# ACT Math

## SAT Question of the Day

The SAT question of the day is a Sentence Completion Question that has already been addressed on this blog:  click here to see an explanation.

## ACT Math Question of the Day

Many ACT math questions are exactly like SAT questions.  Use the same process as you would to answer an SAT question.  Read the question carefully, and identify the bottom line.  Assess your options and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

There are students in a class. If, among those students, p% play at least 1 musical instrument, which of the following general expressions represents the number of students who play NO musical instrument?

Bottom Line:  #kids no musical instrument = ?

Assess your Options:  You could write an equation using the variables that you are given, but many students make mistakes using this method.  Instead, use the strategy of plugging in numbers to make sure that you arrive at the correct answer.

Attack the problem:  When you have a percent problem, use the number 100 for any total that you do not know.  This makes the problem easier because a percent is just a number out of one hundred.  If you start with the number of 100, your answer will automatically be out of 100!

Look up at the problem.  There are students in the class, so let = 100.  You still have another variable, p.  Pick a number for p as well.  It must be less than 100, but not too difficult for this problem, so let’s pick = 30.

Answer the question using the numbers you have chosen.  If you have 100 students and 30 play at least one musical instrument, how many do not play any musical instrument?  70!  100 – 30 = 70.

Now you need to look down at your answer choices and see which choice equals 70 when you plug in = 100 and = 30.

A.  np

B.  .01np

C.

D.

E.  100(1 –p)n

Loop Back: You are just looking for a matching number.

A.  np  = 100(30) = 3,000, not 70

B.  .01np = .01(100)(30) = 30, not 70

C

: Plug in = 100 and cancel the 100 on the top and bottom of the fraction.  You are left with 100 – 30 = 70.  On the actual test, there would be no reason to check any of the other answers, but you can practice working the remaining answer choices now.

D.

=

= –290,000, not 70

E.  100(1 –p)= 100(1 - 30)(100) = –290,000, not 70

For the ACT Question of the Day, visit

http://www.act.org/qotd/

.

To get help preparing for the SAT, PSAT, or ACT Exam, visit www.myknowsys.com!

# Inequalities

## Algebra: Inequalities

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

Always read the question carefully, identifying the bottom line.  Assess your options for reaching the bottom line and use the most efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.

On the line above, if AB < BC < CD < DE, which of the following must be true?

Bottom Line: wotf must be true = ? (which of the following)

Assess your Options:  For a “wotf” question, you will have to look at the answer choices.  Most students will start with “A,” so Knowsys recommends that you start with “E.”  You may also find that this is a good problem to use the strategy of plugging in numbers.

(A) AC < CD
(B) AC < CE
(E) BD < DE

(E) BD < DE  Look up at the figure.  On the figure, does BD look smaller than DE?  No!  It looks slightly larger.  You know that the figure is not drawn to scale, but the figure does give you one possible depiction of the rule.  Use the figure!  If it is possible for BD to be bigger than DE, then this answer is incorrect because you are looking for something that must be true.  Eliminate this choice.

(D) AD < DE  Look up at the figure.  The figure shows you that it is possible for AD to be larger than DE.  Eliminate this choice.

(C) AD < CE  These lengths are very similar on the line.  Break each length down into the parts that compose it so that you can make a precise comparison.  For example, AD contains AB + BC + CD.  CE contains CD + CE.  You now have: AB + BC + CD < CD + DE.  When you have the same thing on both sides of an equation, it cancels.  Eliminate the CD.  You now have AB + BC < DE.
You cannot come to a conclusion about these lengths.  If you want to prove this, try plugging in numbers.  Suppose AB starts at 10 and each portion along this line gets larger by 1.  AB = 10, BC = 11, CD = 12, DE = 13.  Is 10 + 11 < 13?  No.  Eliminate this choice.
(B) AC < CE  This one looks like it could be true, based on the figure.  See if you can prove it.  Break it down into its parts just as you broke down the last answer choice.  AC contains AB + BC.  CE contains CD + DE.  At first it seems as if you cannot compare these either because all of the numbers are different.  Try plugging in the same values as you used before: AB = 10, BC = 11, CD = 12, DE = 13.  Is 10 + 11 < 12 + 13?  Yes!  Will this work for all numbers?  Yes!  You are adding a small number plus a medium number and comparing it to a big number plus an even bigger number.  The former will always be smaller than the latter.  Once you know this, you do not even need to check (A).

(A) AC < CD You can tell from the figure that this does not have to be true.

Loop back:  You solved for what must be true, so you should select the answer you found.

On sat.collegeboard.org, 68% of the responses were correct.

To get help preparing for the SAT exam, visit www.myknowsys.com!

## ACT Question of the Day:

If you have gone 4.8 miles in 24 minutes, what was your average speed, in miles per hour?

Your bottom line here is in miles per hour.  That would be miles over hours.  Your distance (miles) is in the correct unit, but your time (minutes) is not.  You know that there are 60 minutes in an hour.   Find the fraction of an hour that was spent traveling. Take your minutes and put them over the total minutes in an hour:

$\frac{24\: min}{60\: min}=\frac{2}{5}\: of\: an\: hour =.4\: hr$

Now you know that you went 4.8 miles in .4 hours.  How many miles per hour was that?  Divide 4.8 by .4 and you will see that the answer is 12.

Note: You can do this in your head if you realize that this is the same thing as dividing 48 by 4.  This whole problem can be done in seconds if you know your times table all the way up to 12.

(A)  5.0
(B) 10.0
(C) 12.0
(D) 19.2
(E) 50.0

For the ACT Question of the Day, visit http://www.act.org/qotd/.

To get help preparing for the ACT exam, visit
www.myknowsys.com!

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

Always read the question carefully and identify the bottom line.&nbsp. Assess your options for reaching the bottom line – what is the easiest and most time-efficient method to reach the answer? Use that method to attack the problem. When you have an answer, loop back to make sure that you reached the bottom line and did not just solve a portion of the problem.

If $\sqrt{x}=16$, what is the value of $\sqrt{4x}$?

Bottom Line: $\sqrt{4x}=?$

Assess your Options: You might be tempted to find the value of x first, but look at your bottom line. Do you need to know the value of x? No! Don’t waste your time! You just need to know the value of the square root of x multiplied by another number. Use your knowledge of radicals to rearrange your bottom line so that you have fewer steps to solve the problem.

Attack the Problem: Focus on the 4 under the radical. If this question simply asked for the square root of 4, you could easily answer. What is the square root of 4? 2! That value now goes in front of the radical. This could also be written as 2 multiplied by the square root of x. Plug in the value of 16 that you were given for the square root of x. All you have to do to reach a single number is multiply 2 by 16. Here are the steps you just completed:

$\sqrt{4x}= 2\sqrt{x}=(2)(\sqrt{x})=(2)(16)=32$

This method is much easier and faster than finding that x = 256, multiplying 256 by 4, and then taking the square root of 1024. You should not need to waste time typing numbers into a calculator in order to solve this problem.

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

$\sqrt{4x}= 2\sqrt{x}=(2)(\sqrt{x})=(2)(16)=32$

$\sqrt{4x}=?$

(A) 16

(B) 32

(C) 64

(D) 128

(E) 256

The correct answer is (B). On sat.collegeboard.org, 52% of the responses were correct.

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

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 $\sqrt{x-a}=\sqrt{x+b}$ , 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

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

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

# Absolute Value

Happy St. Patrick’s Day!  If you like celebrating holidays, consider tracing the history of a holiday celebration as one of your historical events, or even tracing the history of a holiday up to the present for a current event.  There are many misconceptions about the origins of ideas and traditions, and the way that certain practices came about is fascinating.  Think about how this article about the history of St. Patrick’s day could be used to answer the following SAT essay questions:

(1)  Should people change their decisions when circumstances change, or is it best for them to stick with their original decisions?
(2)  Do you think that ease does not challenge us and that we need adversity to help us discover who we are?
(3)  What motivates people to change?

## Algebra: Absolute Value

Read the following SAT test question and then select the correct answer.
If , which of the following could be true?

Bottom Line: which of the following COULD be true?

Assess your Options:  You could come up with answers to solve this problem, but that will be a waste of time if they are not included in your answer choices.  Instead, look at the answer choices and methodically eliminate incorrect answers.

(A) a = 0
(B) b = 0
(C) a = b
(D) a = -b
(E) a = 1

For a “which of the following” question, Knowsys recommends that you begin with answer choice (E).

Hint:  Instead of thinking of the bars in your equation as absolute value, think of them as simply showing where a positive number will be.  If you do this, you will not have to plug in actual numbers and you can check each answer choice using logic. (This works because an absolute value simply tells you how far a number is from zero.  The double bars only affect negative numbers, making them positive.)

(E) a = 1    Plug a = 1 into your original equation.   Is there any way to start out with the number 1 and subtract a positive number to get the answer 5?  There is not.  Eliminate this answer.

(D) a = -b     This answer choice has a negative sign, but remember that any negative sign will go within the bars and come out a positive number.  So if you plug in b where the variable a is in this equation, you still end up with bb = 5.  Is that possible?  No!  Anything minus itself will be zero.  Eliminate this answer.

(C) a = b     This answer is essentially the same as the last one!  If you plug in b where you have an a, you wind up with bb = 5.  Again, anything minus itself will be zero.  Eliminate this answer.

(B) b = 0     Plug in 0 for the b in your equation.  You now have a positive number minus 0 equals 5.  Is that possible?  Yes!  5 – 0 = 5.  You are finished.  You don’t have to know that a can be either negative 5 or 5, and you don’t have to check the last answer choice.  Let’s check it just for practice.

(A) a = 0     Is there any way to start with 0 and subtract a positive number to get 5?  No!  Eliminate this choice.

On sat.collegeboard.org, 63% 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 $\frac{a}{b}=3$ and $\frac{b}{c}=7$, then $\frac{a+b}{b+c}$ is equal to which of the following?

Bottom line:  $\frac{a+b}{b+c}$

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:

$\frac{a+b}{b+c}=\frac{3b+b}{b+c}=\frac{4b}{b+c}$

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.

$\frac{b}{c}=7$ so  $\frac{c}{b}=\frac{1}{7}$ so $c =\frac{1}{7}b$

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

$\frac{4b}{b+c}=\frac{4b}{b+\frac{1}{7}b}=\frac{4b}{\frac{8}{7}b}$

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:

$\frac{4b}{\frac{8}{7}b}=4b(\frac{7}{8b})=4(\frac{7}{8})=\frac{28}{8}=\frac{7}{2}$

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.

$\frac{b}{c}=7$ so $\frac{b}{1}=7$ 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.

$\frac{a}{b}=3$ so $\frac{a}{7}=3$ so a = 21.

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

$\frac{a+b}{b+c}=\frac{21+7}{7+1}=\frac{28}{8}=\frac{7}{2}$

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

(A) $\frac{7}{2}$
(B) $\frac{7}{8}$
(C) $\frac{3}{7}$
(D) $\frac{1}{7}$
(E) 21

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

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

# Writing Equations

As you prepare for college, one of the best things that you can do for yourself, outside of studying, is to build good relationships with your teachers.  Learning the proper way to ask for help from your teachers can mean the difference between finally understanding a concept and getting written off as a whiner.  Read this article and think about how you can use the given advice not just in the future, but in your classes right now.

## Algebra: Writing Equations

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.  Assess your options for finding the bottom line 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.

A florist buys roses at $0.50 a piece and sells them for$1.00 a piece. If there are no other expenses, how many roses must be sold in order to make a profit of $300? Bottom Line: # roses = ? Assess your Options: You could find the profit from a single rose and then start plugging in answer choices, but that is not the fastest way to solve this problem. A better way to solve this problem is to simply write an equation. You could also solve this problem in a few seconds by using logic. Attack the Problem: Writing an equation will not take you much time. Start by finding the profit from a single rose:$0.50.  (You know that the florist spends $0.50 to make each dollar, so$1.00 - $0.50 =$0.50.)

If each rose brings in a profit of $0.50, then how many must you sell to get$300?  Start by writing the fifty cents, and then use x to represent the unknown number of roses.  Each rose costs the same, so multiply the two numbers.  Together they must all equal $300.$0.50x = $300. (Just divide 300 by .5 to isolate the variable.) x = 600 Loop back: The x represented roses so you found your bottom line. Look down at your answer choices. (A) 100 (B) 150 (C) 200 (D) 300 (E) 600 The correct answer is (E). Alternatively: You can solve this problem in a few seconds. Think about it logically; if you get less than$1 for each rose and you need $300, can you sell 300 roses and get the profit you need? No! You need more than$300 roses.  There is only one answer choice that works.

On sat.collegeboard.org, 71% 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.

Approach all math questions the same way.  Read the question carefully to avoid making careless mistakes.  Identify the bottom line, the question you must solve, and note it on your test.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer addresses the bottom line.

First, 3 is subtracted from x and the square root of the difference is taken. Then, 5 is added to the result, giving a final result of 9. What is the value of x?

Bottom line: x = ?

Assess your options: You could try to plug in answer choices and see which one equals 9, but you may have to write and solve the equation multiple times.  Instead, translate the two sentences into “math” and use algebra to find x.

Attack the problem: Work through the words step by step.  First, 3 is subtracted from x.  Write:

x – 3

The square root of the difference is taken.  That means both numbers involved in the difference are under the radical.

$\sqrt{x-3}$

Then 5 is added and the final result is 9.

$\sqrt{x-3}\, +5=9$

Now that you have your equation written, all you have to do is solve for x:

$\sqrt{x-3}\, +5=9$           (subtract 5 from each side)
$\sqrt{x-3}\, =4$                 (square each side to remove the radical)
$x - 3= 16$
$x = 19$

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

(A) 3
(B) 4
(C) 5
(D) 16
(E) 19

On sat.collegeboard.org, 57% 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.

Always 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; the specific question the problem asked you to solve.

The c cars in a car service use a total of g gallons of gasoline per week. If each of the cars uses the same amount of gasoline, then, at this rate, which of the following represents the number of gallons used by 5 of the cars in 2 weeks?

Bottom line: gal in 2 wks = ?

Assess your Options:  You could try to work backwards from the answer choices by plugging in a number for each variable, but you want to avoid working from the answer choices when you do not have to.  Instead, write an equation using the information that you are given in the problem.

Attack the Problem:  Start with the most basic information that you are given and logically translate the words into a math problem.  You know that c stands for cars and g stands for gallons of gasoline.  If all of the cars use the same amount of gasoline, then the total number of gallons must be divided evenly among each of the cars:

$1\: week = \frac{g}{c}$

Now you know that there are 5 cars.  You might be tempted to put the 5 with the c, but think about it this way: that would mean that the same number of gallons was divided among more cars, so each car was using less gasoline, which is impossible!   If there are more cars, the total amount of gasoline must increase:

$1\: week = \frac{5g}{c}$

Now all you have to do is turn 1 week into 2 weeks by multiplying both sides of your equation by 2:

$2\: week = \frac{10g}{c}$

Loop Back: You found the gallons for 2 weeks, so look down at your answer choices.

(A)
(B)
(C)
(D)
(E)

Alternative method using Knowsys strategies:  If you struggle with writing equations, choose a number to represent the variable you are given in the problem.  You know you have 5 cars, but pick a number to represent the gallons that these cars use.  Any number that is not already in the problem will work; avoid  0 or 1 because multiple equations may work with these choices. Let’s say that g = 10.  In one week, those 5 cars will use 10 gallons.  How many gallons will they use in 2 weeks?  20 gallons!

Plug in the 10 for g and the 5 for c.  10 times 10 is 100, and then if you divide 100 by 5, you get 20.  That matches the answer that you found, so E must be correct.  None of the other answer choices will equal 20.  Strategies are tools to help you – remember that you get the same number of points for the correct answer no matter how you work the problem!

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

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

# Functions

Is there always another explanation or point of view?  Before you answer this released SAT essay prompt, check out this article that is part current event and part historical example with a literary connection thrown in just for fun.  Richard the III was a real king who is best known as a villain in Shakespeare’s work.  Read about what happened to him and why he is appearing in the news now.  There are far too many themes in this article to name them all, so come up with about a dozen ways you could connect this example to an essay prompt.  Then memorize a few of the most interesting facts so that you can use them to support your opinion on any of the themes that show up in your SAT essay prompt.

Note: The identity of King Richard the III has been confirmed.  Read here for details.

## Algebra: Functions

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

Always read the problem carefully, identify the bottom line, and assess your options for solving the problem before you attack the problem.  When you have an answer, loop back to verify that it matches the bottom line.

The function y = f(x), defined for -1.5 ≤ x ≤ 1.5, is graphed above. For how many different values of is f(x) = 0.2?

Bottom Line: # times f(x) = .2

Assess your Options:  Some students will skip this problem, thinking that it requires a lot of time to somehow write a formula for the function from the graph.  However, once you know what you are looking at, this is one of the easiest and fastest problems on the test!  All that you have to do is read the graph!

Attack the Problem:  You know that f(x) = .2 is the same thing as y = .2.  Anytime you see f(x), you can just substitute a y for f(x) if that clarifies the problem in your head.  If y is constant, you know that it will be a horizontal line at .2.  Draw that line on your graph.

Anywhere that the line crosses the function f(x), that function is equal to .2.  Count up the number of intersections between the line that you drew and the original function.  There are four.  That means that f(x) = .2 four times.

(A) None
(B) One
(C) Two
(D) Three
(E) Four

On sat.collegeboard.org, 39% 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!

# 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!

# Equations

## Algebra: Equations

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

You should read every math problem on the SAT carefully.  Identify your bottom line, assess your options for reaching it, and then select the most efficient method to attack the problem.  Once you have an answer, loop back to make sure that it matches your bottom line.

y = x² - 4x + c

In the quadratic equation above, c is a constant. The graph of the equation in the xy-plane contains the points (-2, 0) and (6,0). What is the value of c?

Bottom Line: c = ?

Assess your Options:  For this problem, you are given two points (x, y).  That means that you could plug in the x and y values for either point and solve for c (this is the method that most students will use):

 (-2, 0) (6, 0) 0 = (-2)² - 4(-2) + c 0 = (6)² - 4(6) + c 0 = 4 – (-8) + c 0 = 36 – 24 + c 0 = 12 + c 0 = 12 + c -12 = c -12 = c
Isolating the variable in either of these two equations will get you the correct value of c.  However, notice how many steps there are.  Can you just look at the two points and know the answer?  Yes!  Think about you find the roots of an equation.

Attack the Problem:  Your original equation is already set equal to zero.  You know this because both of the points have a y value of 0.  In order to factor a polynomial, you need two binomials.  Here you already know that x = -2 or 6.  That means that your two binomials are (x + 2) and (x – 6).  Now c is the last number that you would get in your polynomial if you multiplied (x + 2) by (x – 6).  What number is that? -12!

All you had to do was multiply the last two numbers (2 × -6) because every other combination would have an x.  If you don’t see how that works, multiply out (x + 2)(x – 6):

Use FOIL (First, Inner, Outer, Last)
x² + 2x – 6x – 12 (combine like terms)
x² - 4x – 12

When you compare this equation to the original equation, you will see that in place of the c you now have a -12.

Loop Back:  During a test, you would never work through a problem three times (time waster!), so this is where you would check to make sure that you solved for the correct variable.  Look down at your answer choices.

(A) -12
(B) -6
(C) 4
(D) 6
(E) 12

On sat.collegeboard.org, 51% 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!

# 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!

# 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!

# Algebra: Equations

## Algebra: Equations

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

Math problems require you to read carefully. Identify the bottom line, what the question is asking, and assess your options for solving the problem. There are often multiple ways to solve a problem, but you should be as efficient as possible. Choose a method to attack the problem, then loop back to make sure that you solved for the bottom line.

If  $\frac{24}{15} = \frac{4}{n}$, what is the value of 4n?

Bottom Line: 4n =

Assess your Options You could solve for n and multiply by 4, or you could find a way to solve the equation for 4n. The latter method will save you time because you do not have to do the extra step of multiplying by 4. Both ways will work, but on a timed test you should choose the faster method.

Attack the Problem: Go ahead and cross multiply so that you have the equation 24n = 15 × 4. In other words, 24n = 60. Now, if you were solving for n you would have to deal with a decimal or fraction. Instead, look at the equation that you have. You know that 24 and 60 are both multiples of 4, so it would be easier to solve for 4n than to solve for n. What do you have to multiply by 4 in order to get 24? 6. All you have to do is divide each side of your equation by 6, and you will get 4n = 10.

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

(A) 6

(B) 10

(C) 12

(D) 30

(E) 60

Note: If you went to collegeboard.org on November 23rd, a programming error would have given you an incorrect answer. Collegeboard.org corrected this error the following day.

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

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

# Ratios

The election results are in!  The amount of information after an election day can be overwhelming, but limit yourself to one story from the election, and you will have an excellent current event for your SAT essay.  Many people focus on the presidential election, but there are hundreds of other important issues that were brought before the nation.  One group of United States citizens who currently cannot vote for the president of the United States voted about the possibility of becoming the 51st state.  Read this article about what is happening in Puerto Rico, and think about how this prospective state differs from or is similar to other territories that have become states.  Think of the broad themes raised by this story that could relate this article to SAT essay questions.

## 11/8 Algebra:  Ratios

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

Don’t just read the question; read it carefully.  Make sure you know which labels apply to which numbers.  Identify the bottom line.  Assess your options for solving the problem so that you can choose the most efficient method to attack the problem.  Once you have solved the problem, loop back to make sure that you have solved for the bottom line.

In a class of 80 seniors, there are 3 boys for every 5 girls. In the junior class, there are 3 boys for every 2 girls. If the two classes combined have an equal number of boys and girls, how many students are in the junior class?

Bottom Line:  Number of juniors = ?

Assess your options:  You could work backwards by starting with the answer choices, but it might take you a long time to work through all of the possible answers.  Instead, start turning those ratios into actual numbers of students.

Attack the problem:  You know the most about the seniors, so start with them.  You are given a ratio of 3 boys to 5 girls, and you know that the total number of boys and girls must equal 80.  You know that 3 + 5 = 8, so all you have to do is multiply the 3 and the 5 each by 10 and you will have a total of 80 seniors.  There are 30 senior boys and 50 senior girls.

$\frac{senior\: boys}{senior\: girls}=\frac{3}{5}=\frac{30}{50}$

Now that you know the number of senior boys and senior girls, how does that help you find the number of juniors?  Remember that the two classes combined have an equal number of boys and girls.  That means that the senior boys plus the junior boys must be equal to the senior girls plus the junior girls.

$senior\: boys + junior\: boys = senior\: girls + junior\: girls$

Plug in the numbers that you found for the senior boys and girls.

$30 + junior\: boys = 50 + junior\: girls$

What information do you know about the juniors?  You know that there are 3 boys for every 2 girls.  You do not know the total number of juniors, so use an x to represent this number.  What fraction of the total are the boys?  They are actually three fifths of the total number of juniors because you must add the boys and girls to find the total number of juniors.  That means that the girls are two fifths of the total number of juniors.  Plug this into your formula, remembering that anytime you have “of the total” that means that you must multiply by the unknown total.

$30 + \frac{3}{5}x = 50 + \frac{2}{5}x$

Now solve for x.  Rearrange the equation so that you have like terms on the same sides of the equation, and combine those like terms.  Start by subtracting the two fifths of x from each side.

$30+\frac{1}{5}x = 50$

Get those whole numbers together by subtracting 30 from each side.

$\frac{1}{5}x = 20$

To get rid of the fraction, you will need to multiply both sides by 5.  Your answer is x = 100.

Loop Back:  What does x represent?  It represents the total number of juniors, which matches your bottom line.  You are ready to look down at your answer choices.

(A) 72
(B) 80
(C) 84
(D) 100
(E) 120

On sat.collegeboard.org, 44% of the responses were correct.

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

# Algebra

Many of you have been following the news about Hurricane Sandy, and our thoughts and prayers are with those affected by the storm.  For others, it may be easy to hear things like “school is out” and “the power is out” and wish you were in the same situation.  You may think of enjoyable times spent wrapped in blankets and telling stories as a candle or flashlight flickers.  Think for a moment about how important power is for a hospital.  Here is an article about how one hospital responded to the storm.  Think about broad themes such as courage, the fight for life, and the response to danger as you read about this current event.

## 10/30 Algebra

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

If you approach all math problems the same way, you are less likely to make a careless mistake.  Start by reading the problem carefully and identifying the bottom line.  Assess your options for solving the problem so that you do not do more work than you need to.  Then attack the problem and solve it.  Loop back after you have finished to make sure that you found the bottom line.

If  , for which of the following values of x is y NOT defined?

Bottom line:  Although the problem includes an equation for y, you need an x value.  Your answer will be an x value that does something specific to this equation to produce a y that is not defined.  So make a note: x = ?

Assess your options:  You could work backwards and plug in answer choices to find a value that produces a y that is not defined.  This might require you to work the problem numerous times.  Instead, think about your knowledge of number properties.

Attack the problem: Any time a number is divided by zero, it is not defined.  If y is not defined, then it must be equal to something over zero.   Take the bottom part of your fraction and set it equal to zero: (x + 3)(x – 4) = 0.  For which values is this true?  Well, anytime zero is multiplied by a number, the answer is zero.  If either of these binomials is equal to zero, then there will be a zero on the bottom.  So set each binomial equal to zero: x + 3 = 0 and x – 4 = 0.  When you solve both of these equations, you will get two answers: x = -3 and 4.  Both answers will create a zero in the bottom of your fraction.

Loop back:  You found two answers for x that will create a zero on the bottom of your fraction.  Look down at your answer choices to see which one is present.

(A) -4
(B) -3
(C) -1
(D) 2
(E) 3