# Fractions

## Arithmetic: Fractions

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

Use the same process with every SAT question.  Read carefully and identify the bottom line.  Then assess your options for reaching the bottom line and choose the most time efficient method to attack the problem.  When you have an answer, loop back to check that you solved for the bottom line.

$\frac{1}{2}\cdot \frac{2}{3}\cdot \frac{3}{4}\cdot \frac{4}{5}\cdot \frac{5}{6}\cdot \frac{6}{7}=$

Bottom Line: just solve

Assess your options:  When you see a problem like this, get excited!  Some people will multiply all of the numbers, or change the fractions into decimals, but you should recognize a pattern!  Use what you know about fractions to solve this problem in less than 5 seconds.

Attack the problem:   The way you would normally solve the problem is to multiply all of the top numbers and multiply all of the bottom, then simplify the resulting fraction.  There is a faster way!  Although this problem starts out with separate fractions, you can think of the numbers that you are given as factors of the product you would get.  Remember that a number on top of a fraction will cancel if the same number is on the bottom of a fraction. Envision the problem this way:

$\frac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6}{2\cdot3\cdot 4\cdot 5\cdot 6\cdot 7 }=$

Then simply eliminate any numbers that are both on top and bottom!  The 2s cancel.  So do the 3s.  Keep going, and what do you have left?

$\frac{1}{7}$

Loop back:  You solved the original equation, so you are ready to look down at the answer choices.

(A) $\frac{1}{7}$
(B) $\frac{3}{7}$
(C) $\frac{21}{27}$
(D) $\frac{6}{7}$
(E) $\frac{7}{8}$

The correct answer is (A).

On sat.collegeboard.org, 60% of the responses were correct.

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

# Functions

## Link of the Day

As you look for current events to use in an SAT essay, pick news that you can relate to so that it will be easier for you to remember.  For example, take a look at this article about high school graduation rates.  Education and the economy will definitely affect your lifestyle.  Identify other themes that are present in this article.  What does it have to say about change, progress, motivation, rewards, and other words that have shown up in previous SAT essay topics?  There are also a lot of numbers in this article.  Numbers and statistics are fantastic because you can insert them into an essay quickly and seem knowledgeable without taking time away from your argument to explain a whole situation.  Don’t try to memorize all the numbers – focus on a few eye-catching statistics!

## 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) $\frac{27}{4}$

The correct answer is (E).

On sat.collegeboard.org, 61% of the responses were correct.

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

# Fractions

Arithmetic: Fractions

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

Read each question carefully.  Then identify the bottom line and assess your options for finding it.  Choose the most efficient method to attack the problem.  Before selecting your answer, loop back to make sure that you solved for the bottom line.

If  $N \times\frac{5}{14} = \frac{5}{14}\times\frac{7}{9}$then N =

Bottom Line:  N = ?

Assess your options: Normally, you would begin working this problem by multiplying the two fractions on the right and then multiplying them by the reciprocal of the fraction on the left in order to find N.  Before you jump into the problem, think about the properties of multiplication and you will see that there is a much faster way to solve the problem.

Attack the problem:  The commutative property of multiplication tells you that order is not important when you are multiplying; 3 × 5 = 5 × 3.  If you rearrange your equation, you will see that $N \times\frac{5}{14} = \frac{7}{9}\times \frac{5}{14}$ .   When you see the same thing, such as a fraction with 5 over 14, on both sides of the equation, you know that you can ignore that information.  No matter what number or variable you have, if it is the same on both sides of the equation, you will eliminate one side when you eliminate the other.  You can check this fact by multiplying both sides by the reciprocal of $\frac{5}{14}$.  If you multiply both sides by $\frac{14}{5}$, the $\frac{5}{14}$ will cancel on each side and you are left with N = $\frac{7}{9}$.

Loop back:  You solved for your bottom line, N, so you should look down at your answer choices.

(A) $\frac{7}{9}$
(B) $\frac{9}{7}$

(C) 5

(D) 7`

(E) 14

The correct answer is (A).

On sat.collegeboard.org, 82% of the responses were correct.

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

# Fractions

## Link of the Day

Have you ever wanted to try something but thought you wouldn't be any good at it? Don't let that stop you! Some mind-blowing stories of unstoppable dedication have been coming out of the Olympics, especially the tale of Im Dong-Hyun, a South Korean man with 20% vision or less in each eye. This makes him legally 'blind,' but it has not stopped him from setting two world records in archery! Read more here and here. What kinds of essay prompts might you answer with the story of Im Dong-Hyun?

## 7/29 Fractions

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

For every math problem, always be sure to follow the Knowsys Method. This will keep you from falling for traps and trick questions! First, read the question carefully.  You have to understand the information that you are given before you can begin working the problem.  The second step is to identify the bottom line, the question that you must answer.  In this case, you are asked to identify the total number of students who study art.  Next, assess your options to find the most efficient way to solve the problem, attack the problem, and loop back to make sure you answered the correct question.

Every student who studies art in a certain school receives exactly one of the grades A, B, C, or D. If $\frac{1}{5}$ of the students receive A’s, $\frac{1}{4}$ receive B’s, $\frac{1}{2}$ receive C’s, and 10 students receive D’s, how many students inthe school study art?

In this problem, you have fractions for the students who receive A’s, B’s, and C’s, but you have an actual number of students who receive D’s. To find the total number of students, you will need to find out what fraction of the whole is represented by the 10 students who receive D’s. Find out what fraction of the whole is represented by the students who receive A’s, B’s and C’s first. To add these fractions together, you need to find the least common denominator. You don't need to worry much about the 2 since it is a factor of 4. Instead, focus on the 5; the least common denominator will be the product of 4 and 5.

$\frac{1}{5}+\frac{1}{4}+\frac{1}{2}$ then becomes $\frac{4}{20}+\frac{5}{20}+\frac{10}{20}=\frac{19}{20}$

If $\frac{19}{20}$ of the students receive A’s, B’s or C’s, then that only leaves $\frac{1}{20}$ of the students who can receive D’s. Remember that there are 10 students who receive D’s. Think about it this way: $\frac{1}{20}$ of the total number of students is 10 students. For math problems the word “of” indicates that you will need to multiply. So $\frac{1}{20}$ times the total number equals ten. Use x to represent the unknown total, and then solve for x.

$\frac{1}{20}x =10$ Multiply each side by 20 to get rid of the fraction.

1x = 200

You solved for x (the total number of students), so check to make sure that you have found the bottom line. Then match your answer to the answer choices you are given.

(A) 30

(B) 60

(C) 100

(D) 200

(E) 500

(D) is the correct answer.

On sat.collegeboard.org, 60% of responses were correct.

# Fractions

## Link of the Day

Yesterday would have been Alan Turing's 100th birthday. Turing was a mathematical genius who laid the foundations for modern day computer science. By the time he was 26 he was performing cryptanalysis for the Government Code and Cypher School in England. He worked on cryptanalysis throughout WWII and developed many crucial new techniques in code breaking. He is best remembered for the invention of the Turing Machine, a theoretical machine that could solve a complex math problem using only simple calculations and a large amount of storage (a precursor to the modern day computer). You can read more about Alan Turing here. He would make a great historical "Excellent Example" for your essay.

## 6/23 Fractions

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

You probably notice that this problem looks very straightforward. If you have a calculator and you are comfortable with fractions, you are probably thinking that this problem won't be that difficult. If you don't have a calculator however, (or you don't remember how to enter fractions into your calculator), you might think this isn't such an easy problem. Never forget to follow the Knowsys method for Math Problems. Read the problem carefully and identify the bottom line (in this case, you just need to solve the equation). Next, evaluate your options, think about the various ways that you could solve the problem and then select the best choice. Finally, loop back and verify that the answer you choose correctly answers the bottom line. You could enter all these fractions into a calculator. But, because there are so many fractions it will take quite a while (and there is a good chance that you will make an error when you are entering the numbers). There is actually a much easier (and faster) way to solve this problems. If you just look at the first two fractions

$\frac{1}{2}*\frac{2}{3}$

you should see that you could cancel the 2's. Now, look at the second and third fractions

$\frac{2}{3}*\frac{3}{4}$

in this set, you could cancel the 3's.

If you go through the whole equation and cancel as much as you can, you are simply left with

$\frac{1}{1}*\frac{1}{1}*\frac{1}{1}*\frac{1}{1}*\frac{1}{1}*\frac{1}{7}=\frac{1}{7}$

Now, take a look at the answer choices and see which one matches your prediction.

(A)   $\frac{1}{7}$

(B)   $\frac{3}{7}$

(C)  $\frac{21}{27}$

(D)  $\frac{6}{7}$

(E)  $\frac{7}{8}$

The correct answer choices is (A)

On sat.collegeboard.org 63% of the responses were correct.

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

# Fractions

## Link of the Day

College can seem intimidating, and all the more so because it practically involves a different language. Learn the terms around college admission, and you will be able to navigate the process much more easily.

## 4/24 Fractions

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

Remember to always follow the Knowsys Method when approaching math problems: First, read carefully. Then identify the bottom line and assess your options. Next, choose the most efficient method and attack the problem! Finally, loop back to make sure you have the correct answer.

If f(x) = x + ax, and $a=\frac{7}{2}$, what is $f(\frac{3}{2})$?

First, as always, identify the bottom line.

$f(\frac{3}{2})=?$

Next, assess your options. You have variables in the question and numbers in the answer choices, so you could work backward from the answer choices. However, there are also numbers in the question, so you could plug in those numbers to see what that gives you. The second method will be far more efficient.

First, plug in the numbers. $a=\frac{7}{2}$ and $x=\frac{3}{2}$, so plug each of those in where appropriate.

$f(\frac{3}{2})=\frac{3}{2}+\frac{7}{2}(\frac{3}{2})$

Next, following the Order of Operations (PEMDAS), multiply the last two numbers together. When multiplying fractions, simply multiply the numerators and then the denominators.

$f(\frac{3}{2})=\frac{3}{2}+\frac{21}{4}$

Obviously, the next thing to do is to add the fractions together. Remember that when adding fractions, the denominators must match, and you can ONLY add the numerators. You'll need to do something to make the denominators the same.

$f(\frac{3}{2})=\frac{3}{2}(\frac{2}{2})+\frac{21}{4}$

$f(\frac{3}{2})=\frac{6}{4}+\frac{21}{4}$

$f(\frac{3}{2})=\frac{27}{4}$

Now loop back. Is $f(\frac{3}{2})$ what you actually needed to find? Yes, it is! Now look at the answer choices.

$A)\frac{1}{3}$

$B)\frac{3}{2}$

$C)\frac{7}{2}$

$D)\frac{21}{4}$

$E)\frac{27}{4}$

The answer is E.

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

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