# Pattern Problems

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

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 , 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

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 $\frac{2}{3}$. 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:

$\frac{relevant\: outcomes}{total\: outcomes}$

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.

$\frac{r}{g + r}=\frac{r}{36+r}$

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.

$\frac{r}{36+r}=\frac{2}{3}$
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

On sat.collegeboard.org, 47% 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.

Read each question carefully and identify the bottom line to avoid making careless mistakes.  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.

If f(x) = x + ax, and $a =\frac{7}{2}$ what is $f(\frac{3}{2})$?

Bottom Line$f(\frac{3}{2})=?$

Assess your Options:  You could use your graphing calculator to solve this problem, but it would probably take you more time to type in the fractions than to just solve the problem.  You are given a value for each variable in the problem so all you need to do is plug them in.

Attack the Problem:  Start by plugging in the value of a to the function that you were given.

$f(x)=x+ax$
$f(x)=x+\frac{7}{2}x$

Simplify the problem by adding.  Remember that the first x is a whole 1x, but that you must have like terms before you can add fractions.

$f(x)=\frac{2}{2}x+\frac{7}{2}x$
$f(x)=\frac{9}{2}x$

Now solve your function by plugging in the value for x that you were given.

$f(\frac{3}{2})=(\frac{9}{2}\)(\frac{3}{2})$
$f(\frac{3}{2})=\frac{27}{4}$

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

(A) $\frac{1}{3}$
(B) $\frac{3}{2}$
(C) $\frac{7}{2}$
(D)$\frac{21}{4}$
(E) $\frac{27}{4}$

On sat.collegeboard.org, 61% 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.

Use the same method for every math question on the SAT.  Start by reading the question carefully and identifying the bottom line; what do you need to find?  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 it matches the bottom line.

In the xy-plane, the graph of the line with equation y = a intersects the graph of the quadratic function f(x) = x² - 6x + 8 in exactly one point. What is the value of a?

Bottom Line: a = ?

Assess your Options:  You could just try plugging this into your calculator, but if you do not think carefully about what you are doing, you are likely to answer a question that was not asked.  Instead, think through every piece of information that you were given in this problem.

Attack the Problem:  What kind of graph is the function that you are given?  A parabola!  You know this because it has an x².  Picture a parabola in your mind (you know that this is a normal, upward-facing parabola because there is no negative before the x²).  Draw a u-shaped parabola on the xy-axis as part of your scratch work.

Now think about the fact that when y equals a certain number, it creates a vertical line. No matter what y equals, that vertical line will only ever intercept the graph at one point. That's not very useful! However, try flipping the given equation on its head: consider a = y. Remember that a =  is just like x =  and will create a horizontal line. Depending on what x equals, the horizontal line might cross the graph at two points, at no point at all, or at exactly one point--the vertex. You know that you must find the vertex of the parabola, so solve your function for x by setting your polynomial equal to zero and finding the roots of the equation:

x² - 6x + 8 = 0
(x – 2)(x – 4) = 0
(x – 2) = 0 and (x – 4) = 0
x = 2 and x = 4

You just found the two places where the parabola crosses the x-axis: 2 and 4.  All parabolas are symmetrical.  That means that the vertex must be halfway between these two numbers at x = 3.  You found the x value of the vertex, but you need the y value.

Plug in 3 for the x in your original equation:

f(x) = x² - 6x + 8
f(3) = (3)² - 6(3) +8
f(3) = 9 – 18 + 8
f(3) = -1

Loop Back:  When you solve a function for the f(x), you solve for y.  In this problem, you are told that y = a.  You have solved for a, so you are ready to look down at your answer choices.

(A) -3
(B) -1
(C) 1
(D) 3
(E) 4

On sat.collegeboard.org, 31% of the responses were correct.

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

# Scatterplots

## Data Analysis: Scatterplots

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

Always read the question carefully so that you can glean as much information from it as possible.  Identify the bottom line – what is it asking?  Then assess your options and choose the most efficient method to attack the problem.  When you have a solution, loop back to make sure that it matches your bottom line.

The scatterplot above shows the number of items purchased at a grocery store by 28 customers and the total cost of each purchase. How many of these 28 customers bought more than 10 items and spent less than $20? Bottom Line: # of people. Notice that this number must reflect those that meet 2 requirements: buying more than 10 things and spending less than$20.

Assess your Options:  When you have a graph, use the graph!  You can draw directly on it to help you visualize what you need.

Attack the Problem:  The dots represent each person.  Start with the first restriction.  If people must buy more than 10 items, then only the dots to the right of the 10 on the horizontal axis will be counted; those on the line do not count because they are equal to 10 rather than more than 10.  Draw a vertical line on the 10.  Now look at the second restriction.  If the people must spend less than $20, then they must be under the$20 hash mark on the vertical axis.  Draw a line at \$20.  Your graph should look like this:

Count the number of dots in the lower right hand region that you created.  Your answer is 4.

Loop Back:  Each dot represents a customer, a person, so you reached your bottom line.  Look down at your answer choices.

(A) Four
(B) Five
(C) Six
(D) Seven
(E) Eight

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

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

# Rates

## Arithmetic: Rates

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

Use the same process for every math problem so that you are not intimidated by any question.  (1) Read the question carefully.  (2) Identify the bottom line – what is the question asking?  (3) Take the time to assess your options – which methods can you use to solve this problem most efficiently?  (4) Attack the problem and work though it logically.  (5) Loop back to make sure that your answer matches the bottom line – did you complete every step of the problem?

A train traveling 60 miles per hour for 1 hour covers the same distance as a train traveling 30 miles per hour for how many hours?

Bottom line: Make a quick note that you are solving for hours: hrs = ?

Assess Your Options:  You could try to use logic for this problem by thinking that if a train goes more slowly, it must take longer to go the same distance as it did at a faster speed.  Unfortunately, logic will not eliminate all of your answer choices.  Use the distance formula to solve this problem.

Attack the problem:  The distance formula is distance is equal to rate(speed) times time:  D = R × T.  Start with the first train and multiply the rate (60 m/hr) by the time (1 hr) to get the distance:

60 × 1 = 60

The first train traveled 60 miles.  You know that both trains traveled the same distance, so plug in 60 as the distance for the second train. You also know that the rate is 30 and the time is unknown.  That should look like:

30 × T = 60
30T = 60
T = 2

Note:  If you are good at balancing equations, there is an even faster way to do this problem.  Look at the distance equation:  D = R × T.  If the distance for a problem stays the same, but you increase the speed (rate), then you must decrease the time by the reciprocal of the speed increase.  That keeps the equation balanced.  Ex:  If you double the speed, you must halve the time.  In this particular problem you halve the speed (from 60 to 30), so you must double the time.  2 × 1 hour = 2 hours.  This reciprocal rule will always work!

Loop Back:  You solved for the time of the second train, which is already in hours, so you are ready to look at your answer choices!

(A) 3
(B) 2
(C) 1
(D)
(E)

On sat.collegeboard.org, 78% of the responses were correct.

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

# Percents

## Arithmetic: Percents

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

Always read math questions carefully so that you can absorb all the information and avoid mistakes.  Identify the bottom line, what the question is asking you to find, and assess your options for reaching that bottom line.  Choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer matches the bottom line and you have finished all the steps in the problem.

If p percent of 75 is greater than 75, which of the following must be true?

Bottom Line: p = ?

Assess your Options:  It is often tempting to look down at the answer choices before you need them, but they could mislead you since most of them are wrong!.  You could take numbers that fit each answer choice and see if they give you a number greater than 75.  However, by applying what you know about percents, you can solve the problem much faster than you can by trying out 5 different numbers.

Attack the Problem:  There are a number of ways to think about percentages: as percents, decimals, numbers out of a hundred, parts of wholes….  The list continues.  Here is one of the fastest ways to think about the problem:

If you have one hundred percent of something, you have all of it.  So 100% of 75 is going to be 75.  If you want a result that is greater than 75, you are going to need more than 100% of 75.  Therefore, p must be bigger than 100.

Or, if you normally think about percents in terms of decimals, you know that 50% of something is .5.  In order to get a decimal from a percent, you had to move the decimal twice to the left.  So with 100%: 75 × 1.00 = 75.  Try writing an inequality to find the decimal that you would need in order to get a number bigger than 75: 75p > 75.  The p represents the unknown percent of 75 (remember, "of" means multiplication in math).  If you solve the inequality, you get p > 1.  Then you have to move the decimal back in order to get a percent: p > 100.  Your percent must be bigger than 100%.  This method takes much longer than the first one, but it proves that the first method is correct.  The testers realize that students are not used to working with percentages greater than 100, so it is a good idea to review how these work before the test!

Loop back:  You know what p must be greater than, so look down at your answer choices.

(A) p > 100
(B) p < 75
(C) p = 75
(D) p < 25
(E) p = 25

On sat.collegeboard.org, 71% 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 .

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

(A) 0

(B)

(C)

(D)

(E)

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

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

# Rates

## Arithmetic: Rates

Use the same method for every math problem on the SAT.  Read the problem carefully, identify the bottom line, and assess your options for solving the problem.  Choose the most efficient method to attack the problem.  Often there will be multiple steps to a single problem, so when you have an answer, be sure to loop back and verify that it matches the bottom line.

A machine can insert letters in envelopes at the rate of 120 per minute. Another machine can stamp the envelopes at the rate of 3 per second. How many such stamping machines are needed to keep up with 18 inserting machines of this kind?

Bottom Line:  # stamping machines = ?

Assess your Options:  You could try to work backwards from the answers, but there is no need.  It will be faster just to solve the problem.

Attack the Problem:  You have been given two different units of time: minutes and seconds.  There are 60 seconds in a minute.  Changing the minutes to seconds will be easiest, so the letter inserting machine works at a rate of 120 letters per 60 seconds.  120 divided by 60 is 2 letters in envelopes per second.  If there are 18 letter inserting machines, then together they will insert 36 letters in envelopes per second (2 × 18 = 36).

You don’t know how many stamping machines you need, so use x to represent that number.  Stamping machines have a rate of 3 envelopes per second, so each machine will finish 3 envelopes in a second.  You know that the stamping machines must keep pace with the 18 letter inserting machines that finish 36 envelopes per second, so the outcome must be 36.  Write 3x = 36.  When you solve for x,  x = 12.

Loop Back:  You used x to represent the number of stamping machines, your bottom line, so you are ready to look at the answer choices.

(A) 12
(B) 16
(C) 20
(D) 22
(E) 24

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

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

# Coordinate Geometry

A new year symbolizes a new start for many.  Although the world is essentially the same as it was before the clock struck midnight, there is a new optimism about the future.  People want to focus on goals such as peace and prosperity.  Read this current event about an unexpected gesture from North Korea, and then ask yourself what you can expect from 2013.  What themes can you identify in this article that are likely to be part of an SAT essay question?

## Geometry: Coordinate Geometry

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

Always read the question carefully and identify the bottom line.  Assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that it addresses the bottom line.

In the figure, the slope of the line through points P and Q is $\frac{3}{2}$. What is the value of k?

Bottom Line: k = ?

Assess Your Options:  You could start from the point (1, 1) and use the slope to find new points, hoping that by adding 3 to the y value and 2 to the x value you will reach a point that contains a 7 y value.  Unfortunately, it is very easy to make a mistake using this method, such as adding the y change to the x value or vice versa.  Instead, use the information that you are given, the slope, to write an equation.

Attack the problem:  Although you are given the slope, you also know how the slope was obtained.  Think about it:  The slope is rise over run or the change in y over the change in x
$slope=\frac{rise}{run}=\frac{\bigtriangleup y}{\bigtriangleup x}=\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}$
You know two different y values, and two different x values, so you can plug in all the information that you know for the slope.
$slope =\frac{y_{2}\, -\, y_{1}}{x_{2}\, -\, x_{1}}=\frac{7-1}{k-1}$
Now you need to set this formula for slope equal to the value for slope that you were given in the problem, isolate the variable k, and solve for it.
$\frac{7-1}{k-1}=\frac{3}{2}$
$\frac{6}{k-1}=\frac{3}{2}$
3(k – 1) = 6 × 2
3k – 3 = 12
3k = 15
k = 5

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

On sat.collegeboard.org, 51% of the responses were correct.

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

# Multiple Figures

## Geometry: Multiple Figures

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

Use the same steps for every math problem.  First, read the question carefully and identify the bottom line.  Next, assess your options and choose the most efficient method to attack the problem.  Finally, loop back to verify that your answer addresses the bottom line.

In the figure above, if PQRS is a quadrilateral and TUV is a triangle, what is the sum of the degree measures of the marked angles?

Bottom Line:  Sum of degrees of the marked angles = ? (Write Sd = ?)

Assess your Options:  You could try to find the individual angles, but you don’t have enough information to do this.  Instead, use the rules you have memorized about each shape.

Attack the Problem:  You know that TUV is a triangle.  All the angles of a triangle add up to 180 degrees.  You know that PQRS is a quadrilateral.  All the angles of a quadrilateral add up to 360 degrees.  In the image, you can see that all of these angles in each of these two shapes are marked, and you know that you are looking for a sum, so add them together.  180 + 360 = 540.

Loop back: Your answer is in degrees and you have found the total of all the marked angles.  Look down at your answer choices.

(A) 420
(B) 490
(C) 540
(D) 560
(E) 580

On sat.collegeboard.org, 77% of the responses were correct.

For more help with SAT writing, visit www.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
$x=\frac{3}{7}$

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.)

$2\times40\times \frac{3}{7}=Total\; distance$

$\frac{240}{7}=Total\; distance$

$34\frac{2}{7}=Total\; distance$

(A) 30
(B) $30\frac{1}{7}$
(C) $34\frac{2}{7}$
(D) 35
(E) 40

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

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

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

# Writing Equations

## Algebra: Writing Equations

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

Use the same method for all SAT math questions.  Read the question carefully, identify the bottom line, assess your options for reaching the bottom line, and choose the most efficient option to attack the problem.  When you have an answer, loop back to make sure that it matches your bottom line.

A geologist has 10 rocks of equal weight. If 6 rocks and a 10-ounce weight balance on a scale with 4 rocks and a 22-ounce weight, what is the weight, in ounces, of one of these rocks?

Bottom line: Remember to write your bottom line in easy-to-understand shorthand. You could write "weight of 1 rock = ?" but "w = ?" is much shorter.

Assess your options: You could try each of your answer choices in this scenario, but that will waste time because you will most likely need to try multiple answers.  Start by writing an equation so that you only have to solve one problem.

Attack the problem:  On one side of the scale you have 6 rocks and a 10 oz. weight.  You don’t know how much each rock weighs, so you will need to add a variable to represent that number.  There are 6 of that missing weight (w), plus 10 oz.

6w + 10

All of this balances with, is equal to, 4 rocks of the same weight plus 22 oz.

6w + 10 = 4w + 22

Now solve the equation that you wrote by combining like terms and isolating the variable.

2w + 10 = 22
2w = 12
w = 6

Loop Back: You solved for the weight of one rock, so you are ready to look down at your answer choices.

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

On sat.collegeboard.org, 67% of the responses were correct.

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

# Number Line

## Arithmetic: Number Line

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

Work all math problems by reading the question carefully and identifying the bottom line.  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 it satisfies the bottom line.

A, B, C, and D are points on a line, with D the midpoint of segment . The lengths of segments , and  are 10, 2, and 12, respectively. What is the length of segment ?

Bottom Line: distance A to D

Assess your Options:  Drawing out the situation will give you a visual to understand the situation.

Attack the Problem:  Start with what you know.  You have a lot of points named, but the first information that you are given is that D is the midpoint between B and C.

Now you are given three lengths.  You can’t label the ones involving A yet, but you can label the length from B to C.  Remember that D is the midpoint, and you will also know the lengths of B to D and D to C.

Now go back to those other lengths you were given that involved point A.  Point A is 2 units away from C and 10 units away from B.  The only possible location for A is between B and C, but closer to C.

Now that you have all your points labeled, it is time to go back and look for your bottom line.  What is the distance from A to DD to C was 6 units, and A to C was 2 units, so what is 6 minus 2?  The answer is 4.  (You could also use B to A is 10 and subtract the length of B to D, 6, and get the same answer of 4.)

Loop Back:  You solved for the distance from A to D, so you are ready to check your answers.

(A) 2
(B) 4
(C) 6
(D) 10
(E) 12

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

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

# Writing Equations

## Algebra: Writing Equations

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

Read the question carefully so that you don’t miss any important information.  Identify the bottom line and assess your options to find it.  Choose the most efficient method to attack the problem.  Always loop back to make sure that your answer addresses the bottom line.

Milk costs x cents per half-gallon and y cents per gallon. If a gallon of milk costs z cents less than 2 half-gallons, which of the following equations must be true?

Bottom Line: equation

Assess your Options:  The question asks you about "the following equations," so your first instinct is going to be to look down at the answer choices.  Don’t do it!  Most of them are wrong and they are there to distract you from the correct answer.  Instead, write your own equation using what you know from the problem.

Attack the Problem: Start with what you know: “Milk costs x cents per half-gallon and y cents per gallon.”  Make a note:

x = half-gallon
y = gallon

Now look at the conditions that you are given “a gallon of milk costs z cents less than 2 half-gallons.”  The word “costs” is just like the word “is;” it shows you where to put the equal sign.  The words “less than” signal that you will need to subtract the z. Use the variables you have been given to write an equation.

y = 2x – z

Once you have an equation, glance down at your answer choices.  Notice that all of them are set equal to zero, and all the x values are positive.  Set your equation equal to zero and keep the x value positive by subtracting the y variable from each side.

0 = 2xz – y

As you look at your answer choices, realize that when you are adding and subtracting numbers, order does not matter.  In fact, all of the answers have the variables arranged alphabetically.  Do the same to your equation.

0 = 2x – y – z

Loop Back:  You can be confident in your answer because you reached it by writing your own equation.

(A) x – 2y + z = 0
(B) 2xy + z = 0
(C) x – y – z = 0
(D) 2x – y – z = 0
(E) x + 2y – z = 0

On sat.collegeboard.org, 56% 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.

Approach all math questions the same way so that you can be confident in your method.  Start by reading the question carefully and making a note of the bottom line – the answer that you must find.  Then, assess your options and choose the most efficient method to attack the problem.  When you have found an answer, loop back to make sure that it is your bottom line.

In the triangles above, 3(y – x) =

Bottom Line: 3(yx) = ?  (Don’t solve for x or y and think that you are finished!)

Assess your Options:  The wonderful thing about geometry questions is that there is often more than one way to get to an answer.  The tricky thing is that using some geometry rules will take longer than others.  For example, you could use the rule that all degrees in a triangle add up to 180 degrees.  Then you would write out an equation to solve for the missing variables in each triangle.  This is the method used on collegeboard.org.  However, if you have special triangles memorized, you can save a lot of time.

Attack the Problem:  The first triangle is a right isosceles triangle.  You know this because it has one right angle, and the other two angles are equal.  This is a special triangle that is very common, so you should memorize the fact that its angles measure 45, 45, and 90 degrees.  The x is equal to 45.

Now look at the second triangle.  It is an equilateral triangle.  You know this because all three angles are equal.  You should memorize the fact that all the angles in an equilateral triangle equal 60 degrees.  The y is equal to 60.

Now that you know what the x and y are, plug these numbers into your equation.

3(yx) =
3(60 – 45) = 45

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

(A) 15
(B) 30
(C) 45
(D) 60
(E) 105

On sat.collegeboard.org, 77% of the responses were correct.

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

# Writing Equations

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

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

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

# Ratios

Freedom of the press has long been a hallmark of liberty in the United States.  However, many of the released SAT essay prompts have to do with balancing public and private lives, knowledge as a burden, and the abundance of information available through better technology.  Here is a current event that relates to all of these ideas:  a judge in the UK is calling for an independent group to regulate the press.  Think carefully about this current event, and decide where you stand on the issues that are raised.  If you decide to use this as one of your five current events, you will need to prepare a list of relevant details about this news story and a list of the broad topics that would let you know that this example relates to your essay prompt.

## Arithmetic: Ratios

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

Use the same method for every math question on the SAT.  Start by reading the question carefully and identifying the bottom line.  Next, assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that it matches the bottom line that you were asked to find.

Miguel is 180 centimeters tall. At 2:00 p.m. one day, his shadow is 60 centimeters long, and the shadow of a nearby fence post is t centimeters long. In terms of t, what is the height, in centimeters, of the fence post?
Bottom line: fence post = ?
Assess your Options:  Collegeboard.org uses a method that includes drawing the person and the post and then creating two right triangles.  That is a waste of time.  All you need to do is recognize that this is a ratio problem and set up your ratios correctly.
Attack the problem:  Set up the labels that you will use in your ratios so that you do not get confused about which number represents the length of the actual person and which number represents the length of his shadow.  You can set it up as actual divided by shadow:
$\frac{actual\: height}{shadow} = \frac{person}{his\: shadow} = \frac{post}{its\: shadow}$

Plug in the values that you know from the problem.  The only value that you do not know is the height of the post.  Leave that as a question mark so that you know which variable you must isolate.

$\frac{actual\: height}{shadow}= \frac{180}{60} = \frac{?}{t}$
All you need to do now is solve for the height of the post.  If you divide 180 by 60, the answer is 3.  You will need to multiply both sides by t to isolate the variable.  (Remember, you are solving for the question mark!)  Then you have your answer.

$3=\frac{?}{t}$

$3t = ?$
Loop Back:  The ? represented the height of the post in your original ratio, so you solved for your bottom line.  Look down at your answer choices.

(A) t + 120
(B) $\frac{t}{3}$

(C) 3t

(D) $3\sqrt{t}$
(E) $(\frac{t}{3})^{2}$