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SAT Math: Algebra

If 7 less than 4 times a number is 2 more than 3 times that number, what is the number?

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 number = ? 

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, it is most straightforward to translate the statement from English to math and then solve. 

Attack the problem.  

The statement translates to:  4x – 7 = 3x + 2

The bottom line is:  what is x?

When we solve for x we get x = 9. 

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

The correct answer is 9.  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.  

Subscribe to Knowsys SAT & ACT Blog by Email

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.  

Subscribe to Knowsys SAT & ACT Blog by Email

 

SAT Math: Data Analysis

Averages

The average (arithmetic mean) of the weights of 26 stones is p pounds.  In terms of p, what is the total weight of the stones, in pounds?  

A. 26 + p
B. p - 26 
C. p / 26
D. 26 / p
E. 26p

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.   total weight  of the 26 stones = ?

Assess your options.  You could pick a number for p and then use it to find the total weight or you could leave p as a variable.  Either way, you need to use the "average formula," which you should definitely memorize.  Every time you see the word "average" on the exam, you should immediately think:  "average = the sum divided by the number."  Let's use that formula to solve this problem quickly and easily. 

The Average Formula

average formula.gif

Attack the problem.  

We are looking for the "total weight," which is the same thing as the SUM.  So, let's rearrange to get:

average * number = sum 

In this particular problem we know the average = p and the number = 26.  Let's just plug in to get:  

p(26) = sum

  26p = sum 

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

The correct answer is E.

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.  

Subscribe to Knowsys SAT & ACT Blog by Email

SAT Math: Algebra

Functions

Let the function be defined by f(x) = 5x - 10.  If 2f(m) = 20, then what is the value of f(3m)? 

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.  f(3m) = ?

Assess your options.  This function problem is very straightforward.  All you have to do is plug in the information given to you and solve. 

Attack the problem. 

2f(m) = 20, so f(m) = 10

If f(x) = 5x - 10, then f(m) = 5m - 10.  Set that equal to the value you already know for f(m) and solve for m.

10 = 5m - 10
20 = 5m
4 = m

Now plug in the value of m to find f(3m).

f(3m) = f(12) = 5(12) - 10 = 50

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

The correct answer is 50.

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.  

Subscribe to Knowsys SAT & ACT Blog by Email

 

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.  

Subscribe to Knowsys SAT & ACT Blog by Email

SAT Math: Data Analysis

Median

Screen Shot 2014-01-23 at 4.56.27 PM.jpg

The table above shows the number of students enrolled at Allenville Middle School from 2005 to 2011.  If the median enrollment over these seven years was 789, and no two years had the same number of students enrolled, what is the lowest possible value for y?

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.   Review the definitions of important terms.  The median is the middle number in a set of numbers.

Identify the bottom line.  lowest possible y = ?

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

Attack the problem.  Whenever you are working with medians, you should start by putting the list of numbers in order.

737  755  776  789  804  811  

Now you need to determine where y should fall in the list.  The problem tells you that 789 is the median,  and no two years had the same enrollment, so y must be greater than 789.  The smallest that y can be, therefore, is 790.

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

The correct answer is 790.

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.  

Subscribe to Knowsys SAT & ACT Blog by Email

SAT Math: Geometry

Circles

Screen Shot 2014-01-20 at 11.24.40 AM.jpg

Knowsys Method

Read the problem carefully.  This problem deals with sectors, which are sections of a circle.  If you think of a circle as a pie, then sectors are slices of that pie.  

Identify the bottom line.  degrees in 1/6 of a circle - degrees in 1/9 of circle = ?

Assess your options.  If a sector makes up 1/6 a circle, then that sector must contain 1/6 of the 360 degrees in the circle.  Use that logic to figure out how many degrees are contained in each sector mentioned in the problem, then subtract the smaller number of degrees from the larger number to find the difference. 

Attack the problem.

Screen Shot 2014-01-20 at 11.30.49 AM.jpg

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.  

Subscribe to Knowsys SAT & ACT Blog by Email

SAT Math: Algebra

Symbol Problems

Screen Shot 2014-01-15 at 4.02.34 PM.jpg

A. -16
B. -4
C. -2
D. -1
E. -3/4

Knowsys Method

Read the problem carefully.  If you have never seen this problem type before, it might look strange to you.  Don't worry, though.  We will show you just how simple these problems are!

Identify the bottom line. c = ?

Assess your options.  At Knowsys, we call these problems "symbol problems" because they use fnon-mathematical symbols like stars, hearts, smiley faces, etc.  To solve a symbol problem, all you have to do is plug in the information given to you.  Use the following steps to ace these problems on the SAT. 

1. Write out the symbol problem defined in the question.  For instance,

&a = a + 2

2. Directly below that, write out the new version of the symbol problem, aligning the symbol in the original with the symbol in the new version.

&a = a + 2

&b =

&7 =

3. Substitute the variable(s) or number(s) in the new version for the variables in the original. 

&a = a + 2

&b = b + 2

&7 = 7 + 2 = 9

Attack the problem.  Let's apply those steps to the symbol problem above.  First, write out the original symbol problem defined in the question.

Screen Shot 2014-01-15 at 4.02.38 PM.jpg

You will have to deal with the right and left sides of the new version of the symbol problem separately.  Start with the left side.  Write that part out below the original, and substitute the variables into the original. 

Screen Shot 2014-01-15 at 4.02.44 PM.jpg

Now, repeat the same step with the right side. 

Screen Shot 2014-01-15 at 4.02.50 PM.jpg

Use FOIL, distribute the -2, and combine like terms.

Screen Shot 2014-01-15 at 4.02.59 PM.jpg

Now, set the solutions you have found for each side equal to each other and solve for c.

Screen Shot 2014-01-15 at 4.03.07 PM.jpg

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

The correct answer is D.

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.  

Subscribe to Knowsys SAT & ACT Blog by Email

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.  

Subscribe to Knowsys SAT & ACT Blog by Email