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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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ACT Math: Data Analysis

Directions:  Solve each problem and choose the correct answer.

On the first 5 of 6 semester tests, each of which is worth 100 points, a student has earned the following grades:  75, 87, 83, 89, and 94.  What score must the student earn on the sixth 100-point exam to earn an average test grade of 89 for all 6 tests?

     A.  86
     B.  89
     C.  94
     D.  99
     E.  The student cannot earn an average of 89.

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.   last score needed so that average of 6 is 89?

Assess your options.  You could sub in the different answers for the 6th test score or you could leave the missing number 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 6th test, which is part of the sum.  Here's what we are given:  

20140210 ACT average question.gif

All we have to do now is solve for x.  

89 * 6 = 428 + x

    534 = 428 + x

     106 = x 

Hmmm.  Since the test is only worth 100 points, can the student earn a 106?  Nope.  So, our answer is E.

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: 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.  

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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:

Screen Shot 2013-11-20 at 3.25.28 PM.jpg

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. 

Screen Shot 2013-11-20 at 3.22.17 PM.jpg

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.

Screen Shot 2013-11-20 at 3.26.53 PM.jpg

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

The correct answer is (E).
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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