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Course: Digital SAT Math>Unit 3

Lesson 3: Percentages: foundations

Percentages | Lesson

A guide to percentages on the digital SAT

What are percentages?

A percentage is a ratio out of $100$ that represents a part-to-whole relationship. Percent ($\mathrm{%}$) 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
You can learn anything. Let's do this!.

How do we calculate percentages?

Finding a percentage

Khan Academy video wrapper
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:
$\mathrm{%}=\frac{\text{part}}{\text{whole}}\cdot 100$
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 $24$.
$\frac{21}{24}\cdot 100=87.5\mathrm{%}$
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 $150\mathrm{%}$ of $8$ ?

Finding complementary percentages

Since all parts of a whole should add up to $100\mathrm{%}$, 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 $56\mathrm{%}$ 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, $45\mathrm{%}$ 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 $1,300$ 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.
Season$\mathrm{%}$ of respondents
Winter$16$
Spring$22$
Summer
Fall$28$
What percentage of respondents said summer was their favorite season? (Ignore the $\mathrm{%}$ symbol when entering your answer. For example, if the answer is $12\mathrm{%}$, enter $12$.)
What number of respondents said summer was their favorite season?

What forms can percentages take?

Converting percentages to decimals and fractions

Khan Academy video wrapper
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, $50\mathrm{%}$ is equivalent to the following:
• The ratio $50:100$, which reduces to $1:2$.
• The fraction $\frac{50}{100}$, which reduces to $\frac{1}{2}$.
• The decimal value $0.5$.
Note: a useful shortcut for converting percentages to decimals is to remove the $\mathrm{%}$ 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 $112\mathrm{%}$ of value $x$, we could simply multiply $x$ by the decimal equivalent, $1.12$.
Example: What is $25\mathrm{%}$ 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 $=$
• "of" means multiplied by
• "percent" means divided by $100$
So:
$36$ is what percent of $60$? → $36=\frac{x}{100}\cdot 60$

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 $\mathrm{%}$ sign). So, if the answer is $50\mathrm{%}$, you should simply enter $50$.

Try it!

Try: switch between forms of percentages
In $2019$, $29\mathrm{%}$ 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 $125\mathrm{%}$ 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 $125\mathrm{%}$ 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

Khan Academy video wrapper
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 $100$.
$\mathrm{%}\phantom{\rule{0.167em}{0ex}}\text{change}=\frac{\text{difference}}{\text{initial}}\cdot 100$
Example: The price of a vacuum was reduced from $\mathrm{}200$ to $\mathrm{}170$. 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 $\mathrm{}40$ after a $20\mathrm{%}$ 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 $\mathrm{}63.89$. Due to a rate increase, her monthly bill is now $\mathrm{}68.86$. To the nearest tenth of a percent, by what percent did the amount of the customer’s internet bill increase?
• $63.89$ is the
.
• $68.86$ 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 $\mathrm{%}$ symbol when entering your answer. For example, if the answer is $12.4\mathrm{%}$, enter $12.4$.)

Your turn!

Practice: Calculate a percentage
Where Do People Get Most of Their News?
SourcePercent of those surveyed
Internet/social media$43\mathrm{%}$
Television$23\mathrm{%}$
Newspapers$15\mathrm{%}$
Radio$11\mathrm{%}$
Others/none of the above$8\mathrm{%}$
The table above shows a summary of $1,800$ 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?
Choose 1 answer:

Practice: USE COMPLEMENTARY PERCENTAGES
Tara has read $95\mathrm{%}$ of the books she owns. If Tara owns $160$ books, how many of her books has she NOT read?
Choose 1 answer:

Practice: CALCULATE PERCENT INCREASE
The price of a particular brand of headphones was $\mathrm{}7$ in $2016$. In $2018$, the price of the same brand of headphones was $\mathrm{}10$. What is the approximate percent increase in the price of the headphones?
Choose 1 answer:

Practice: Calculate percent change
YearSubscriptions sold
$2018$$10,500$
$2019$$12,390$
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.
$p\mathrm{%}=\frac{p}{100}$
A shortcut for converting percentages to decimals is to remove the $\mathrm{%}$ symbol and move the decimal point left $2$ places.
When translating word problems:
• "what" means $x$
• "is" means $=$
• "of" means multiplied by
• "percent" means divided by $100$
The sum of all parts of a whole is $100\mathrm{%}$.
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.
$\mathrm{%}\phantom{\rule{0.167em}{0ex}}\text{change}=\frac{\text{difference}}{\text{initial}}\cdot 100$

Want to join the conversation?

• Why do they teach such basic concepts with so much complexity?
I now have to come to think of Indian system of teaching math much simpler and easier to understand.
(80 votes)
• Rewarr, Arey bhai. They are trying to break it down man. Here we get the simple questions so that we will understand the question better. If they give us the real deal most of us will chicken out and won't do well. Hence they give simple ones and make us confident and strong with the methodology and strategy.
(90 votes)
• is practising from khan academy enough for the DSAT?
(21 votes)
• hopefully
(37 votes)
• i thought finding for the difference is (initial-final) so why do they make it vice versa in others
(7 votes)
• whether you do final-initial, or initial-final, it doesnt matter. The value doesnt change but only the signs do.

for example, 6-4 = 2
4-6 = -2
what's important is the positive value of the difference and the original amount

An example :
a banana cost 5 dollars on Monday, but on Tuesday it was for 8 dollars. Find the % change in price

here, youll get 3 or -3 as the difference, depending on the method you used. The positive value of the difference is 3 and the original value is 5

therefore, 3/5 x 100 = 60%

Thus, there was 60 % increase in the price of bananas
(24 votes)
• i miss the old khan academy practice sat math
(12 votes)
• What was it like?
(1 vote)
• In last question why we added 4460.4 to 12,390 in last step?
(4 votes)
• To find the amount in 2020 ,
basically, the amount in 2020 was the amount in 2019 plus 36 percent of that same amount in 2019
You could also do 1.36 x 12390 to make the work a bit shorter :D
(15 votes)
• Why are the advanced and medium lessons the same?
(1 vote)
• Because the difference between the levels is not the method for the questions but the difficulty of the questions.
(16 votes)
• Why do we calculate 36% of 12390, instead of the initial 10500?
(7 votes)
• Are there any questions banks available for the Digital SAT yet?
(3 votes)
• Download the app Bluebook from the College Board website and you'll find practice tests there.
(7 votes)
• its so good
(5 votes)
• Hey guys, this isn't necessarily a question but a tip, I'll be taking the Digital SAT this weekend and a quick way to find an answer to questions like:

"What is 33% of 8?"

What you could do is divide both values by 10 and multiply them together. For example:

3.3 x 0.8 = 2.64

Hoped this help, and anyone who will be taking the test. Good luck and YOU GOT THIS!
(5 votes)