# 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!

# Percents

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## 8/7 Percents

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

When you work a percent problem, it is especially important to read the problem carefully because small words like “of” make all the difference in describing a percentage.  Once you have understood the question and identified the bottom line, assess your options and choose a method to attack the problem.  After you have finished the problem, loop back to verify that the answer addresses the bottom line.

The population of Norson, the largest city in Transitania, is 50 percent of the rest of the population of Transitania. The population of Norson is what percent of the entire population of Transitania?

Your bottom line is a percentage: the number of people in Norson as a percent of the entire population of the country of Transitania.  At the top of your scratch work, write N = ?%  Next, assess your options.  You don’t know how many people are in the country of Transitania, so solving the problem algebraically would be challenging. Instead, pick a number so that the problem will be more concrete.  When picking numbers on a percent problem, you should always pick 100 because 100 is the easiest number to use; the answer you find is already out of 100 so you will never need an extra step to find the correct percentage. Now you're ready to attack the problem.

If you assume that there are 100 people total in all of Transitania, you still do not know how many people are in the city of Norson.  Use the variable “N” to represent the people in the city.  The problem tells you that the population of this city “is 50% of the rest of the population of Transitania.”  You know that Norson has half as many people as the rest of the population.  Think about it this way: for every person in Norson, there must be 2 people outside of the city.  Write an equation representing this knowledge and solve it.

N + 2N = 100

3N = 100

N = 33.33

You found that 33.33 people out of 100 live in the city of Norson, so you know that the percentage of people who live in Norson is 33.33%.  This matches the bottom line that you needed to find.

(A)               20%

(B)               25%

(C)               30%

(D)               33 1/3%

(E)                50%

On sat.collegeboard.org, 38% of answers were correct.

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

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## 6/20 Algebra: Roots and Radicals

If , which of the following must be true?

Step 1 of the Knowsys method for math is always to read the question carefully. Then, identify the bottom line. In this case, you need to select the answer choice that is true given the formula above. Step 3 is to evaluate your options. Think about what you could do, and what you should do. Since you have variables in the problem and in the answer choices, you could pick numbers for two of the variables, solve the equation for the third variable, and then look at the various answer choices. However, since this problem looks fairly straightforward, that's not necessary. It is faster and easier to simply manipulate and simplify the equation and then look at the answer choices (so this is what you should do). Note that if you did get stuck in the simplifying process, you could always go back and try the other method.

Start by squaring both sides of the equation.

$\sqrt{x-a}=\sqrt{x+b}\therefore (\sqrt{x-a})^{2}=(\sqrt{x+b})^{2}\therefore x-a=x+b$

Now, simply subtract x from both sides of the equation.

$x-a=x+b \therefore -a=b$

Lastly, add a to both sides of the equation.

$-a=b\therefore a+b=0$

Now, you know that a is equal to -b and you also know that the value of x doesn't matter (since it was eliminated). Take a look at the answer choices and see which one must be true. Don't forget to check answer choice (E) first and then work backwards. On "which of the following" questions, the test makers know that you will probably start with answer choice (A) and work your way down. Because they want you to as much time as possible, they usually put the correct answer near the end (not always, but usually).

$(A) a=0$

$(B) b=0$

$(C)a+ b=0$

$(D)a-b=0$

$(E)a^{2}+b^{2}=0$

Answer choice (E) would only be true if both a and b were equal to 0. Since you are looking for the answer choice that must be true, (E) won't work. You know that (D) is incorrect because it does not match the simplified form of the equation. (C) matches your prediction exactly. (B) and (A) both could be true, but they do not have to be true so neither of them is the correct answer choice.

The correct answer choice is (C).

On sat.collegeboard.org 57% of the responses were correct.

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

# Equations and Factoring

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## 6/5 Equations and Factoring

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

This problem is a great chance to put the Knowsys method for math to use. Remember, the first step is to read the problem carefully. If you started reading this question and felt a sinking feeling in your stomach when you saw 8x or the words "positive integer" stop and take a moment to reread the problem. It's really not as difficult as it first appears to be. Step 2 of the Knowsys method is to identify the bottom line. In this case we are looking for what the result must be a multiple of.  Next we assess our options. We could write out a formula and try to simplify it or we could pick numbers. In this case it is probably faster to write out a formula but we will work the problem using both methods so that you know how to use either one. Step 4 is to attack the problem. Believe it or not, attitude makes a big difference. Relax, trust in the method, and go step by step. Don't worry if you aren't sure exactly how the problem will work out.
If we write out the formula described in the problem we get the following
$8x+2x-5=$

We can simplify the equation like so

$8x+2x-5=10x-5=5(2x-1)$

If we look at the fully factored form of the equation we can see that we are going to multiply whatever we have in the parenthesis by 5. That means that our result must be a multiple of 5. Now we look at the answers below.

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

You can see that answer (B) matches our prediction exactly. As I mentioned earlier, there is one other way to solve this problem. We can pick numbers for X. The easiest number to pick for X is 1. If we do that the formula becomes

$8x+2x-5=8(1)+2(1)-5=8+2-5=5$

In this case, our formula gives us the number 5. Since we need an answer that must be true, we can eliminate any answers that do not fit and in this case only (B) works (don't forget that step 5 of the Knowsys math method is to loop back and verify that your answer matches the bottom line). Picking numbers can be a great strategy if you get stuck on a problem but it is generally faster to solve the problem by writing out the formula.

On sat.collegeboard.com 68% of the responses were correct.

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

# Slope

Write it Down! This infographic linked today from www.coolsiteoftheday.com discusses the importance of taking notes, a few different methods, and the potential benefits and drawbacks of taking notes digitally or the old--fashioned way. Did you know that your brain actually processes information differently while you're taking notes? This is a good resource to bookmark and revisit when you notice that your class notes are less than helpful--it might be time to try out a different method.

## 5/15 Slope

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

Remember to read carefully, identify the bottom line, assess your options, attack the problem, and loop back. When you use this method, you will get more problems right and you will move faster through the test.

In the xy-plane, line l passes through the points (a, 0) and (0, 2a), where a > 1. What is the slope of line l?

First, read carefully. You have two points on a line, which means you can visualize that line if you wish. Picking a number for a might make that easier if the variable trips you up. Next, identify the bottom line. The question asks for the slope of line l, so at the top of your scratch work write "slope = ?"

Now assess your options. Since you need to find the slope of the line, a good place to start is with the formula for slope: rise over run. There are two choices here; you can use a as a variable or you can pick a number for a. Using a directly involves fewer steps because you don't need to plug in the value, but manipulating the variable can be confusing for some and can cost time. Which tool you choose to solve the problem is up to your personal preference.

Either way, the first step in the problem is to set up your formula. Since a must by greater than 1, I'll use 2.

$\frac{rise}{run}=\frac{2a-0}{0-a}$                                                             $\frac{rise}{run}=\frac{2(2)-0}{0-(2)}$

$\frac{rise}{run}=\frac{2a}{-a}$                                                                    $\frac{rise}{run}=\frac{4}{-2}$

$\frac{rise}{run}=-2$                                                                    $\frac{rise}{run}=-2$

Now loop back to make sure that you answered the right question. Your bottom line asks for the slope, so you found the change in y-coordinates (rise) and the change in x-coordinates (run), divided one by the other and reduced. That is the slope, so -2 is the answer you need.

A) -2

B) $\frac{-1}{2}$

C) 2

D) -2a

E) 2a

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

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

# Integers

Pythagoras, best known to high school students for his Pythagorean Theorem, actually discovered much more than that one formula. Even if you are not mathematically inclined, the beginning of this paper has some interesting notes on how the Pythagoreans--the followers of Pythagoras--lived.

## 3/28 Integers

Always attempt to solve the problem before looking at the answer choices. Read carefully, then identify the bottom line--what the question is actually asking--and mark it at the top of your scratch work. Assess you options by asking "What could I do?" to open your toolbox, then "What should I do?" to select the best way to solve the problem. Attack the problem fearlessly, then loop back to the bottom line to check whether what you found is the correct answer.

If p is an odd integer, which of the following is an even integer?

At the top of your scratch work, write even = ?

Next, ask "What could I do?" You could think through each answer choice abstractly, determining that if p is odd then... but that is difficult and gets confusing quickly. You could pick a number for p, then use that number to find a value for each answer choice. The smallest odd number is the best for this. Pick one. Since this question includes the phrase "which of the following," the answer is very likely to be D or E. Start at the bottom and work your way up.

E) $p^{2}-p$
If p = 1, then $1^{2}-1=0$. 0 is neither positive nor negative, but neutral; however, it is still even. This distinction confuses some students, so make sure you know it. Now loop back to the bottom line. $p^{2}-p=0$, so it is even, so it is the answer. On the SAT, you could continue on from this point or check the other answers.

D) $(p-2)^{2}$
$(1-2)^{2}=(-1)^{2}=1$ is odd.

C) $p^{2}-2$
$1^{2}-2=1-2=-1$ is odd.

B) $p^{2}$
$1^{2}=1$ is odd.

A) $p-2$
$1-2=-1$ is odd.

On sat.collegeboard.org, 62% of responses were correct.

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

# Picking Numbers

## Mathematics: Standard Multiple Choice

Multiple-choices directions are really simple, aren't they? The questions on the test are longer, but they don't say much more than this. Remember that you are allowed to use your calculator, but that it can get you in trouble. I don't actually own one, and I use the one on my computer as little as possible.
The weights of 10 bags of apples range from 2.75 pounds to 3.15 pounds. If w is the weight, in pounds, of one of these bags, which of the following must be true?
After reading carefully, the next step is to identify the bottom line. Here, the question asks "Which of the following must be true?" and that's extremely hard to put in mathematical shorthand, so the best choice here is to write weight=? at the top of your scratch work.
Next, assess your options. You could translate this problem into an inequality, but that could be complicated because the answers are all absolute value. Instead, pick numbers for w and see which inequality below works out. To make sure that the answer choice will always be true, try three different numbers. When picking numbers, always choose numbers that are easy to work with and that are distinct from one another. In this case, picking 3 and then picking 3 again won't tell you anything about the problem.   32.8, and 3.1 should work nicely. I picked 3 first because it is a whole number, and those are always easier to deal with. 2.8 and 3.1 are near the ends of the given range, so they will make sure you get the right answer.
Take a look at the answer choices:
(A)
(B)
(C)
(D)
(E)

You are probably accustomed to starting with A, but starting with E is a better habit. It will save you some time on this problem; you can eliminate E immediately. 3-10= -7, and 7 is too large an absolute value.

Look at D: 3-0.2= 2.8  D doesn't work either.

You can eliminate C right away because adding to the numbers you chose will not give a number with an absolute value less than 0.2--it would be closer to 6.

Now try B. 3-2.95= .05 That works with the inequality given in the answer choice. Plug in the other numbers you selected: 2.8-2.95= -.15 is also within the range. 3.1-2.95= .15  B fits the range given in the problem, so it is the right answer.

43% of those who attempted this question at sat.collegeboard.org got it right.

Want more help with math? Visit myknowsys.com!