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## SAT (Fall 2023)

### Course: SAT (Fall 2023)>Unit 6

Lesson 5: Problem Solving and Data Analysis: lessons by skill

# Percents | Lesson

## What are percentages, and how frequently do they appear on the test?

A percentage is a ratio out of 100 that represents a part-to-whole relationship. Percent (percent) means parts per hundred.
In this lesson, we'll learn to:
1. Calculate percentages using part and whole values
2. Switch between equivalent forms of percentages
3. Calculate percent change
On your official SAT, you'll likely see 1 to 3 questions about percentages.
You can learn anything. Let's do this!.

## How do we calculate percentages?

### Finding a percentage

Finding a percentSee video transcript

### Calculating a percent value

There are two values that are important for finding a percentage: a part and a whole. To calculate a percentage, use the following formula:
percent, equals, start fraction, start text, p, a, r, t, end text, divided by, start text, w, h, o, l, e, end text, end fraction, dot, 100
For example, say you took a quiz in math class and got 21 out of the 24 questions correct. We could calculate the percentage of questions you got correct as follows:
• The part is 21.
• The whole is 24.
start fraction, 21, divided by, 24, end fraction, dot, 100, equals, 87, point, 5, percent
If we have any two of the part, the whole, and the percentage, we can solve for the missing value!
Note: be careful when identifying the part and the whole; the part won't necessarily be the smaller number!

Example: What is 150, percent of 8 ?

### Finding complementary percentages

Since all parts of a whole should add up to 100, percent, we can also use percentages to determine the value of any missing parts.

Example: A bag is filled with red and blue marbles. If there are 25 marbles in the bag, and 56, percent of the marbles are blue, how many red marbles are there in the bag?

### Try it!

Try: Identify parts and wholes
Of the juniors at South High School, 45, percent played for a school sports team. If 117 juniors played for a school sports team, how many students are in the junior class?
• 45 is the
.
• 117 is the
.
• We need to solve for the
.

Try: Find complementary percentages
A group of 1, comma, 300 people took a survey in which they had to select their favorite of the four seasons (winter, spring, summer, and fall). The incomplete table below shows the results.
Seasonpercent of respondents
Winter16
Spring22
Summer
Fall28
What percentage of respondents said summer was their favorite season? (Ignore the percent symbol when entering your answer. For example, if the answer is 12, percent, enter 12.)
What number of respondents said summer was their favorite season?

## What forms can percentages take?

### Converting percentages to decimals and fractions

Converting percents to decimals & fractions exampleSee video transcript

### Switching between forms of percentages

We can use equivalent forms of percentages interchangeably and choose the one(s) that best suit our purpose.
For example, 50, percent is equivalent to the following:
• The ratio 50, colon, 100, which reduces to 1, colon, 2.
• The fraction start fraction, 50, divided by, 100, end fraction, which reduces to start fraction, 1, divided by, 2, end fraction.
• The decimal value 0, point, 5.
Note: a useful shortcut for converting percentages to decimals is to remove the percent symbol and move the decimal point 2 places to the left.
Decimal equivalents for percentages are highly useful when making calculations. For example, if we wanted to find 112, percent of value x, we could simply multiply x by the decimal equivalent, 1, point, 12.

Example: What is 25, percent of 364 ?

### Translating percentage word problems

You'll frequently see percentages referenced in word problems. Luckily, there's an easy way to translate these word problems into arithmetic:
• "what" means x
• "is" means equals
• "of" means multiplied by
• "percent" means divided by 100
So:
36 is what percent of 60? → 36, equals, start fraction, x, divided by, 100, end fraction, dot, 60

### In what form should I enter my answer?

Questions on the SAT may ask for "what percent" and require you to enter that value into the answer field.
In these instances, you should not enter decimal or fractional equivalents, but instead enter the percent value as an integer (without a percent sign). So, if the answer is 50, percent, you should simply enter 50.

### Try it!

Try: switch between forms of percentages
In 2019, 29, percent of Major League Baseball starting pitchers were left-handed.
The equivalent fraction is
.
The equivalent decimal is
.

Try: use decimal equivalents in calculations
In testing two engines, a mechanic finds that Engine A puts out 125, percent of the horsepower put out by Engine B. If Engine B puts out 120 horsepower, how many horsepower does Engine A put out?
First, we can convert 125, percent to the decimal
.
We can then multiply that decimal by
to find the number of horsepowers put out by Engine A.
Engine A puts out
horsepower.

## How do we calculate percent changes?

### Percentage word problems

Percent word problem: guavasSee video transcript

### Calculating percent change

We're often asked to calculate by what percent a quantity changes relative to an initial value: the percent discount on jeans, the percent increase in population, etc. When calculating a percent change from an initial value to a final value:
1. Find the difference between the initial and final values.
2. Divide the difference by the initial value.
3. Convert the decimal to a percentage by multiplying the quotient by 100.
percent, start text, c, h, a, n, g, e, end text, equals, start fraction, start text, d, i, f, f, e, r, e, n, c, e, end text, divided by, start text, i, n, i, t, i, a, l, end text, end fraction, dot, 100

Example: The price of a vacuum was reduced from dollar sign, 200 to dollar sign, 170. What was the percent reduction in price?

If we have any two of the percent change, the initial value, and the final value, we can solve for the missing value! And remember: decimal equivalents for percentages are highly useful when making calculations.

Example: The price of a pair of shoes is dollar sign, 40 after a 20, percent discount. What is the price of the shoes before discount?

### Try it!

Try: Identify Initial value, final value, and percent change
A customer’s monthly internet bill was dollar sign, 63, point, 89. Due to a rate increase, her monthly bill is now dollar sign, 68, point, 86. To the nearest tenth of a percent, by what percent did the amount of the customer’s internet bill increase?
• 63, point, 89 is the
.
• 68, point, 86 is the
.
• We need to solve for the
.
The difference between initial and final values is
.
The percent change (to the nearest tenth of a percent) is
. (Ignore the percent symbol when entering your answer. For example, if the answer is 12, point, 4, percent, enter 12, point, 4.)

Practice: Calculate a percentage
Where Do People Get Most of Their News?
SourcePercent of those surveyed
Internet/social media43, percent
Television23, percent
Newspapers15, percent
Others/none of the above8, percent
The table above shows a summary of 1, comma, 800 responses to a survey question. Based on the table, how many of those surveyed get most of their news from either newspapers or the radio?

Practice: USE COMPLEMENTARY PERCENTAGES
Tara has read 95, percent of the books she owns. If Tara owns 160 books, how many of her books has she NOT read?

Practice: CALCULATE PERCENT INCREASE
The price of a particular brand of headphones was dollar sign, 7 in 2016. In 2018, the price of the same brand of headphones was dollar sign, 10. What is the approximate percent increase in the price of the headphones?

Practice: Calculate percent change
YearSubscriptions sold
201810, comma, 500
201912, comma, 390
The manager of a streaming video service received the report above on the number of subscriptions sold by the service. The manager estimated that the percent increase from 2019 to 2020 would be double the percent increase from 2018 to 2019. To the nearest integer, how many subscriptions did the manager expect would be sold in 2020 ?

## Things to remember

Percent means parts per hundred.
p, percent, equals, start fraction, p, divided by, 100, end fraction
A shortcut for converting percentages to decimals is to remove the percent symbol and move the decimal point left 2 places.
When translating word problems:
• "what" means x
• "is" means equals
• "of" means multiplied by
• "percent" means divided by 100
The sum of all parts of a whole is 100, percent.
When calculating a percent change from an initial value to a final value:
1. Find the difference between the initial and final values.
2. Divide the difference by the initial value.
3. Convert the resulting decimal to a percentage.
percent, start text, c, h, a, n, g, e, end text, equals, start fraction, start text, d, i, f, f, e, r, e, n, c, e, end text, divided by, start text, i, n, i, t, i, a, l, end text, end fraction, dot, 100