# 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!

# Sequence Problems

## Arithmetic: Sequence Problems

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

# Multiples

## Arithmetic: Multiples

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

Approach each problem the same way so that you feel confident about your ability to solve it.  Start by reading the question carefully and identifying your bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer addresses the bottom line.

Add 8x to 2x and then subtract 5 from the sum. If x is a positive integer, the result must be an integer multiple of

Bottom Line:  multiple of = ?

Assess your Options:  You have to write an equation for this problem, but after doing so you can use logic or the strategy of plugging in numbers to find possible answers to the equation.  Both methods are quick and will result in the correct answer.

Attack the Problem:  Your first step is to translate all the words you are given into an equation. If you add 8x to 2x, you get 8x + 2x.  Then subtract 5.  You should have:

8x + 2x – 5

Always simplify as much as possible before moving to the next step.  Here, you can combine like terms.

10x – 5

Now go back to the other information that you are given.  The variable x must be a positive integer.  Plug in the smallest possible value for x, and you will get the smallest possible result of this equation.  Plug in x = 1.

10(1) – 5 = 5

Now, multiples will always get larger, so there are other possible answers to this equation.  However, this is the smallest answer and you are looking for what the result “must” be an integer multiple of.  Multiples are simply the product of a number and an integer.  5 is a prime number, so the only thing that the answer must be a multiple of is 5.

(If you want to make sure you are on the right track, plug in x = 2.  The answer is 15.  15 is still a multiple of 5.  Any positive number that you plug in will still be a multiple of 5 because when you subtract 5 from a multiple of 10, you will always get a number ending in a 5.)

Loop Back:  You found that the answer must be a multiple of 5.  Look down at your answer choices.

(A) 2
(B) 5
(C) 8
(D) 10
(E) 15

On sat.collegeboard.org, 68% 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!