# 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!

# Algebra 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, and assess your options for solving the problem.  Choose the most efficient way to solve the problem, and attack it!  Finally, loop back to make sure that you solved for the bottom line.

If  and , then t exceeds s by

Bottom line:  You need to know how much t exceeds s.  So the question you are asked is really, “How much bigger is t than s?”  Your bottom line is t – s.

Assess your options:  Normally you would simplify s before plugging the s value into the equation for t, then find the difference between t and s.  However, doing this will give you some ugly fractions and take a lot of time.  Instead, try starting with your bottom line and plugging in everything that you know.

Attack the problem:  Start with the t – s and plug in the equations for both of these:

$t-s=(1+\frac{1}{2}s)-(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32})$

Then plug in s one more time:

$t-s=[1+\frac{1}{2}(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32})]-(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32})$

This looks terribly ugly, but keep calm and use the order of operations.  You always multiply or divide before you add or subtract, so your first job is to distribute the half within the brackets by multiplying it by every number within the first set of parentheses:

$t-s=(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64})-(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32})$

Look at the new equation that you have.  How many things cancel out when you start subtracting the second group of numbers from the first?  Almost everything!  You are left with a single fraction:

$t-s=\frac{1}{64}$

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

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

On sat.collegeboard.org, 41% of the responses were correct.

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

# Algebra Equations: Substitution

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

For every SAT math problem, read the problem carefully so that you know exactly what information you are given.  Then identify the bottom line, the information that you must find.  Assess your options for solving the problem, and choose the most efficient method to get to the answer.  Attack the problem to find the answer, and loop back to your bottom line to make sure that your answer matches what you were supposed to find.

If   and x = 12, then x – y =

Bottom Linex – y = ?

Assess your options:  You could try to work backwards from the answer choices, but that would require you to know the values for and y.  Instead, use arithmetic to solve for and y, then plug those numbers into the last equation to find the bottom line.

Attack the problem: You are given the fact that x = 12.  You can create another equation to solve for x: if you multiply both sides of the first equation by y, you have = 3y. Now plug in your x value and you will have 12 = 3y. When you divide both sides by 3, you will find that 4 = y.  Return to the last equation and plug in your values for x and yx – y becomes 12 – 4 = 8.

Loop back: The last equation was your bottom line, so you are ready to look at the answer choices.

(A)  3

(B)  5

(C)  6

(D)  8

(E)  9

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

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

# Writing Equations

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

Math questions should be read carefully so that you understand each part of the problem.  Take a moment to make a note of the bottom line, the answer you are asked to find, to make sure that you do not accidently solve for the wrong variable or do more work than the problem requires.  Assess your options for solving the problem, choose the most efficient method, and then attack the problem!  Always loop back to make sure that your answer matches the bottom line.

There are n students in a biology class, and only 6 of them are seniors. If 7 juniors are added to the class, how many students in the class will not be seniors?

For this question it is easiest to look at one piece of information at a time.  Your bottom line is the number of students who are not seniors.  Start with the n first.  The n represents the total number of students in the class.  Then you have 6 students from the class that are seniors.  Since you want the number that are not seniors, you must subtract the 6 from the n:
n – 6

Now look at your second piece of information.  7 juniors are added to the class.  Because they are not seniors, they will be added to the number of students who are not seniors:
n – 6 +7

You are not given any more information.  Simplify the equation that you have written:
n + 1

Even though you have not found an exact number, this equation represents the number of students in the class that will not be seniors.  Now look down at your answer choices.

(A) n – 3
(B) n – 2
(C) n – 1
(D) n + 1
(E) n + 2

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

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

# Writing Equations

## 8/19 Writing Equations

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

Always take the time to read math questions carefully so that you will not make careless mistakes.  Identify the bottom line, which is the question you must answer, and assess your options for reaching the bottom line.  Choose the most efficient method to solve the problem and then attack it.  Do not forget to loop back and make sure that you solved for the bottom line, especially when you get a problem that requires multiple steps.

The price of 10 pounds of apples is d dollars. If the apples weigh an average of 1 pound for every 6 apples, which of the following is the average price, in cents, of a dozen such apples?

The bottom line that you are looking for is the cost of 12 apples in cents.  You can make a note of this by writing 12app = ¢?  Now ask yourself what you could do, and what you should do.  You could choose a number for the variable d and plug it into all of the answer choices, but then you would have to work several problems to find matching answers.  Instead, try working with the information you have, setting up the information in simple equations.

Your bottom line asks for the cost of 12 apples, but you were given information about 6 apples.  6 apples weigh 1 lb.  It is easy to change 6 to 12 by doubling it.

6 apples = 1 lb
12 apples = 2 lb

Now you have the dozen apples, so you must determine how many cents they cost.  You know that 10 lbs = d dollars.  Start by changing the dollars into cents, because you know you must end with cents.  To change dollars into cents, you must multiply the dollars by 100.

10 lb = d (solving for dollars)
10 lb = 100d (solving for cents)

Now you have the correct monetary unit, but you also still have an equation that solves for 10 lbs.  Your dozen apples is only 2 lbs.  To get from 10 to 2, divide both sides of your equation by 5.

10 lb = 100d
2 lb = 20d

Now put all the information that you have together to make sure that you solved for the bottom line:

12 apples = 2 lb = 20d

(A)

(B)

(C)

(D)

(E)

On sat.collegeboard.org, 36% of responses were correct.

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

# Equations

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

Read the question carefully, identify the bottom line (what the question is asking), and assess your options for solving it. You want to be as efficient as possible when solving math questions, so for most problems you should not look at the multiple choice answers before attacking the problem with the method you have chosen. Always loop back at the end of the problem to make sure that your answer addresses the bottom line.

If $(t-2)^{2}=0$, what is the value of (t + 3)(t + 6)?

You must find the value of (t - 3)(t + 6). In order to do this, you must first find the value of t. Paraphrase the question in your mind: “If this is true, then solve this.” This question is already set up in two steps for you.  Solve the first equation and you will have the key to solving the second part of the problem because there is only one variable involved: t.

Think about the first equation logically. Something squared is equal to zero, so what can be multiplied by itself and equal zero? The only possible answer is zero! The squared portion of the problem must be equal to zero.

$(t-2)^{2}=0$ and $0^{2}=0$, therefore t - 2 = 0.

When you add the 2 to both sides of your new equation, you will see that = 2. Now that you know the value of t, you have all the information that you need to solve the second part of the problem with simple arithmetic.

(t + 3)(t + 6)

(2 + 3)(2 + 6)

(5)(8)

40

Loop back to make sure that the answer you found answers the question you were asked. The problem asked for the value of (t + 3)(t + 6), and that is exactly what you found. Finally, match your answer to the correct answer choice.

(A) 40

(B) 18

(C) 9

(D) 4

(E) It cannot be determined from the information given.

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

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

# Area of a Triangle

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## 6/2 Area of a Triangle

What is the area of the triangle in the figure above?

As always, the first step is to read the problem carefully and identify the bottom line. We are looking for the area of the triangle above and we can use the fact that it has been drawn on the coordinate plane to figure out the base and the height. Remember that since there is nothing on the drawing that says "figure not drawn to scale", the figure must be drawn to scale. That means that we could "eyeball" it if we needed to. In this case we can see that the triangles height is 3 and the base is 5. All that we need to solve the problem is the formula for the area of a triangle.

$a=\frac{1}{2}b\times h$

We simply plug in our values for h and b to get

$a=\frac{1}{2}(5)\times (3)=\frac{1}{2}(15)=7.5$

We now look at the answer choices below to see which one matches our prediction.

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

The correct answer choice is (B).

On sat.collegeboard.com 70% of the responses were correct.

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

# Algebra: Substitution

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## 5/30 Algebra: Substitution

Always follow the Knowsys Method for math problems. It will save you time not only on the SAT, but also on math tests and quizzes in your school classes. Read the question carefully and identify the bottom line. Assess your options. Ask what you could do and then what you should do, and solve the problem quickly and efficiently once you have decided on a strategy. Finally, loop back to double check that you answered the question correctly

If A,B, and C are numbers such that  and , then  is equal to which of the following?

At first this problem may seem to be challenging but it is important that we tackle it step by step. Most of the more difficult problems on the SAT consist of 3 or 4 steps, each of which is fairly easy on their own. As long as we focus on the next step and don't panic, most of the "hard" problems will actually turn out to be fairly straightforward. The first step is always to identify the "bottom line". In this problem, we are looking for what  is equal to.
Since the equation currently has 3 different variables we cannot solve it as it is. In order to eliminate some of the variables we will need to use substitution. We start with first equation

$\frac{a}{b}=3$

We can manipulate this equation to give us

$a=3b$.

We then substitute this into the original equation to get

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

Now we work on eliminating c. By solving the second equation we get the following.

$\frac{b}{c}=7 \therefore \frac{b}{7}=c$

Substituting this into our equation now gives us

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

At this point, you may be starting to panic because the equations look so complicated. If so, take a deep breath and relax, we are almost done. Remember that with fractions, dividing is the same as multiplying by a reciprocal. We can now solve our equation since we only have one variable.

$\frac{4b}{\frac{8b}{7}} = \frac{4b}{1}\times \frac{7^{}}{8b}=\frac{28b}{8b}=\frac{7}{2}$

Now all that's left to do is look at the answers and find which one matches our solution.

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

The answer is A and that matches our prediction exactly!

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

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

# Rates

The Industrial Revolution is well-known as a time of explosive economic growth and invention. Here are five top inventions--and why they're important! Any of them would make a noteworthy Excellent Example in your SAT essay.

## 4/6 Rates

Always stop and read carefully before you do anything else. Make sure that you mark the bottom line and carefully label each step of your scratch work. It's easy to let confidence trick you into thinking it's safe to skip steps, but on a high-stakes test like the SAT, writing down each step and making sure that every answer is right can be the difference between acceptance and rejection at the school of your dreams. Once you find an answer, loop back to make sure that you have answered what the question asked. Then, and only then, should you select your answer from among the answer choices.

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 stamping machines are needed to keep up with 18 inserting machines of this kind?

When you read this question carefully, one thing that should jump out at you is the fact that you have two different units of measurement. Immediately think, "I could convert both of these to seconds." Look for the bottom line: How many stamping machines do you need?

s = ?

Next, focus on your options. What could I do? What should I do? You could, as previously mentioned, convert one rate so that they use the same unit of measurement. But what could you do after that? You could guess, if you feel that this problem is too hard and that your time is better spent elsewhere, but guessing is only allowed on the test. No guessing during practice! The next option is to write an equation to describe what is happening in this problem, and then solve for s.

Let's start by converting the inserting machine's rate into seconds.

i: 120/60 = 2

Each inserting machine can stuff 2 envelopes per second.

Next, set up a formula to compare the number of envelopes that the machines can prepare. Each inserter can stuff 2 envelopes each second, and each stamper can stamp 3 envelopes per second. Plug the 18 inserting machines n for i.

2i = 3s

2(18) = 3s

Now solve for s.

$\frac{2(18)}{3}=s$

2(6) = s

12 = s

Loop back to make sure that what you found was the correct answer. s represents the number of stamping machines needed to prepare an equal number of envelopes as i inserting machines. Each inserting machine completes 2 envelopes per second, while each stamping machine finishes 3. Logically, you need fewer stamping machines than inserting machines. Look at the answer choices:

A) 12

B) 16

C) 20

D) 22

E) 24

On sat.collegeboard.org, 59% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Exponents

## Mathematics: Exponents

According to the Knowsys Method, the first thing you should do when facing a math problem is to read carefully, then note the bottom line, assess your options, and attack the problem.

If $x^{-2}=16$, what is the value of $x^{2}$?

At the top of your scratch work, write the bottom line.  $x^{2}=?$

In most problems you would want to solve for x, but here you need to solve for $x^{2}$ instead. How do you do that?

First, notice that the given exponent is very similar to what you need to find. To correct negative exponents, convert each term to its reciprocal. Since you do not have a fraction to start with, simply put 1 in the numerator.

$\frac{1}{x^{2}}=16$

To get the variable out of the denominator, multiply both sides by $x^{2}$

$x^{2}(\frac{1}{x^{2}})=(16)x^{2}$

After canceling out $x^{2}$ on the left-hand side of the equation, you are left with $1=16x^{2}$  Divide both sides by 16 to isolate $x^{2}$

$\frac{1}{16}=\frac{16x^{2}}{16}$

$\frac{1}{16}=x^{2}$

Now loop back to your bottom line and check whether you answered the correct question.

$x^{2}=?$

Look at the answer choices to make a final selection:

$A) \frac{1}{16}$

$B) \frac{1}{4}$

$C) 2$

$D) 4$

$E)12$

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

Want more help with math? Visit myknowsys.com!

# Algebra Equations: Solve

## Mathematics: Standard Multiple Choice

Read the question carefully and note the bottom line at the top of your scratch work. The nice thing about short questions like this is that they make the first two steps of the Knowsys Method absolutely effortless. At the top of your scratch work, write Q=?

Next, assess your options. What could I do? What should I do? The two most obvious choices here are to work backwards by plugging in the answers or to rearrange the equation to solve for Q. It will be faster on this problem to solve for Q.
Obviously, you can combine like terms here to give you 66Q=6

Next, to isolate Q, divide both sides by 66.

$Q=\frac{6}{66}$

Reduce the fraction to its lowest terms.

(C) 10
(D) 11
(E) 20