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Roots and Radicals

Algebra: Roots and Radicals

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

Always read the problem carefully and determine the bottom line, the question that you must answer.  Assess your options for solving the problem and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that you completed all the necessary steps and solved for the bottom line.

If  , which of the following must be true?

Bottom Line: Which of the following . . . ?

Assess your Options:  Many "Which of the following . . . " questions require you to look at the answer choices to solve the problem, but you should always check to see whether you can simplify the equation that you have been given.  Instead of jumping to the answer choices, work the equation into a form that is not as intimidating.

Attack the Problem:  The original equation has a square root on each side.  How do you get rid of these square root signs?  Square both sides of the equation, and the roots will cancel out.  You are left with:

xa = x + b

You just showed that when something is on both sides of the equation, you can cancel it out.  There is a positive x on both sides of the equation.  If you subtract it from one side, you must subtract it from the other, and the x is eliminated.  You are left with:

-a = b

This looks fairly simple, so glance down at your answer choices.  All of them are set equal to 0.  Set your equation equal to zero by adding an a to each side.

0 = b + a

Remember, it doesn’t matter what order you use when adding two variables. 

Loop Back:  You put your answer in the same form as the answers on the test, so now all you have to do is match your answer to the correct one!

(A) a = 0
(B) b = 0
(C) a + b = 0
(D) a b = 0
(E) a² + b² = 0

The correct answer is (C).


On sat.collegeboard.org, 54% of the responses were correct.

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

Circles

Link of the Day

Anytime something new happens to something very old, the result is a rich current event that could be interpreted in many different ways.  The Catholic Church has chosen a new pope, and for the first time ever, the pope is from the Americas.  Look for broad themes in this article that would make it easy to relate this current event to an SAT essay topic.

Geometry: Circles

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

Read the question carefully and identify the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that your answer matches the bottom line.
graphic
In the figure above, a shaded circle with diameter line C D is tangent to a large semicircle with diameter line A B at points C and D. Point is the center of the semicircle, and line A B is perpendicular to line C D. If the area of the large semicircle is 24, what is the area of the shaded circle?

Bottom Line: A sm =? (What is the area of the small, shaded circle?)

Assess your Options:  There are two good ways to approach this problem.  Both ways require you to know the formula for the area of a circle. On collegeboard.org you will find a method that is especially efficient for students who are good at writing equations.  The method used here will focus on geometry skills and estimation in order to avoid the mistakes that often come with working more abstract formulas.

Attack the Problem:  You know the most about the large circle, so start there.  A semicircle is just half of a whole circle.  Therefore, to find the area of the whole circle, you would simply double the 24. 

24 × 2 = 48

If you know the area of the large circle, you can use the area formula to find out more information.  The area of a circle is   Plug in the area you just found to find the radius. 

Note: working backwards using the area formula for a circle is difficult, because using pi will always result in icky decimals.  If you glance at your answer choices, all of them are whole numbers.  You can estimate pi as 3 instead of 3.14 in order to keep this problem as easy as possible.

48 = 3r²
16 = 
4 = r

You now have the radius for the big circle.  Now look back up at the diagram.  The radius for the big circle is also the diameter for the little circle!  If the diameter of the little circle is 4, the radius will be half of that.  Once you know that the radius of the little circle is 2, you are ready to find the area!


A = 3 × 2²

A = 3 × 4
A = 12

Loop Back:  You found the area of the small circle, so you are ready to look at your answer choices.

(A) 8
(B) 10
(C) 12
(D) 14
(E) It cannot be determined from the information given.

The correct answer is (C).


On sat.collegeboard.org, 56% of the responses were correct.

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

Sequence Problems

Arithmetic: Sequence Problems

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

Always read the question carefully and identify the bottom line.  Then assess your options and use the most efficient method to attack the problem.  When you have an answer, loop back to make sure that you solved for the bottom line.

8, a, 14, b, 20, …
The first term of the sequence above is 8. Which of the following could be the formula for finding the nth term of this sequence for any positive integer n?

Bottom Line: You want a formula to describe this number sequence.

Assess your Options:  You could try to write a formula, but you will have a hard time doing that because you do not know the second and fourth terms in your pattern.  You also do not need to find numbers for the variables a and b in order to solve this problem.  Instead, use the answer choices to help you find an answer.

Attack the Problem: The first thing to do is realize that n is not a variable that you have to find algebraically; the nth term just describes the number of that term in the sequence, like the first, second, third, fourth, or fifth.  Therefore:
8,   a,  14,  b,   20, …
1,   2,   3,   4,     5

That means that when you plug in 1 to the formula, you should always get 8, when you plug in 3, you should always get 14, and when you plug in 5, you should always get 20.

Take a look at your answer choices:

(A) 2n + 6
(B) 3n + 5
(C) 5n + 3
(D) 6n + 2
(E) 6n + 5

You could start by plugging in 1 and finding out which of these equals 8, eliminate any that do not, and then try plugging in 3 and then 5 (this method is used on collegeboard.org).  However, just by looking at the numbers (a lot of 2s and 6s and a lot of 3s and 5s) you should be able to tell that a lot of these will equal 8.  To save time, start by plugging in the biggest term you know, the fifth, and see which answer choices equal 20.

(A) 2(5) + 6 = 16
(B) 3(5) + 5 = 20
(C) 5(5) + 3 = 28
(D) 6(5) + 2 = 32
(E) 6(5) + 5 = 35

Note: if you use logic, you do not even have to work out (C), (D), and (E) because the product of the first two numbers is larger than 20 before you even add to them.

Only one answer choice results in the correct 5th term of 20.  You don’t need to check any other numbers!

Loop back:  You found the only formula that will work for every number in the sequence, so select that answer.

(A) 2n + 6
(B) 3n + 5
(C) 5n + 3
(D) 6n + 2
(E) 6n + 5

The correct answer is (B).


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

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

Coordinate Geometry

Geometry: Coordinate Geometry

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

Approach every question the same way to minimize mistakes.  Start by reading the question carefully and identifying the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches your bottom line.

In the xy-plane, line l passes through the points (0, 0) and (2, 5). Line m is perpendicular to line l. What is the slope of line m?

Bottom line: slope m = ?

Assess your Options:  You could draw out a graph and solve this visually, but that is a waste of time if you know the formula to find the slope of a line.

Attack the Problem:  You are given the most information about line l, so start with that line.  You should have the formula for slope memorized:


It is easiest just to think about slope as the change in y-values over the change in x-values.  If you look up at the original points that you have been given, from zero the y-values go up to 5 and the x-values go up to 2.  You now have 5 over 2.

The slope of line l is .

At this point, some students will think they are finished and select answer (D).  However, your bottom line was the slope of line m!  The problem tells you that line m is perpendicular to line l.  In order to find a perpendicular line, you must take the opposite reciprocal of the first line; in essence you must flip the sign (negative or positive) and the numbers (a fraction or whole number).

The slope of line m is .

Loop Back:  You solved for your bottom line, so look down at your answer choices.

(A) 
(B) 
(C) 
(D) 
(E) 5

The correct answer is (B). 


On sat.collegeboard.org, 47% of the responses were correct.

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

Equations

Algebra: Equations

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

Always read the question carefully and identify the bottom line so that you do not waste time finding something unrelated to the question.  Assess your options for solving the problem and choose the most efficient method to attack the problem.  When you have an answer, take a second or two to loop back and make sure that your answer matches the bottom line.

If a, b, and c are numbers such that  and , then  is equal to which of the following?

Bottom line:  

Assess your Options:  There are two ways that you can solve this equation, and both will arrive at the correct answer.  You can solve it algebraically by substituting information into the equation, or you can pick your own numbers for the variables.  Choose the method that is easier and faster for you.

Attack the problem:  If you are going to solve a problem algebraically, always look for ways to simplify the problem that you are given.  In this case, you will want to get rid of unnecessary fractions.  Look at the first piece of information that you are given.  If a divided by b is 3, you can get rid of the fraction by multiplying each side of the equation by b.

Now you have a = 3b.

Look at the numerator (the top part of the fraction) of your bottom line.  You can now make sure that there is only one variable in this portion of the equation.   Substitute 3b for a.  Now you have 3b + b, which will simplify to 4b

Here are the steps you just completed:


Look at the denominator of your equation.  How can you simplify b + c?  You might be tempted to substitute 7c for b, but remember your goal is to get to a number without a variable.  If you have the same variable in the top and bottom, the two variables cancel. Therefore, you need to find what c is equal to in terms of b

When you are given the information that b divided by c is 7, then you know that c divided by b is 1 over 7.  You flip both equations.  Solve for c by multiplying both sides of the equation by b.

 so   so 

Plug this information into your bottom line equation and combine like terms.


A fraction over a fraction is ugly, but remember that dividing by a fraction is the same thing as multiplying by the reciprocal of that fraction.  In other words:


Notice that the variable b moves to the bottom of the second fraction and cancels out.  You solved the equation!

Alternatively:  If you dislike algebra, use the strategy of picking numbers to solve this problem.  You want to get rid of ugly fractions, and the best way to do that is to put a number over 1.  You cannot just put b = 1 because b affects two different equations and you might end up with numbers that are difficult to use in your other equation.   However, c is on the bottom of a fraction in one equation.  Pick c = 1.  Plug 1 into the second piece of information with c and solve for b.

  so  so b = 7.

The variable b must equal 7. Now plug that into the first piece of information that you were given.  If b is 7, then a must equal 21.  

 so  so a = 21.

Now that you have numbers for a, b, and c, plug those into your bottom line equation:


Bottom Line:  As soon as you have a value to represent your bottom line, look down at your answer choices.

(A) 
(B) 
(C) 
(D) 
(E) 21

The correct answer is (A).


On sat.collegeboard.org, 42% of the responses were correct.

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

Sets

Arithmetic: Sets

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

Approach each math question on the SAT the same way.  Read the question carefully to be sure you take into account all of the information as you solve it, and be sure to identify and note the bottom line.  Assess your options for solving the problem, and then choose the most efficient method to attack the problem.  Never forget to loop back and make sure that your final answer solves for the bottom line, the question that you were asked.

If S is the set of positive integers that are multiples of 7, and if T is the set of positive integers that are multiples of 13, how many integers are in the intersection of S and T?

Bottom Line: # of intersections = ?

Assess your Options:  When you have a question that asks about number properties, ignore your answer choices!  If you look down and see a 0, you could think to yourself that both 7 and 13 are prime, so they have nothing in common.  Are you looking for factors?  No!  You are looking for multiples.  Think through all of the information that you are given before looking at the answer choices.

Attack the Problem:  A set is just a collection of data.  You are given two different sets and asked to find the intersections, the data that the two have in common.  The only restriction on both sets is that all of the numbers must be positive.

Now think about what multiples are.  Multiples are the product of a number and an integer.  So Set S contains 7, 14, 21, 28… and continues in this manner into infinity.  Set T contains 13, 26, 39, 52… and continues in this manner into infinity.

If you keep listing numbers in each set, it will take you forever to find the answer to this problem.  Instead, think logically about where you know you must have multiples that match.  For example, if you multiply 7 times 13, you will find a number that belongs in both sets.  If you multiply 14 times 13, you will find another intersection.  Notice that you can keep doing this because you will never reach infinity.  The answer to this problem is that there are an infinite number of intersections between S and T.

Loop Back:  You found your bottom line, so look down and see which answer choice it matches.

(A) None
(B) One
(C) Seven
(D) Thirteen
(E) More than thirteen

The correct answer is (E).


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

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

Coordinate Geometry

Geometry: Coordinate Geometry

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

Always be sure to read the question carefully and make a note of the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

In the xy-plane, line l passes through the points (a, 0) and (0, 2a), where a > 1.  What is the slope of line l?

Bottom Line: slope of l = ?

Assess Your Options: You could select a number larger than 1, plug it in for the variable a, and then work the problem.  However, if you peek down at the answer choices, notice that some have a variable still in the problem.  It will take you longer to plug in a number than to work the problem using the variables.

Attack the Problem: Your bottom line is a slope, so use the formula for the slope of a line. The formula for slope of a line is:



To find the change in y coordinates, subtract the first y-value from the second y-value.  Do the same with the x values:



The variable will cancel when you simplify the problem.  Your answer is -2.

Loop Back:  You found the slope of the line, so you are ready to look down at your answer choices.

(A) -2
(B)
(C) 2
(D) -2a
(E) 2a


On sat.collegeboard.org, 42% of the responses were correct.

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

Triangles

Geometry: Triangles

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

Always read the question carefully so that you don’t misapply any information.  Identify the bottom line and assess your options for solving the problem.  Choose the most efficient method to attack the problem.  When you think you have the answer, loop back to make sure that it matches the bottom line.

2-16-2013-MRO_CG_5.png

If triangle ABC above is congruent to triangle DEF (not shown), which of the following must be the length of one side of triangle DEF?

Bottom Line: side of DEF = ?

Assess your Options: Many students go straight to the Pythagorean Theorem whenever they see a right triangle.  This formula, a² + b² = c², will not help you in this case because you do not know a or b.  Instead, use your knowledge of special triangles to solve this problem.

Attack the problem:  As soon as you see that this is a 30° – 60° – 90° triangle, you should think about the sides that relate to this special triangle.  Those sides, which you should have memorized, are x - x√3 – 2x.  Remember that the longest side has to be across from the biggest angle, the 90° angle.  That is your 2x.  This triangle has a 12 in that position.  Solve for x.

2x = 12
x = 6

Now you know that the side across from the 30° angle, AB, must be 6.  Label it.  Look at the side across from the 60° angle.  AC must be x√3.  You know that x = 6, so this side must be 6√3.  Label it.  You now know all the sides of triangle ABC:

x - x√3 – 2x
6 -6√3 – 12

Your bottom line is a side on triangle DEF, not on triangle ABC.  However, the problem tells you that ABC is congruent to DEF.  Congruent triangles have the same shape and size; they are basically the same triangle with different labels.  That means that the side lengths from triangle DEF will match the lengths you already found for ABC.

Loop back: You took into account all of the information that you were given and solved for your bottom line.  Look down at your answer choices.  One of the three side lengths you found will be there.

(A) 18
(B) 24
(C) 3√6
(D) 6√3
(E) It cannot be determined from the information given.

The correct answer is (D).

On sat.collegeboard.org, 38% of the responses were correct.

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

Pattern Problems

Link of the Day

How important is history?  Should people take more responsibility for solving problems that affect their communities or the nation in general?  That second question is a previous SAT essay question.  Before you answer it, read here and here about how the people of Mali reacted to a threat against manuscripts as old as the 13th century.  What important themes do you see in these articles that would be easy to write about as a current event example?  Write down details and facts that could help to support an opinion on a broad topic.


Arithmetic: Pattern Problems

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

Work each math problem by reading the question carefully and identifying the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that you finished all of the steps to match your bottom line.

If it is now 4:00 p.m. Saturday, in 253 hours from now, what time and day will it be? (Assume no daylight saving time changes in the period.)

Bottom Line: 253 hours from now = ?

Assess your Options: You could try to count the hours elapsed from the answer choices, but that will be time consuming if you don’t guess the right one first and end up working the problem five times.  Instead, use logic to methodically work through this problem.

Attack the Problem:  You are given 253 hours, but you know that there will also be a change in the day.  There are only 24 hours in a day.  Find out how many days there are in this time period by dividing.  Now, this problem is similar to the pattern problems in your Knowsys book.  You want to know how many days and hours have passed.  Instead of dividing with your calculator, use long division to find out how many days pass and how many hours remain. 253 ÷ 24 is 10 with a remainder of 13.  That means that there are 10 days and 13 hours that pass.

Continue to think about this logically.  If it is 4:00 p.m. on Saturday and a week passes, it will be the same day.  So 7 days will get you back to the same place.  Then you have 3 of your 10 days still to go.  Count 3 days from Saturday, (Sun, Mon, Tues), and you are now at 4:00 p.m. on Tuesday.

That accounts for all of the days that have passed, but you still have 13 hours.  If you add 12 hours to 4:00 p.m., it becomes 4:00 a.m. on the next day, Wednesday.  Add 1 more hour and you get 5:00 a.m. on Wednesday. 

Loop Back:  You accounted for all of the 253 hours by counting out 10 days and 13 hours.  Look down at your answer choices.

(A) 5:00 a.m. Saturday
(B) 1:00 a.m. Sunday
(C) 5:00 p.m. Tuesday
(D) 1:00 a.m. Wednesday
(E) 5:00 a.m. Wednesday

The correct answer is (E).


On sat.collegeboard.org, 49% of the responses were correct.

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

Functions

Algebra: Functions

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

Always use the same process for math problems on the SAT.  Read carefully and make a note of the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to be sure it matches your bottom line.  

If the function f is defined by function f of x = (x minus a) times (x minus b) over (x minus c), where 0 < a < b < c, for which of the following values of x is f undefined?

I. a
II. b
III. c

Bottom Line: For which value(s) of x is f undefined?

Assess your Options: You could pick numbers, but that will get confusing with three variables.  You could just start plugging in the variables a, b, and c for x and then simplify the function, but you will end up wasting time.  Time is precious on the SAT!  Start with the information that you are given and think about it logically.

Attack the Problem:  Always think about the information that you are given before you jump into the problem.  The inequality that you are given simply tells you that all of your variables are positive numbers.  A function or a fraction is undefined whenever it is divided by zero because you cannot divide by zero. 

Think about it logically:  do you care what is on the top of the fraction?  No!  Focus on the bottom of the fraction.  How can you make x c = 0?  The variable that you are changing in this problem is x.  If you set x = to c, then cc = 0. 

Note:  You do not know whether a or b is equal to c, so you cannot assume that ac or bc would equal 0.  If you plug those variables in, you still have a lot of variables on the bottom!

Loop Back:  You found the only answer that will work out of the three that you were given.  Look down at your answer choices.

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

The correct answer is (C).


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

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

Probability

Data Analysis: Probability

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

Always read each question carefully and make a note of the bottom line (what you are trying to find).  Assess your options to find the best strategic method and use that method to attack the problem.  When you have an answer, loop back to verify that the answer matches the bottom line.

A jar contains only red marbles and green marbles. If a marble is selected at random from the jar, the probability that a red marble will be selected is . If there are 36  green marbles in the jar, how many red marbles are there in the jar?

Bottom Line:  You want to know how many red marbles there are, so use r to represent red and just write r = ?

Assess your Options:  You could try to work backwards from the answer choices to find a number that, when combined with 36, makes the right fraction.  That won’t be any faster than just solving the problem.  Use the probability formula.

Attack the Problem:  The probability formula is:



In this problem, you know the red marbles are the relevant outcome, while the red and green marbles together are the total (all that is in the jar).  Use g for the green marbles.  There are 36 green marbles.



You have already been given the probability that a red marble will be selected.  Set the formula that you created equal to the probability that you were given.  Then solve for r with cross-multiplication.


3r = 2(36 + r)
3r = 72 + 2r
r = 72

Loop Back:  You solved for your bottom line, so you are ready to look at the answer choices.

(A) 18
(B) 24
(C) 54
(D) 72
(E) 108

The correct answer is (D).


On sat.collegeboard.org, 47% of the responses were correct.

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

Probability

Data Analysis: Probability

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

Read each question carefully to avoid making any mistakes. Identify the bottom line (what the question is asking) and assess your options for reaching it by asking yourself “What could I do?” and “What should I do?” Choose the most efficient method to attack the problem and find an answer. Last, loop back to make sure that your answer addresses the bottom line.

If a number is chosen at random from the set {-10, -5, 0, 5, 10}, what is the probability that it is a member of the solution set of both 3x – 2 <10 and x + 2?

Bottom Line: Prb = ?

Assess Your Options: You cannot solve for a probability until you know whether each number in the set meets the requirements that you are given. You could plug numbers from the set into each inequality and see if they work, but it is much faster to simplify the inequalities before you begin working with them.

Attack the Problem: Simplify the inequalities by solving both for x.

3x – 2 < 10
3x < 12
x < 4

x + 2 > -8
x > -10

You now know that x must be less than 4, but greater than -10. The question asked you to find a number that fits both of these solution sets. Look at the original set that you were given. The only two answers that are between -10 and 4 are -5 and 0 (-10 does not work because it cannot be equal to negative -10; it has to be greater than -10). You found 2 numbers out of 5 that you were given that work. To write this as a probability, you must set the number of relevant outcomes over the number of total possible outcomes. Your answer is 2 over 5.

Loop Back: You found a probability matching the restrictions you were given. Look down at your answer choices.

(A) 0

(B) 1 over 5

(C)2 over 5

(D)3 over 5

(E)4 over 5

The correct answer is (C).


On sat.collegeboard.org, 50% of the responses were correct.

For more help with SAT math, visitwww.myknowsys.com!

Rates

Arithmetic: Rates

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

Using the same method with every math problem to minimize mistakes.  Read the question carefully.  Identify the bottom line and assess your options for finding it.  Choose the most efficient method to attack the problem.  Once you have an answer, loop back to make sure it addresses the bottom line.

A woman drove to work at an average speed of 40 miles per hour and returned along the same route at 30 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?

Bottom Line: Td = ?  (Total distance)

Assess your Options: Remember that speed is really a rate.  There are 4 key rate scenarios: separation, overtake, round trip, and meet in between--this one is a round trip.  You can figure all of these out by using the distance formula (rate × time = distance), but it can be difficult to keep track of which scenario you have unless you treat all of them the same way.  Knowsys recommends that you use a chart to quickly organize your thoughts so that you can be sure that you accounted for all of the information in the problem. (Spoiler: many students make mistakes on these types of problems!  You do not get any extra points for ignoring the chart, so use it!)  

Attack the Problem:  Here is the chart that you should use with all rate scenarios:


1
2
Total
Rate



Time



Distance




Start filling in the information that you know.  The first trip was at a rate of 40 miles per hour and the second trip was at a rate of 30 miles per hour.  The total time was 1 hour. 


Trip 1
Trip 2
Total
Rate
40
30

Time


1
Distance




If you don’t know the time for the first trip, choose a variable to represent the unknown.  Put an “x” in that box.  You know that the time for the trips together must total 1 hour (x + ? = 1).  Therefore, the second trip is equal to 1 minus x


Trip 1
Trip 2
Total
Rate
40
30

Time
x
1 – x
1
Distance




You already know that rate × time = distance, so multiply the two columns representing the trips. 


Trip 1
Trip 2
Total
Rate
40
30

Time
x
1 – x
1
Distance
40x
30(1 – x)


Before you start worrying about the total number of miles, remember that this person is using the same route each time.  That means the distance traveled each time is an equal length.  Set the distances equal to each other.

40x = 30(1 – x)
40x = 30 – 30x
70x = 30


If you know x, you can now find a number value for each part of your chart.  What was the bottom line?  You need to find the total number of hours.  You could plug x into both distances and add them up; however, there is an even faster method.  Take the first distance and multiply it by 2.  (Remember that the distances are the same.)







Loop Back:  You solved for your bottom line, so look down at your answer choices.

(A) 30
(B) 
(C) 
(D) 35
(E) 40

The correct answer is (C).


On sat.collegeboard.org, 38% of the responses were correct.

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

Lines

Geometry: Lines and Angles

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

Work all math problems the same way.  Read the problem carefully, identify the bottom line, and assess your options for solving the question.  Choose the most efficient method to attack the problem, and loop back to make sure that your answer matches the bottom line.

Four distinct lines lie in a plane, and exactly two of them are parallel. Which of the following could be the number of points where at least two of the lines intersect?

I. Three
II. Four
III. Five

Bottom Line: # intersections

Assess your Options: You could just start drawing any combination that you can think of, but try to think of the particular answer choices that you are given.  Examine options I, II, and III independently.

Attack the Problem: Think first about option I.  Go ahead and draw out two horizontal parallel lines.  The other lines cannot be parallel to these lines or to each other because the problem says that there are "exactly" two parallel lines.  How could you create three intersections?  One way is to make those next two lines into an “X” and put the middle of the X on one of the preexisting parallel lines.  If you extend the legs of the X out far enough (remember these are lines, not line segments), they will cross the other parallel line in two places.  You have created an image with 3 intersections while following all of the stipulations.

Now turn your attention to option II.  If you have two parallel lines, those lines will never cross.  You know that the other two lines cannot be parallel because this situation has "exactly two" parallel lines.  If you have two lines that are not parallel, no matter how close their slopes are, eventually they must cross.  That is why representing these lines as an X is a good idea.  Any combination of the lines other than the above combination will result in 1 intersection between the lines that are not parallel and 4 intersections where these lines cross the parallel lines for a total of 5 intersections.  There is no way to get only 4 intersections.

Look at option III.  You already thought about it conceptually while examining option II, but you can prove this possibility by drawing a picture.  Draw two horizontal parallel lines.  You can keep the other two lines as an X, but move the middle of the X off of the parallel lines.  Whether the middle of the X is between the parallel lines, above them, or below them, you will now have 5 intersections while following all of the stipulations. (This also confirms your thinking about option II.)

Loop Back:  You examined each option separately, so you are ready to look down at your answer choices.

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III

The correct answer is (D).


On sat.collegeboard.org, 29% of the responses were correct.

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

Writing Equations

Link of the Day

New things can be exciting, but also scary.  Several years ago, Y2K (the year 2000) frightened many people.  Now people are worried about the end of the Mayan calendar on Dec 21, 2012.  Take a look at this article to see how people are reacting to rumors about the end of the world.  How could you use this current event on an SAT essay?  It would easily relate to questions about whether the world is getting better, how people understand themselves and those in authority, feelings and rationality, and many other topics.  Make sure to pick out specific details to mention in your essay if you choose this as one of your current event examples!

Algebra: Writing Equations

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

Always read math problems carefully so that you don’t miss an important piece of information.  Identify the bottom line, and assess your options for reaching it.  Choose the most efficient method to attack the problem.  Many problems have multiple steps, so be sure to loop back and make sure that you solved for the bottom line.

The stopping distance of a car is the number of feet that the car travels after the driver starts applying the brakes. The stopping distance of a certain car is directly proportional to the square of the speed of the car, in miles per hour, at the time the brakes are first applied. If the car’s stopping distance for an initial speed of 20 miles per hour is 17 feet, what is its stopping distance for an initial speed of 40 miles per hour?

Bottom Line: d (distance) = ?

Assess your Options:  You have to decide how to use the information in this problem; in other words, you need to write an equation.  Plugging in the answer choices will take a lot of guess work.  Instead, carefully work through each piece of information that you are given.

Attack the Problem:  You have probably worked with distance, rate, and time before.  One formula that is often used in Knowsys classes is distance = rate × time.  This problem is asking you to write a similar equation.  The problem says: “The stopping distance of a certain car is directly proportional to the square of the speed of the car, in miles per hour, at the time the brakes are first applied.”  In other words, you know that distance is (is means equals in math) directly proportional to something.  Now pay particular attention to the part that says “directly proportional.  This phrase just means that when the distance gets bigger, so does the other side of your equation.  For that to happen, you need another constant number on the other side of the equation.  Your distance is equal to some constant number times speed squared.  Your formula should look like this:

distance = constant number × speed²

Now that you have written an equation to show what is happening in this problem, you are ready to look at the next piece of information.  Plug in the first situation in which an initial speed of 20 miles per hour results in a distance of 17 feet.

d = c × s²
17 = c × 20²

Now you can solve for c by isolating the variable.  Use your calculator when it will be faster than mental math.

17 = c × 400  (divide each side by 400)
.0425 = c

Now you have enough information to find your bottom line. Plug in the second situation in which the car is going 40 miles per hour and solve for the distance.

d = c × s²
d = .0425 × 40²
d = .0425 × 1600
d = 68

Loop Back:  You solved for the stopping distance of a car traveling 40 mph, just as the question asked.  You are ready to look at your answer choices.

(A)  34 feet
(B)  51 feet
(C)  60 feet
(D)  68 feet
(E)  85 feet

The correct answer is (D).
On sat.collegeboard.org, 39% of the responses were correct.

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

Group Problems

Arithmetic: Group Problems

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

Work all math problems the same way so that you can approach even the most difficult problems with confidence.  Start by reading the question carefully.  Many problems have several steps, so you must identify the bottom line: what is the question asking?  Assess your options and choose the most efficient method to attack the problem.  Finally, loop back to make sure that your answer matches the bottom line.

In a community of 416 people, each person owns a dog or a cat or both. If there are 316 dog owners and 280 cat owners, how many of the dog owners own no cat?

Bottom Line: just dogs = ?
When you get to the step where you look at the answers, notice that (E) comes from not reading carefully.  Yes, there are 316 total dog owners, but some of them also own cats.  You must find how many own only dogs.

Assess your Options:  You could try to work backwards using the answer choices, but trying to think about the steps of a problem backwards often leads to mistakes.  You could also realize that this is a problem involving two overlapping groups and draw a Venn Diagram.  Forget those methods because the fastest method is to use the Group Formula.  Take a moment now to memorize this formula if you have not already done so: Total = Group 1 + Group 2 + Neither – Both.

Attack the Problem:  Plug all the information that you know into the formula.  How many total people are there? 416.  Then there are your two groups: Those who own dogs and those who own cats.  Plug in the numbers 316 and 280 to represent these groups.   Now, the problem tells you that “each person owns a dog or a cat or both,” so how many people own neither animal?  Zero.  The only thing that you are not given in the problem is how many people own both a dog and a cat.  Your formula should now look like this:

Total = Group 1 + Group 2 + Neither – Both
416 = 316 + 280 + 0 – B  

That B represents the unknown Both, but you can now solve for it because it is the only variable left in your equation.  Start by simplifying the problem.

416 = 316 + 280 + 0 – B  
416 = 596 – B   (add B to each side to make it positive)
416 + B = 596   (subtract 416 from each side)
B = 180

You have finished one step, but you have not yet reached your bottom line!  Do not look at the answer choices yet or you will be tempted to pick a wrong answer!

You just solved for the number of people who own both a cat and a dog.  How do you find the number of people who own just a dog?  Take the number who own both and subtract it from the total number of dog owners.  Remember that the total number of dog owners was given in the problem as 316.

316 – 180 = 136

Loop Back:  Your second step solved for those who just own dogs, so you are ready to look at your answer choices.

(A) 36
(B) 100
(C) 136
(D) 180
(E) 316

The correct answer is (C). 


On sat.collegeboard.org, 46% of the responses were correct.

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

Functions

Link of the Day

How do medical breakthroughs happen?  How do you feel about animal testing?  A few dogs that were once paralyzed are now walking again with the aid of some cells from healthy dogs.  Scientists recognize that they are not ready to apply their findings to humans with spinal cord injuries, but they are hopeful about the future.  This development would make a great current event example for your SAT essay.  If you choose to use it as a current event, take notes detailing the facts in this article and how you could apply them to a wide variety of topics.

Algebra: Functions

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

Take the time to read math questions carefully because this will save you from wasting time as you solve the problem.  Start by identifying the bottom line and assessing your options for reaching it.  Choose the most efficient method to attack the problem.  When you think you have an answer, loop back to make sure that it matches the bottom line.
  • f(2n) = 2f(n) for all integers n
  • f(4) = 4
If f is a function defined for all positive integers n, and f satisfies the two conditions above, which of the following could be the definition of f?

Bottom line: f(n) = ?

Assess your options:  You could start by plugging in f(4) to see which of your answer choices results in the number 4, then check any that do against the first condition.  This is the method recommended by collegeboard.com.  However, this requires multiple steps as you compare each answer choice to both conditions.  Instead, take a moment to think logically about the two conditions that you are given.

Attack the problem:  Start with the second condition because it is already in a format that is easy to use.  If f(4) = 4 and you can use the same variable to represent numbers that are the same,  that is the same as saying that f(n) = n.  Now look at the first condition and think about it logically.  If you multiply the variable within the function by 2, that gives you the same number as multiplying the result of the function by 2.  In order for those two numbers to be the same, the final result of the function has to match the number that is plugged into the function.  In other words, f(n) = n.  Both conditions give you the same definition of the function.

Loop Back:  You found a simple way to define both of the conditions for f(n), so look down at your answer choices.

(A)  f(n) = n - 2
(B)  f(n) = n
(C)  f(n) = 2n
(D)  f(n) = 4
(E)  f(n) = 2n – 4

On sat.collegeboard.org, 46% of the responses were correct.

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

Triangles

Geometry: Triangles

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

You should start by reading the problem carefully and identifying your bottom line.  Then assess your options and choose the most efficient method to attack the problem.  Finally, loop back to make sure that the answer you found matches the bottom line that you set out to find.

Which of the following CANNOT be the lengths of the sides of a triangle?

Bottom Line: The word “CANNOT” tells you that you are looking for something that is not true.

Assess your Options:  Problems that ask you to find what cannot be true are often impossible to predict; you will have to look at the answer choices to determine whether they work.  Go ahead and look at the answers.

(A) 1, 1, 1
(B) 1, 2, 4
(C) 1, 75, 75
(D) 2, 3, 4
(E) 5, 6, 8

You could try to use logic and your experience with triangles to eliminate some choices.  For example, answer choice A is not the answer because you know there is such a thing as an equilateral triangle and the sides 1, 1, and 1 would create that kind of triangle.  However, after that point you would probably just be guessing.  If you know the Triangle Inequality Theorem, you can systematically check each answer choice.

Attack the problem:  When you are dealing with three sides of a triangle and you do not know that the triangle is a right triangle, you should always think of the Triangle Inequality Theorem.  This theorem states that for any triangle, side x is less than the sum and greater than the difference of the other two sides.  In other words, each side of the triangle must be less than the other two sides added together and greater than the difference of the other two sides.  If you have a triangle with sides x, y and z, you would write the theorem this way:
|yz| < x < y + z. For the subtraction part you can use absolute value or just always do the bigger side minus the smaller side; the result will be the same. The easiest way to think about the theorem is this: for any triangle, other sides subtracted < one side < other sides added.

Start by checking (E).  It has sides 5, 6, and 8.  Plug these sides into your formula by using the first side, 5, as your x.  Make sure that 5 is greater than the difference of the other two sides, but smaller than the other two sides added together.

|y – z| < x < y + z
8 - 6 < 5 < 6 + 8 
2 < 5 < 14
This is true!  (E) works as a triangle.

Now check (D). It has sides 2, 3, and 4.  Plug those numbers into the Inequality Theorem.
 |y – z| < x < y + z
4 – 3 < 2 < 3 + 4
1 < 2 < 7
This is true!  (D) works as a triangle.

Now check (C).  It has sides 1, 75, and 75.  Plug those numbers into the Inequality Theorem.
|y – z| < x < y + z
75 – 75 < 1 < 75 + 75
0 < 1  < 150
This is true!  (C) works as a triangle.

Now check (B).  It has sides 1, 2, and 4.  Plug those numbers into the Inequality Theorem.
|y – z| < x < y + z
4 – 2 < 1 < 2 + 4
2 < 1 < 6.
Is 2 less than 1?  No!  This is false.  You cannot have a triangle with these three side lengths.

Loop Back:  Your bottom line was to find an answer choice that cannot be a triangle, so you are finished!

The correct answer is (B).

On sat.collegeboard.org, 49% of the responses were correct.

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

Quadrilaterals

Geometry: Quadrilaterals 

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

Math questions should always be read carefully.  You will also avoid making errors by identifying the bottom line and assessing your options for solving the question.  Choose the most efficient method to attack the problem.  When you have finished, loop back to be sure that even if there were multiple steps, you reached the bottom line.

The length of a rectangle is increased by 20%, and the width of the rectangle is increased by 30%. By what percentage will the area of the rectangle be increased?

Bottom Line: % change = ?

Assess your Options:  You could work this problem without picking any numbers; however, picking easy numbers will allow you to think about the problem in a more concrete way and avoid errors.

Attack the Problem:  One of the easiest numbers to work with is one.  Think of your original rectangle as having a length of one and a width of one.  The formula for area of a rectangle is length times width.  If L × W = A, for your first rectangle you have 1 × 1 = 1.   The area of the original rectangle is one.

Then think about the changes that occur to that rectangle.  The length increases by 20%.  In order to find 20% of 1, all you have to do is move the decimal over twice to .2.  The new length is 1.2.  Use the same method to find the new width, and an increase of 30% becomes 1.3.  The area of the rectangle after the change is 1.2 × 1.3 = 1.56

The formula for percent change would require you to find the difference between these two areas and divide that by the original number.  You use the same formula whether you are looking for an increase or a decrease.  Notice that your original number is one, so dividing by one will not change your answer.  All you need to do is find the difference between the areas: 1.56 – 1 = .56.  What is .56 as a percent?  Your answer is 56%.

Loop Back:  You found the percent change, which was your bottom line.  Look down at your answer choices.

(A) 25%
(B) 36%
(C) 50%
(D) 56%
(E) 60%

The correct answer is (D).


On sat.collegeboard.org, 34% of the responses were correct.

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

Coordinate Geometry

Geometry:  Coordinate Geometry

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

Use the same method for all the math questions on the SAT.  First, read the question carefully to avoid making mistakes.  Identify the bottom line and assess your options for reaching it.  Next, choose an efficient method to attack the problem.  Finally, loop back to make sure that your answer addresses the bottom line.  Many problems have multiple steps.

If the graph of the function f in the xy-plane contains the points (0, -9), (1, -4), and (3, 0), which of the following CANNOT be true?

Bottom Line:  You are looking for something false.

Assess your Options:  You could try drawing an xy-plane and graphing the points to help you visualize the question, but your graph may be inaccurate without graph paper.  Instead, try to find the relationship between the three points.

Attack the problem:  To find the relationship between these points, you will need to find the slope of the line between each point.  The formula for slope is:  

 




Check the slope of the line between (0, -9) and (1, -4):


Then check the slope of the line between (1, -4) and (3, 0):


The function in this problem has a very steep slope between the first two points, but becomes less steep between the second two.  This is a “which of the following” question, so start with answer (E) as you work through your answer choices.

(A) The graph of f has a maximum value.
(B) y ≤ 0 for all points (x, y) on the graph of f.
(C) The graph of f is symmetric with respect to a line.
(D) The graph of f is a line.
(E) The graph of f is a parabola.

(E) The function could be a downward facing parabola if it continues to the right.  You are only given three points, but there could be many more points on this function. 

(D)  In geometry, a line is always straight, without any curves.  Notice that there are different slopes connecting the three points.  You cannot draw one straight line through all three of these points, so this choice cannot be true.

Loop Back:  Your goal was to find an answer choice that was false.  You did so, so you are finished!  If you have extra time, you can check the other answer choices and see that they are all possible, depending on how you draw the rest of the function.  (E), (C), (B), and (A) could all describe a downward facing parabola with the equation y = -(x – 3)².

The correct answer choice is (D).


On sat.collegeboard.org, 42% of the responses were correct.

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