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SAT Math: Arithmetic

The sum of 9 consecutive integers is 3150.  What is the value of the least of these integers?

 

Knowsys Method

Read the problem carefully.  This is a grid in question.  Instead of bubbling in a letter, you will bubble in your answer.  You should always guess on grid in questions because, unlike multiple choice questions, grid in questions do not have a wrong answer penalty.

Identify the bottom line.   the value of the least integer = ? 

Assess your options.  You could start guessing and trying, but the more effective method will be to use a method.  In this type of problem, there is a method that will make it super simple.  Do you know it?  

Attack the problem.  

The issue of the least or the greatest or any integer in between does not matter until we actually have a starting point.  These questions will always involve an odd number of integers (here, 9).  So, all we have to do is divide the sum by the number of integers to get the average integer. Sound familiar?  It should!  This is just another use of the average formula.  

The Average Formula: 

average formula.gif

In this particular problem we know the sum = 3150 and the number is 9.  Let's just plug in to get:

20140211 SAT average question.gif

When we solve for x, we get 350.

That tells us that the middle number (= the median) = 350.  Since we want the LEAST integer, count back 4 more:  349, 348, 347, 346.  Found it!  

Note:  If the problem had asked for the GREATEST integer, we would have counted up 4:  351, 352, 353, 354.  

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

The correct answer is 346.  Grid it in.

Level = Medium  

Want some help reviewing the math concepts you need to master?  Try these Knowsys resources:

  1. Knowsys Pre-Algebra Flashcards
  2. Knowsys Algebra I Flashcards
  3. Knowsys SAT & ACT Math Practice book.  

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SAT Math: Arithmetic

Groups

At a certain restaurant, guests have a choice of either mustard or mayonnaise on their sandwiches.  At lunch yesterday, the restaurant served a total of 100 sandwiches.  55 of the sandwiches had mustard, 37 mayonnaise, and 18 had neither.  How many of the sandwiches had both mustard and mayonnaise?

A. 0
B. 2
C. 8
D. 10
E. 12

Knowsys Method

Read the problem carefully.  Be sure to keep the numbers in this problem straight so that you do not make any careless mistakes in your work. 

Identify the bottom line.  # of sand. with must. and mayo = ?

Assess your options.  You could solve this problem by creating a Venn diagram, but there is a faster method.  If you memorize "the group formula," you can solve these problems quickly and easily without drawing out a diagram. 

The Group Formula

2 groups:

Total = Group 1 + Group 2 + Neither - Both

3 groups:

Total = Group 1 + Group 2 + Group 3 + Neither - Both - 2(All)

What do all these terms mean?

Total = Total number of items.  In this problem, it is the total number of sandwiches.

Group 1, 2, 3 = The number of items in each group.  In this problem, the two groups are the sandwiches with mustard and the sandwiches with mayo.

Neither = The number of items that do not fall into one of the 2 (or 3) groups.  In this problem, it is the sandwiches that have neither mustard nor mayo on them.

Both = The number of items that fall into 2 groups.  In this problem, it is the sandwiches that have both mustard and mayo.

All = (Used for 3 group problems ONLY!) This is the number of items that fall into all 3 groups (as opposed to the number of items that fall into only 2 out of the three groups).  There are only 2 groups in the problem at hand, so this term does not apply. 

Attack the problem.  Sub the numbers from the problem into the group formula to figure out how many sandwiches have both mustard and mayo.

Total = Group 1 + Group 2 + Neither - Both

100 = 55 + 37 + 18 - Both

100 = 110 - Both

10 = Both

So, 10 sandwiches have both mustard and mayo.

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

The correct answer is D.

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

Percents

A local electronics store marks up its items 30% from their wholesale cost.  At its annual Spring sale, the store discounts the prices of all items by 20%.  The sale price is what percent of the wholesale cost?

A. 90%
B. 100%
C. 104%
D. 110%
E. 117%

Knowsys Method

Read the problem carefully.  Make careful note that the price is originally INCREASED by 30%, then it is DECREASED by 20%.  Also, notice that the problem asks, "the sales price is what percent OF the wholesale cost," not what percent OFF the wholesale cost.  That is an important distinction.  For instance, if a shirt is discounted by 30%, then the price is 30% OFF, but 70% OF the original price. 

Identify the bottom line.  sale price = what % of wholesale cost?

Assess your options.  You can solve the problem by working solely with percents, or you can pick numbers.  We recommend the latter method because it is more concrete.

Attack the problem.  Whenever you are picking a number to use in a percent problem, it is easiest to use 100.  So, let's say that the wholesale cost of an item is $100.  That cost is marked up 30% to create the regular price at which the item is sold to customers.  There are two ways to find that price. 

100 + (.3)(100) = 130

100 x 1.3 = 130

The regular price of the item (before the sale) is $130.  At the sale, this price of $130 is marked down by 20%.  There are two ways to find the sale price.

130 - (.2) (130) = 104

130 x .8 = 104

The sale price is $104, and the wholesale cost was $100.  104 is 104% of 100, so the answer is C.

Note: Look at answer choice D.  NEVER fall for this trap: 100 + 30 - 20 = 110.  Remember that the price goes up by 30%, then decreases by 20% off the NEW price.  

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

The correct answer is 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: 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.  

Screen Shot 2013-12-23 at 10.21.01 AM.jpg

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.

The correct answer is D.

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?

Screen Shot 2013-12-10 at 4.52.26 PM.jpg

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:

Screen Shot 2013-12-10 at 4.22.48 PM.jpg

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

The correct answer is C
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: Arithemetic

The Number Line

Screen Shot 2013-11-27 at 9.19.57 AM.jpg

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.

Screen Shot 2013-11-27 at 9.19.36 AM.jpg

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

Screen Shot 2013-11-27 at 9.19.45 AM.jpg

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

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

The correct answer is B.
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: 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.

The correct answer is (B).

 

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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