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Proportions

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

The Knowsys Method is to read the question carefully and identify the bottom line.  Think about your options for solving the problem and choose the most efficient method.  Then attack the problem, solve it, and loop back to make sure that you solved for the bottom line.

A gasoline tank on a certain tractor holds 16 gallons of gasoline. If the tractor requires 7 gallons to plow 3 acres, how many acres can the tractor plow with a tankful of gasoline?
First, find the bottom line.  Your bottom line asks how many acres can be plowed with one tankful of gasoline.  A tankful is 16 gallons and you have information about how many gallons it takes to plow a certain number of acres.  Consider your options: you could use basic arithmetic to estimate the answer, you could solve for one gallon of gasoline and then multiply it by 16, or you could use a proportion.  Use a proportion because it only requires you to set up one problem.  Always remember that the long way is the wrong way!  Set up your proportion showing acres per gallon and use an x to mark your bottom line.   Then attack the problem using cross-multiplication to solve for the bottom line.







Check back to make sure that the x you solved for matches your bottom line, then look down at your answer choices.

(A) 6 and (6 over 7)

(B) 7 and (1 over 6)

(C) 7 and (1 over 3)

(D) 10 and (2 over 3)

(E) 37 and (1 over 3)



The correct answer is (A).


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

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

Probability

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

Read the question carefully, paying attention to the bottom line, the information that you must find.  Assess your options for solving the problem and attack the problem using the most efficient method possible.  In most cases you will not need to look at the answer choices until you have found your answer and double checked that it corresponds with your bottom line.

math image

In the figure above, the length of line A B is 2x and the length of  line B C is 3x.  If a point is chosen at random from line A C, what is the probability that the point will lie on line B C?

Start by labeling the line with the information that you are given because it helps to have a visual.  Part of the line is 2x while part of the line is 3x.  Your bottom line is a probability, the likelihood of an event occurring.  Probability is expressed as the number of relevant outcomes divided by the total possible outcomes, so you will need to find the length of the whole line (your total).  

The total line length of the whole line from point A to C is: 2x + 3x = 5x.  

The point that is randomly chosen can be anywhere within the 5x length, so that number represents the whole.  The relevant part of the line is from point B to C (3x) because you are asked to find the probability that the point is between B and C.   When you plug in those numbers into the equation for probability, you have 3x divided by 5x.  Notice that you can simplify your answer because the variable cancels out.  Now look down at your answer choices.

  

(A) 2 over 5
(B) 1 over 2
(C) 3 over 5
(D) 2 over 3
(E) 5 over 6

The correct answer is (C).


On sat.collegeboard.org, 69% of the responses were correct.

For more help with the writing section of the SAT, visit www.myknowsys.com!

Proportions

Link of the Day

Using current events in your SAT essay will impress your readers with the idea that you are informed about the world around you.  However, you are not limited to current events on this planet.  For example, have you heard the whimsical story about the Mars rover taking pictures of itself?  If not, follow this link to read the article.  This is a cute story, but it also relates to science, planning, creativity, and even self-awareness – topics that have shown up on the SAT.   With just a little more research, you could use the Mars rover as one of your excellent examples for the essay.  

9/9 Proportions

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

Read the question carefully to glean as much information as possible.  Then identify your bottom line, assess your options for solving the problem, and attack that question in the most efficient way possible.  When you have found an answer, loop back to make sure that you have answered the bottom line.

A group of workers can harvest all the grapes from 10 square meters of a vineyard in ½ minute. At this rate, how many minutes will the group need to harvest all the grapes from100 square meters of this vineyard?

Did you make a note of the bottom line?  The bottom line is the number of minutes it takes to harvest 100 square meters.  There are a number of ways to think about this problem, but one of the easiest ways is to set up a proportion.  You could set it up as time over area.  Write the ½ as .5 just so that you do not get confused between the fraction and the proportion.



When you set up the problem like this, you can easily see that in the denominator you would need to multiply the first 10 by 10 to get the second denominator of 100.  Anything you do to the bottom of a fraction you must also do to the top.  Multiply the .5 by 10 and you will get 5.  You solved for minutes just as the question asked you to do.  Look down at your answer choices.

(A) 5
(B) 10
(C) 20
(D) 50
(E) 60

The correct answer is (A).


On sat.collegeboard.org, 69% of the responses were correct.

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

Logic

Link of the Day

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

The correct answer is (B).


On sat.collegeboard.org, 48% of the responses were correct.

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

Writing Equations

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

Math questions should be read carefully so that you understand each part of the problem.  Take a moment to make a note of the bottom line, the answer you are asked to find, to make sure that you do not accidently solve for the wrong variable or do more work than the problem requires.  Assess your options for solving the problem, choose the most efficient method, and then attack the problem!  Always loop back to make sure that your answer matches the bottom line.

There are n students in a biology class, and only 6 of them are seniors. If 7 juniors are added to the class, how many students in the class will not be seniors?

For this question it is easiest to look at one piece of information at a time.  Your bottom line is the number of students who are not seniors.  Start with the n first.  The n represents the total number of students in the class.  Then you have 6 students from the class that are seniors.  Since you want the number that are not seniors, you must subtract the 6 from the n:
n – 6

Now look at your second piece of information.  7 juniors are added to the class.  Because they are not seniors, they will be added to the number of students who are not seniors:
n – 6 +7

You are not given any more information.  Simplify the equation that you have written:
n + 1

Even though you have not found an exact number, this equation represents the number of students in the class that will not be seniors.  Now look down at your answer choices.

(A) n – 3
(B) n – 2
(C) n – 1
(D) n + 1
(E) n + 2

The correct answer is (D).


On sat.collegeboard.org, 66% of the responses were correct.

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

Functions

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

All math problems require the same approach.  Read the question carefully and identify the bottom line that you must find.  Assess your options for working the problem, choose the most efficient strategy, and attack the problem.  Always make sure that your answer addresses the bottom line, especially in problems with multiple variables.

If the function f is defined by f(x) = 2x + 3, and if f(a) = 11, what is the value of a?

This question may look complicated because it uses functions, but it can be worked very quickly.  You are given a function using the variable x:
f(x) = 2x + 3

However, your bottom line requires you to solve for another variable: a. Think about the first information you are given in terms of a by substituting an a for every x:
f(a) = 2a + 3

Now plug in your last piece of information, f(a) = 11, and solve the problem using simple arithmetic.
11 = 2a + 3
8 = 2a
4 = a

Your bottom line was to solve for a so you are finished!  Take a look at the answer choices.

(A) 4
(B) 7
(C) 11
(D) 17
(E)25

The correct answer is (A).


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

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

Multiple Figures

Link of the Day

SAT geometry questions mention basic shapes such as squares and cubes or circles and spheres that are all around us in the natural world.  One sphere that people have always looked towards at night is the Moon.  Right now, people around the world are remembering the life of Neil Armstrong, the first man to set foot on the Moon.  Neil Armstrong is an excellent historical figure to mention in your SAT essay.  Review a few facts about the life of this famous man here.  See how Americans are responding to his death here.

8/28 Geometry: Multiple Figures

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

Geometry questions often require you to add labels to a diagram, so you must be especially careful to note exactly which information you are given when you read the question.  As always, make a note of the bottom line, assess your options for efficiently solving the problem, attack the problem, and loop back to make sure that you have answered the bottom line.  Writing what you know neatly will often help you see new ways to work with the shapes you are given.

graphic

In the figure above, O is the center of the circle and triangle A B O is equilateral. If the sides of triangle A B O are of length 6, what is the length of line B C?

Geometry problems can be difficult if you are not sure how to attack the problem.  Think of these kinds of problems as puzzles; use the pieces of information and the rules that come to your mind.  There are multiple ways of arriving at the correct answer, but this is one of the fastest ways to get there.

The first information that you are given is about an equilateral triangle (Triangle ABO).  Identify the equilateral triangle and label all of the interior angles 60̊°.  All equilateral triangles only have angles of 60°.  You are also given the information that the sides of this triangle have a length of 6.  Label all the sides of this triangle as well.

Now look at the information a little differently.  The two triangles inscribed on the circle form a single larger triangle.  You labeled the length of one side as 6 (Side AB).  Look at Side AC.  Line AO forms the radius of the circle, as does Line OC, so both must be the same length.  Your total length of Side AC must be 12.

Here is a rule you should memorize: any triangle that has the diameter of a circle as one of its sides will be a right triangle.  The diameter forms the hypotenuse, so the opposite angle (in this case Angle B) must be 90°.  Once you know two sides of any right triangle, you can find the third.  Before you pull out the Pythagorean Theorem, notice that Triangle ABC is a special triangle.  Angle A is 60° and Angle B is 90°, so Angle C must be 30°.  For any 30-60-90 triangle, the corresponding sides will be x, x√3, and 2x.  In this case, your x = 6 and your 2x = 12, so what is the missing side?  Label the missing side 6√3 and look up at the question to see whether you have found your bottom line.  Then match your answer to the answer choices.

(A) 3√3
(B) 4√3
(C) 6√3
(D) 9
(E) 12

The correct answer is (C).


On sat.collegeboard.org, 53% of the responses were correct.

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

Logic

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

The Knowsys math method calls for you to read each question carefully, but you should be especially conscious of the possibility of misreading when your question involves a graph.  Be sure that you understand each label on the graph and take the time to read any additional information given in the question.  Identify the bottom line and asses your options for solving the problem in an efficient manner.  Select your method, attack the problem, and loop back to make sure that the answer you found fits the bottom line, that it answers the question that you were asked.

 graphic

The bar graph above shows the number of people in attendance at each of the four meetings of the Maple Street Block Association that were held in 2011. Only members of the Block Association can attend the meetings, and no members joined or left the Block Association during 2011. Based on the bar graph, what is the least number of members the Maple Street Block Association could have had in 2011?

 Your bottom line is the least number of members belonging to the association.  You must use logic to determine which information in this graph is relevant.  The question is meant to mislead you, because as soon as you see the word “least,” you are likely to look for the smallest value on this bar graph.  Before you jump to any conclusions, think about what the bar graph represents.  At any of these four meetings, all of the members could have shown up, or only some of the members.  In other words, it is possible for people to be absent, but it is not possible for people to be at these meetings without being members.  All of the people present at any meeting must be members.  The greatest number of people who came to any meeting is 72.  Therefore, there cannot be fewer than 72 members in this club.  Look down at your answer choices.

(A) 61
(B) 65
(C) 67
(D) 72
(E) 268

The correct answer is (D).


On sat.collegeboard.org, 43% of the responses were correct.

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

Number Properties

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

Read the question carefully so that you are sure you understand what you are being asked.  You must identify the bottom line that you will solve for; note it at the top of your scratch work.  Assess your options to find the most efficient way to solve the problem, and then attack the problem.  Be sure to write out your scratch work clearly so that you do not make careless mistakes.  Your last step is to loop back to make sure that your answer matches the bottom line.  You cannot get a question right if you solved for an answer that you were not asked to find.

For how many positive two-digit integers is the ones digit greater than twice the tens digit?

You must find out how many numbers fit the given requirements.  There is no formula to find numbers in which one digit is more than twice the other, so you must think about this question logically.  Your only option is to methodically check positive integers to see which numbers will work.

You know that you need a positive two digit integer, so your first digit, at the very least, must be a 1.  Your second digit must be more than twice the first.  The number 1 multiplied by 2 is 2, so the number 12 will not fit the requirement of having a ones digit greater than twice the tens digit.   However, any number that begins with a 1 and has a second digit larger than 2 will work.  List all the numbers that begin with 1 and fit the requirements of this problem:

13, 14, 15, 16, 17, 18, 19

After 19, you must start each number with a 2, so find out what the second digit must be.  Again, it must be bigger than twice the first digit, so it must be larger than 4.  24 will not work, so start with 25 and list all of the numbers that fit the requirements of this problem:

 25, 26, 27, 28, 29

Follow this procedure for numbers beginning with 3. 3 times 2 is 6, so only numbers larger than 36 will work.

37, 38, 39

Move on to numbers that start with a 4.  4 times 2 is 8, so the second digit must be greater than 8.  This time there is only one number that fits the requirements:

49

Now you have reached numbers beginning with the digit of 5.  5 times 2 is 10.  You cannot have a value as your second digit that is more 10, so any number larger than 49 will not work.

Count up all of the numbers that you have found that fit the requirements of this problem.  That number will satisfy your bottom line.

(A) 16
(B) 20
(C) 28
(D) 32
(E) 36

The correct answer is (A).


On sat.collegeboard.org, 40% of the responses were correct.

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

Writing Equations

Link of the Day

Today’s SAT question talks about apples, but many people are talking about a different kind of apple right now.  Your cell phone, sitting innocuously in your pocket, could be your link to an interesting current event example for your next essay.  The companies Apple and Samsung are currently arguing about patent infringement.  Think about the themes of technology, development, creativity, and competition.  What do you think about this case?  Are you curious about which company will win?

8/19 Writing Equations

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

Always take the time to read math questions carefully so that you will not make careless mistakes.  Identify the bottom line, which is the question you must answer, and assess your options for reaching the bottom line.  Choose the most efficient method to solve the problem and then attack it.  Do not forget to loop back and make sure that you solved for the bottom line, especially when you get a problem that requires multiple steps.

The price of 10 pounds of apples is d dollars. If the apples weigh an average of 1 pound for every 6 apples, which of the following is the average price, in cents, of a dozen such apples?

The bottom line that you are looking for is the cost of 12 apples in cents.  You can make a note of this by writing 12app = ¢?  Now ask yourself what you could do, and what you should do.  You could choose a number for the variable d and plug it into all of the answer choices, but then you would have to work several problems to find matching answers.  Instead, try working with the information you have, setting up the information in simple equations.

Your bottom line asks for the cost of 12 apples, but you were given information about 6 apples.  6 apples weigh 1 lb.  It is easy to change 6 to 12 by doubling it.

6 apples = 1 lb
12 apples = 2 lb

Now you have the dozen apples, so you must determine how many cents they cost.  You know that 10 lbs = d dollars.  Start by changing the dollars into cents, because you know you must end with cents.  To change dollars into cents, you must multiply the dollars by 100.

10 lb = d (solving for dollars)
10 lb = 100d (solving for cents)

Now you have the correct monetary unit, but you also still have an equation that solves for 10 lbs.  Your dozen apples is only 2 lbs.  To get from 10 to 2, divide both sides of your equation by 5.

10 lb = 100d
2 lb = 20d

Now put all the information that you have together to make sure that you solved for the bottom line:

12 apples = 2 lb = 20d

Look at your answer choices.

(A) 20 times d

(B) (50 times d) over 3

(C) 5 times d

(D) (5 times d) over 3

(E) d over 20

The correct answer is (A).


On sat.collegeboard.org, 36% of responses were correct.

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

Functions

Link of the Day

Many schools in the United States participate in campaigns to keep children from smoking.  However, there are countries that are taking even greater measures to make smoking unattractive.  Cigarette packaging in Australia will no longer display colorful logos, but instead will display images depicting the dangers of smoking.  As you read this article, think about whether or not you agree with these measures, and then think about the themes that might relate this current event to an SAT essay topic.

Also, if you are a senior who dreads the college application process, take a look at this checklist and remember to breathe in the next few months!

8/16 Functions

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

Read each math question carefully so that you can identify exactly what is being asked of you.  Once you have identified the bottom line, assess your options to find an efficient way to solve the problem.  Finally, attack the problem, solve it, and loop back to make sure that your answer addresses the bottom line that you were asked to find.

 math image
Which of the following could be the equation of the function graphed in the xy-plane above?

You have been given a graph, and you must find the equation that has been graphed.  You could plug all of the answer choices into your calculator, but that would take a long time and you risk making a typo.  Instead, break the graph down into its most basic components.  What shape that you have often seen does this graph most resemble?  It looks like a parabola opening upwards, so you know that f(x) = x² will be part of your equation.

Picture the f(x) = x² parabola in your mind.  It passes through the origin at (0,0).  However, the graph in this problem would extend past the point (0,0) into the negative numbers if you continued the basic curve of the parabola.  To translate the function down on the graph, you would need to subtract a number from the original function.  Now you have f(x) = x² - n, where n = any number.

There is one more step.  The basic curve of the normal parabola has been reflected across the x-axis in this problem so that all the values of the parabola are now positive.  What can you do to make sure that all of the numbers in a function are positive?  Take the absolute value of the function.  Now you have f(x) = |x² - n|.  Look down at your answer choices.

(A) y =  (-x)² + 1
(B) y = -x² + 1
(C) y = |x² + 1|
(D) y = |x² - 1|
(E) y = |(x – 1)²|

(A), (B), and (C) cannot be the answers because they all add to the equation and would result in a parabola that has been shifted above the x-axis.  (E) will not be symmetric to the y-axis, and the graph that you have remains symmetric to the y-axis; it has not been shifted to the right or the left. The (x – 1)² part of the equation in (E) shifts the entire parabola away from its original position on the y-axis.  (D) is the only answer that matches the equation you wrote for this graph.

The correct answer is (D).


On sat.collegeboard.org, 39% of the answers were correct.

For more help, visit www.myknowsys.com!

Solids

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

The Knowsys method requires you to read each math question carefully and identify the bottom line.  You must also assess your options to find the best way to attack the problem, solve it, and loop back to make sure that you solved for the bottom line.

A right circular cylinder has height 6 and volume 54π. What is the circumference of its base?

Make a note of the fact that you are solving for the circumference of the base by writing C  = ? under the problem.  You should know that C = 2πr, and based on that you should realize that this problem will require more than one step.  You cannot solve for the circumference of the base without knowing the radius of the base circle.  You do know the volume of the cylinder, so you can use that information to find out more about the base circle.  You should have the formula for the volume of a cylinder memorized: Volume = πr²h. Plug in the values you already know to solve for the radius.

V = πr²h
54π = πr²6   (divide each side by 6π)
9 = r²    (take the square root of both sides)
3 = r

You now have the information that you need for the circumference formula.  Be careful not to look down at your answers yet, because even though you solved part of the problem, you have not yet found the bottom line.

C = 2πr
C = 2π3
C = 6π

Now that you have an answer, match your answer to the answer choices.

(A) 2π
(B) 3π
(C) 6π
(D) 9π
(E) 18π

The correct answer is (C).


On sat.collegeboard.org, 41% of the responses were correct.

For more help, visit www.myknowsys.com!

Number Properties

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

In the SAT math section, you must read every problem carefully and identify the bottom line.  Assess your options before solving the problem so that you are able to choose the most efficient method of solving the problem.  Then attack the problem to find your answer, and loop back to make sure that your answer addresses the bottom line.

The sum, product, and average (arithmetic mean) of three integers are equal. If two of the integers are 0 and -5, the third integer is…

This problem asks you to find one unknown integer.  You could try to write equations or plug in the answer choices to solve this problem.  However, these methods will take you longer than thinking logically about the properties of the numbers involved.

You know that the sum, product, and average of three integers must be equal.  One of the numbers that you are given is a zero.  Zero multiplied by any other number will always be zero, so the product must be zero.  That means that the sum and the average of these three numbers must also be zero. What do you have to add to 0 and -5 in order to get zero? The only possible answer is a positive 5.  Additionally, if you add 0, -5, and 5 together and then average them, your sum already equals zero, so zero divided by 3 will still be zero.  The product, sum, and average are all 0 when the missing third integer is 5.

0 + -5 + x = 0
x = 5

(0 + -5 + x) / 3 = 0
x = 5

Before looking at the answer choices, check to make sure that your answer fits the bottom line that you were asked to find.

(A) -5
(B) 0
(C) 2
(D) 5
(E) 10

The correct answer is (D).


On sat.collegeboard.org, 72% of responses were correct.

For more help, visit www.myknowsys.com!

Percents

 

Link of the Day

 

College loans may seem like the only way to get through your education, but they can be risky. Ben Bernanke, chairman of the Federal Reserve, warns that over-investing in college loans can lead to economic ruin. Instead of loans that you will have to pay back, look for scholarships, grants, and federal aid money that other agencies are willing to pay because they see it as an investment in their future. Scholarships and awards are given out for a dizzying variety of reasons, so make your search thorough! Do you have good grades? Do you have a hobby or sport that you're good at? Is there a cause you've volunteered for? Do you or does someone in your family have a rare disease? Some scholarships are restricted to members of certain racial or ethnic minorities; others are based on religious affiliation. Check if your parents' employers have any kind of scholarship fund. Sites like fastweb.com and findtherightscholarship.com can help in your scholarship search. 

 

8/7 Percents

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

 

When you work a percent problem, it is especially important to read the problem carefully because small words like “of” make all the difference in describing a percentage.  Once you have understood the question and identified the bottom line, assess your options and choose a method to attack the problem.  After you have finished the problem, loop back to verify that the answer addresses the bottom line.

 

The population of Norson, the largest city in Transitania, is 50 percent of the rest of the population of Transitania. The population of Norson is what percent of the entire population of Transitania?

Your bottom line is a percentage: the number of people in Norson as a percent of the entire population of the country of Transitania.  At the top of your scratch work, write N = ?%  Next, assess your options.  You don’t know how many people are in the country of Transitania, so solving the problem algebraically would be challenging. Instead, pick a number so that the problem will be more concrete.  When picking numbers on a percent problem, you should always pick 100 because 100 is the easiest number to use; the answer you find is already out of 100 so you will never need an extra step to find the correct percentage. Now you're ready to attack the problem. 

If you assume that there are 100 people total in all of Transitania, you still do not know how many people are in the city of Norson.  Use the variable “N” to represent the people in the city.  The problem tells you that the population of this city “is 50% of the rest of the population of Transitania.”  You know that Norson has half as many people as the rest of the population.  Think about it this way: for every person in Norson, there must be 2 people outside of the city.  Write an equation representing this knowledge and solve it.

 

 

N + 2N = 100

3N = 100

N = 33.33

You found that 33.33 people out of 100 live in the city of Norson, so you know that the percentage of people who live in Norson is 33.33%.  This matches the bottom line that you needed to find.

(A)               20%

(B)               25%

(C)               30%

(D)               33 1/3%

(E)                50%

The correct answer is (D).

 

On sat.collegeboard.org, 38% of answers were correct. 

 

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

Lines and Angles

Geometry: Lines and Angles

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

For every math problem, you should use the Knowsys method: read the question carefully, identify the bottom line, assess your options, attack the problem, and loop back to verify that the answer you found addresses the bottom line. 

math image

In the figure above, x = 60 and y = 40. If the dashed lines bisect the angles with measures of x° and y°, what is the value of z?

Geometry questions often include figures with multiple variables.  When you are assessing your options, realize that you can estimate values with figures that are drawn to scale, but that figures that are not drawn to scale may be misleading and estimation may result in a wrong answer.  When you are prepared to attack your problem, it is especially important to write your scratch work so that you can see how each number you find relates to the figure.  The easiest way to do that is to add the values you find to the figure.

The bottom line that you are solving for is z, but the information you are given is about x and y. First look at x.  Your ability to solve this problem hinges on your knowledge that “bisect” means “divides in half.” You know that x totals 60, so half of 60 is on each side of the dashed line that bisects x

60 ÷ 2 = 30

Likewise, you know that y totals 40, so half of 40 is on each side of the dashed line that bisects y.

40 ÷ 2 = 20

Now look at z. This variable overlaps half of x and half of y.  You just solved for each of these, so add them together.

30 + 20 = 50

Loop back to make sure that you solved the question that was asked and then match your answer choice to the answers that are given.

(A) 25
(B) 35
(C) 40
(D) 45
(E) 50

The correct answer is (E).


On sat.collegeboard.org, 81% of responses were correct.


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

Equations

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

Read the question carefully, identify the bottom line (what the question is asking), and assess your options for solving it. You want to be as efficient as possible when solving math questions, so for most problems you should not look at the multiple choice answers before attacking the problem with the method you have chosen. Always loop back at the end of the problem to make sure that your answer addresses the bottom line.

If , what is the value of (t + 3)(t + 6)?

You must find the value of (t - 3)(t + 6). In order to do this, you must first find the value of t. Paraphrase the question in your mind: “If this is true, then solve this.” This question is already set up in two steps for you.  Solve the first equation and you will have the key to solving the second part of the problem because there is only one variable involved: t.

Think about the first equation logically. Something squared is equal to zero, so what can be multiplied by itself and equal zero? The only possible answer is zero! The squared portion of the problem must be equal to zero.

 and , therefore t - 2 = 0.

When you add the 2 to both sides of your new equation, you will see that = 2. Now that you know the value of t, you have all the information that you need to solve the second part of the problem with simple arithmetic.


(t + 3)(t + 6)

(2 + 3)(2 + 6)

(5)(8)

40

Loop back to make sure that the answer you found answers the question you were asked. The problem asked for the value of (t + 3)(t + 6), and that is exactly what you found. Finally, match your answer to the correct answer choice.

(A) 40

(B) 18

(C) 9

(D) 4

(E) It cannot be determined from the information given.

The correct answer is (A).


On sat.collegeboard.org, 58% of responses were correct.


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

Fractions

Link of the Day

Have you ever wanted to try something but thought you wouldn't be any good at it? Don't let that stop you! Some mind-blowing stories of unstoppable dedication have been coming out of the Olympics, especially the tale of Im Dong-Hyun, a South Korean man with 20% vision or less in each eye. This makes him legally 'blind,' but it has not stopped him from setting two world records in archery! Read more here and here. What kinds of essay prompts might you answer with the story of Im Dong-Hyun?

7/29 Fractions

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

For every math problem, always be sure to follow the Knowsys Method. This will keep you from falling for traps and trick questions! First, read the question carefully.  You have to understand the information that you are given before you can begin working the problem.  The second step is to identify the bottom line, the question that you must answer.  In this case, you are asked to identify the total number of students who study art.  Next, assess your options to find the most efficient way to solve the problem, attack the problem, and loop back to make sure you answered the correct question.

Every student who studies art in a certain school receives exactly one of the grades A, B, C, or D. If of the students receive A’s, receive B’s, receive C’s, and 10 students receive D’s, how many students inthe school study art?

In this problem, you have fractions for the students who receive A’s, B’s, and C’s, but you have an actual number of students who receive D’s. To find the total number of students, you will need to find out what fraction of the whole is represented by the 10 students who receive D’s. Find out what fraction of the whole is represented by the students who receive A’s, B’s and C’s first. To add these fractions together, you need to find the least common denominator. You don't need to worry much about the 2 since it is a factor of 4. Instead, focus on the 5; the least common denominator will be the product of 4 and 5.

then becomes


If of the students receive A’s, B’s or C’s, then that only leaves of the students who can receive D’s. Remember that there are 10 students who receive D’s. Think about it this way: of the total number of students is 10 students. For math problems the word “of” indicates that you will need to multiply. So times the total number equals ten. Use x to represent the unknown total, and then solve for x.

Multiply each side by 20 to get rid of the fraction.

1x = 200

You solved for x (the total number of students), so check to make sure that you have found the bottom line. Then match your answer to the answer choices you are given.

(A) 30

(B) 60

(C) 100

(D) 200

(E) 500


(D) is the correct answer.


On sat.collegeboard.org, 60% of responses were correct.

Triangles

Link of the Day

Yesterday's Question of the Day about Red Cloud piqued my interest, so I decided to look him up for today's Link of the Day. Red Cloud was an amazingly successful war leader of the Lakota Indians, assaulting several United States Army forts along the Bozeman Trail in the 1860's. By the end of the decade, the US agreed not only to abandon its forts in Lakota territory, but also to guarantee Lakota control over a vast land area, including the western half of modern South Dakota and parts of Montana and Wyoming. Unfortunately, Red Cloud's victories did not last, and eventually the white settlers reclaimed and broke apart the Lakota holdings. Red Cloud's tireless efforts to protect his people and his culture would make an outstanding Excellent Example for your essay.

Let's take this a step further: What kind of essay prompt could you answer with the story of Red Cloud? Please respond in the comments! 

7/26 > Triangles

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

Whenever you approach a math problem, remember to follow the Knowsys method. Rather than charging in, take a moment to read the problem carefully and identify the bottom line. Consider the best way to approach the problem--what could I do? What should I do? Then attack the problem and, finally, loop back to the top and make sure you answered what the question was actually asking. The last and easiest step is to match your answer to the provided answer choices.

In triangle ABC, the length of side is 2 and the length of side is 12. Which of the following could be the length of side ?

First, note the bottom line at the top of your scratch work.

 = ?

Next, consider your options. What does the problem tell you? What strategies, formulas, or theorems do you know that could help you solve it? In this case, the problem tells you that you are dealing with a triangle and supplies two side lengths. With so little information, you really only have one tool that can help you: the Triangle Side Lengths Inequality.

The Side Lengths Inequality states that any side of a triangle must be less than the sum and greater than the difference of the other two sides. When you think through it, this actually becomes fairly obvious. If one side were longer than the other two sides put together, the shape could no longer be a triangle. It would fold flat into a line. If one side were too short, it would not be able to "reach" the other sides and the triangle would just be three line segments rather than a closed shape. The Side Lengths Inequality is usually expressed this way:



For simplicity's sake, you can rename the sides of the triangle in the problem x, y, and z rather than shuffling As, Bs, and Cs around. (Be sure to note this in your bottom line!) Now that you've chosen the most efficient way to solve the problem, attack it ruthlessly!

First, take the side lengths you are given and plug them into y and z.



Next, perform some simple arithmetic to solve for x.





Now you've narrowed down the range of possible values of x. Loop back to double-check the bottom line. If you remembered to update it earlier, it should look something like this:

x =  = ?

Since you've found the possible values of x, you've also found the possible lengths of side . The last step is to find an answer choice that matches what you found. 

(A) 6

(B) 8

(C) 10

(D) 12

(E) 14

Note that the inequality uses "less than" signs, not "less than or equal to" signs. That means that side  cannot equal 10 or 14; it must be 12. The answer is D.


On sat.collegeboard.org, 45% of responses were correct.


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

Quadratic Equations

Read the following question and select the correct answer choice.

As always, remember to follow the Knowsys method for math. Read the problem carefully and identify the bottom line (what you are looking for). Then, consider your options. How could you solve it? How should you solve it? Next, attack the problem using the method that you selected. Finally, loop back and verify that your answer matches the bottom line.


If (x + y)^2 =x^2 + y^2, which of the following statements must also be true?
Roman numeral 1x = 0
Roman numeral 2.(x minus y)^2 = x^2 + y^2
Roman numeral 3x times y = 0

This problem is going to be a tricky one. On the actual SAT, this would probably be one of the last problems in a math section (and since problems go in order from easiest to hardest, this gives you a clue that this problem is a difficult one). This means that you need to take your time solving this problem. If you think you have found the answer in 20 seconds, you have probably fallen for a trap (a common wrong answer that the test makers put in the answer choices to trick you). Take your time and follow the Knowsys method to avoid traps.

Start by reading the problem carefully and identifying the bottom line. You are looking for the statements that must be true. That means you will need to evaluate each step carefully to find out if it must be true. A good way to test if a statement must bet true is to try and prove it false. If you can't prove it false, then it must be true.

Now consider your options. Because there are variables in both the problem and the answer choices, you could pick numbers for the variables and test the answer choices. However, since the formula given to you could be expanded, it's probably a better idea to expand the formula first and see what you can deduce from that.







Notice that the only way for this formula to be true is if 2xy = 0. In other words, either x or y must be zero. Now, take a look at the three statements and try to prove them false, given that either or y must be zero. If you can prove that it is false, eliminate it.

Roman numeral 1x = 0

I could be true, since either x or y must be zero. However, it does not need to be true. x could have some other value as long as y is zero. Since it is possible for this statement to be false, you can eliminate it.

Roman numeral 2.(x minus y)^2 = x^2 + y^2

II is a little bit more difficult to evaluate. However, if you look closely, you should notice that it looks a lot like the original equation you were given. If you expand the equation, you get the following:







Once again, notice that as long as 2xy = 0, this equation is true. In other words, since you already know that x or y must be zero, this equation must be true.

Roman numeral 3x times y = 0

III also has to be true because you already know that either x or y must be zero.

You now know that (II) and (III) must be true. Choose the answer choice that matches your prediction.

(A) None
(B) I only
(C) II only
(D) III only
(E) II and III

The correct Answer Choice is (E).

On sat.collegeboard.org, 33% of the responses were correct.

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

Pronouns

Link of the Day

Did you know that pi has the approximate value of 22/7? That's why some people celebrate Pi Day today (many people also celebrate Pi Day on March 14 - 3/14). Pi is the ratio of the circumference of a circle to its diameter. Because it's an irrational number (one that cannot be expressed as an exact ratio) it is impossible to calculate the exact value of pi. As of October 2011, pi has been calculated to 10 trillion digits (although you really only need to know the value to 3 digits for the SAT). You can read more about pi and Pi Day here.

7/22 Identifying Sentence Errors

The following sentence contains either a single error or no error at all. If the sentence contains an error, select the one underlined part that must be changed to make the sentence correct. If the sentence contains no error, select choice E.

When you are working an Improving Sentences question, read the question carefully and focus on the underlined portion. If you identify an error, make a prediction about how you could fix the error. The correct answer won't always match your prediction, but making a prediction will help you to identify the underlying grammar concept being tested.

The largest European type of newt grows to about seven inches, while the largest American type, the California newt, it grows to about six inches. No error

Take your time and read the question carefully until you find something that stands out. (A) doesn't have any error. The adjective is correctly used and clearly modifies one noun. (B) also doesn't have an error. The word "type" is correctly used (also, the word "of" is the correct preposition to use here). (C) is part of a clause and it is idiomatically correct. (D) should stand out to you.  It's awkward because there is an unnecessary pronoun that makes the comparison unclear. Without the pronoun "it", the comparison becomes "The largest European newt grows to about seven inches . . ." while the other "grows to be about six inches". Since you have identified an error, (E) cannot be the correct answer.

The correct Answer Choice is (D).

On sat.collegeboard.org 81% of the responses were correct.

Want more help with grammar? Visit www.myknowsys.com!