# 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!

# 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!

# 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