Percents

Arithmetic: Percents

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

Always read math questions carefully so that you can absorb all the information and avoid mistakes.  Identify the bottom line, what the question is asking you to find, and assess your options for reaching that bottom line.  Choose the most efficient method to attack the problem.  When you have an answer, loop back to verify that the answer matches the bottom line and you have finished all the steps in the problem.

If p percent of 75 is greater than 75, which of the following must be true?

Bottom Line: p = ?

Assess your Options:  It is often tempting to look down at the answer choices before you need them, but they could mislead you since most of them are wrong!.  You could take numbers that fit each answer choice and see if they give you a number greater than 75.  However, by applying what you know about percents, you can solve the problem much faster than you can by trying out 5 different numbers.

Attack the Problem:  There are a number of ways to think about percentages: as percents, decimals, numbers out of a hundred, parts of wholes….  The list continues.  Here is one of the fastest ways to think about the problem:

If you have one hundred percent of something, you have all of it.  So 100% of 75 is going to be 75.  If you want a result that is greater than 75, you are going to need more than 100% of 75.  Therefore, p must be bigger than 100.

Or, if you normally think about percents in terms of decimals, you know that 50% of something is .5.  In order to get a decimal from a percent, you had to move the decimal twice to the left.  So with 100%: 75 × 1.00 = 75.  Try writing an inequality to find the decimal that you would need in order to get a number bigger than 75: 75p > 75.  The p represents the unknown percent of 75 (remember, "of" means multiplication in math).  If you solve the inequality, you get p > 1.  Then you have to move the decimal back in order to get a percent: p > 100.  Your percent must be bigger than 100%.  This method takes much longer than the first one, but it proves that the first method is correct.  The testers realize that students are not used to working with percentages greater than 100, so it is a good idea to review how these work before the test!

Loop back:  You know what p must be greater than, so look down at your answer choices.

(A) p > 100
(B) p < 75
(C) p = 75
(D) p < 25
(E) p = 25

The correct answer is (A).

On sat.collegeboard.org, 71% of the responses were correct.

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

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

Math questions should always be read carefully.  You will also avoid making errors by identifying the bottom line and assessing your options for solving the question.  Choose the most efficient method to attack the problem.  When you have finished, loop back to be sure that even if there were multiple steps, you reached the bottom line.

The length of a rectangle is increased by 20%, and the width of the rectangle is increased by 30%. By what percentage will the area of the rectangle be increased?

Bottom Line: % change = ?

Assess your Options:  You could work this problem without picking any numbers; however, picking easy numbers will allow you to think about the problem in a more concrete way and avoid errors.

Attack the Problem:  One of the easiest numbers to work with is one.  Think of your original rectangle as having a length of one and a width of one.  The formula for area of a rectangle is length times width.  If L × W = A, for your first rectangle you have 1 × 1 = 1.   The area of the original rectangle is one.

Then think about the changes that occur to that rectangle.  The length increases by 20%.  In order to find 20% of 1, all you have to do is move the decimal over twice to .2.  The new length is 1.2.  Use the same method to find the new width, and an increase of 30% becomes 1.3.  The area of the rectangle after the change is 1.2 × 1.3 = 1.56

The formula for percent change would require you to find the difference between these two areas and divide that by the original number.  You use the same formula whether you are looking for an increase or a decrease.  Notice that your original number is one, so dividing by one will not change your answer.  All you need to do is find the difference between the areas: 1.56 – 1 = .56.  What is .56 as a percent?  Your answer is 56%.

Loop Back:  You found the percent change, which was your bottom line.  Look down at your answer choices.

(A) 25%
(B) 36%
(C) 50%
(D) 56%
(E) 60%

The correct answer is (D).

On sat.collegeboard.org, 34% 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%

The correct answer is (D).

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

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