:

a = b + c

AND

a = d + e

Therefore:

b + c + d + e = 2a

Loop back.  You found the solution to the bottom line.

This is a hard level problem.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Algebra

A.  7
B.  13
C.  49
D.  91
E.  364

## Knowsys Method

Read the problem carefully.  Note that an integer is a whole number, not a fraction.

Identify the bottom line.  Least potential value of x = ?

Assess your options.  You could plug in each of the answer choices and find the smallest choice that results in an integer, but there is a much quicker way to solve this type of problem.

Attack the problem.

Loop back.  Did you solve for the bottom line? Yes.

This is a medium-level problem.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Arithemetic

## The Number Line

The letters a, b, c, and d represent numbers on the number line above.  Which of the following expressions has the greatest value?

A. a + b
B. c + d
C. a + d
D. c - b
E. a - b

## Knowsys Method

Read the question carefully.  There are two important things to note about this question.  First, you know that the image is drawn to scale because you do not see any indication otherwise.  Second, the question asks you for the expression with the GREATEST value.

Identify the bottom line.  Which expression has the greatest value?

Assess your options.  You could use the number line to solve this problem, or you could determine the value of each letter and solve the expressions mathematically.  The latter option is more concrete, so we will pursue that option.

Attack the problem.  You can easily determine the number value of each letter if you note that the marks on the number line represent fourths.  Write down the value of each letter.

Now simply plug these numbers into each of the expressions to determine which one has the greatest value.

Choose the expression with the greatest value, which is answer choice B.

Loop Back.  Did you find the expression with the greatest value?  Yes.

This is a hard level question.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Data Analysis

## Descriptive Statistics

If the sum of a list of temperatures is divided by the average (arithmetic mean) of the temperatures, the quotient is z.  What does z represent?

A. The average of the temperatures
B. The sum of the temperatures
C. Half the average of the temperatures
D. Two times the sum of the temperatures
E. The number of temperatures

## Knowsys Method

Read the problem carefully.  Be sure that you consider the meaning of all the terms in this problem.  For instance, it is important to remind yourself that a quotient is the result when one number is divided by another.

Identify the bottom line.  What does z represent?

Assess your options.  There are two ways to solve this problem.  Your first option is to “think” though the problem in order to figure out what the variable z refers to.  Your second option is to pick numbers, which will make the problem much more concrete if you have trouble thinking about mathematical concepts abstractly.  The latter option is the easier of the two, so that is the one we will illustrate here.

Attack the problem.  The first thing you should do any time you see an average problem is to write out the average formula:

Next, pick two numbers: a larger one to be the sum of all the temperatures, and a smaller one to be the number of temperatures. Pick easy numbers, and make sure that the smaller number is a factor of the larger number so that it will divide in evenly.  Let’s use 12 as the sum of the temperatures and 3 as the number of temperatures.  If you plug these two numbers into the average formula, the average of the temperatures will come out to be 4.

The problem asks you to find the quotient (z) when the sum of the temperatures is divided by the average of the temperatures.  Plug in the numbers you have chosen to find the result.

The result is 3, which is the number of temperatures.  Pick the answer choice that matches with this solution.

This is a medium-level problem.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Algebra

## Exponents

Read the question carefully.  This question requires you to know exponent rules.  It is important for you to note that a and b are positive integers and that you are asked for a in terms of b.

Identify the bottom line.  a = ?

Assess your options.  Unfortunately, this problem is going to be difficult for you to solve if you have not memorized exponent rules.  If the rules you need to solve this problem do not immediately pop into your head, go back and review the exponent rules.

Attack the problem.  There are two rules that apply to this problem:

The first thing you have to do is get all the bases to look the same.  Look back at the original equation:

The number 9 is three squared, so you can rewrite the equation like this:

Now apply the first exponent rule to the left side of the equation:

Now you can apply the second rule.  If the base is 3 on both sides, then the exponents must be the same number.  Set them equal to one another and solve for the variable.

Loop back.  You found a in terms of b, so now you just need to select the correct answer choice, which is (B).

This is a medium level question.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Geometry

## Knowsys Method

Read the question carefully.  Make note that the figure is not drawn to scale, so the sizes of the angles in the picture do not reflect their true measurements.

Identify the bottom line.  a = ?

Assess your options.  Visualize the point in the center as the center of a circle.  The number of degrees around the center point is 360, just like the number of degrees around the center of a circle.  That means that all of these angle measurements should add together to equal 360 degrees.  If you set that up as an equation, then you can solve for a.

Attack the problem.  Set up your equation and find the value of a.
2a + 4a + 5a + 7a = 360
18a = 360
÷18    ÷18
a = 20

Loop back. Did you find the value of a?  Yes.  All that is left is to select the letter of the correct answer.

This is a medium level question.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Arithmetic

## Consecutive Integers

If the sum of 5 consecutive positive integers is 145, what is the value of the largest integer?

Note: In the math section of the SAT, you will encounter questions that do not have answer choices.  Instead of bubbling in a letter, you will bubble in your answer.  These questions are called grid in questions, and you should always guess an answer for them because there is no penalty for getting the question wrong!

## Knowsys Method

Read the question carefully.  First, let's identify key terms.  An integer is a whole number (positive, negative, or zero).  Consecutive integers follow in a sequence and have a difference equal to 1 between each number.  An example of five consecutive integers is 3, 4, 5, 6, 7.  Be sure to note that you are looking for the largest number in this series of consecutive integers.

Identify the bottom line.  The largest number in a series of consecutive integers that all add up to 145.

Assess your options.  There are two methods that you could use to solve this problem.  The first method is to use division to find the middle number in the series.  The second method is to use an equation.  See both methods explained below.

Attack the problem.

Method 1: Division

Any time you see a problem set up like this one, you can divide the sum by the number of integers to get the value of the middle integer.  Divide 145 by 5 to get 29.  Then add 2 to 29 to get 31, the largest integer.

Method 2: Set up an equation

You know that there is a difference of 1 between each integer, so if the largest integer in the series is n, then the next largest integer would be n - 1, and the third largest would be n - 2, and so on.

Set up an equation in which all five integers add up to 145.  Your equation should look like this:

n + (n - 1) + (n - 2) + (n - 3) + (n - 4) = 145

Combine like terms and solve for n.

5n - 10 = 145
+10    +10
5n = 155
÷5     ÷5
n =  31

Loop back.  Have you found the largest number in the series of consecutive integers?  Yes.  Now that you are sure that you have solved for the bottom line, bubble in your answer choice.

The answer to this question is: 31

This is a medium level question.

Want some help reviewing the math concepts you need to master?  Try out the Knowsys Pre-Algebra Flashcards, the Knowsys Algebra I Flashcards, and the Knowsys SAT & ACT Math Practice book.

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# SAT Math: Arithmetic

## Inclusive Numbers

Tammy will be on vacation from July 3rd through July 18th, inclusive.  How many days will Tammy be on vacation?

A.       15

B.       16

C.       17

D.       20

E.        21

## Knowsys Method

Read the question carefully.  If you miss the word “inclusive,” you will get the wrong answer.

Identify the bottom line.  You need to know how many days.  Make a note in your scratch work that this is the question that you must answer.  D = ?

Assess your options.  You could try to count out the days, but you might make a mistake and there is an easier method.  For inclusive numbers you must include the first day in your count, not simply start counting at the first day.  The fastest way to get the answer is to (1) Subtract the smaller number from the larger number and (2) add 1 to the difference.

Attack the problem.  Once you know what to do, don’t hesitate!  Start plugging in the numbers from your problem.

18 – 3 = 15      (Many students will stop here.  However, you need to include the first day in your count)

15 + 1 = 16

Loop Back.  You solved for the number of days, and you made sure that you have the inclusive number.  You are ready to look down at your answer choices.