Computation with Percentages

 

Hello, and welcome to this video about computation with percentages! Today we’ll learn how to solve different types of problems involving percentages and how this skill can help you in the real world.

Before we get started, let’s review a few things. First, a ratio is a comparison of two or more numbers. Some ratios are part-to-whole ratios, comparing a part of something to the whole group. A percentage is a part-to-whole ratio that’s expressed as a fraction of \(100\). For example, the percentage \(80%\) means \(80\) out of \(100\): \(\frac{80}{100}\). In this case, \(80\) is the part and \(100\) is the whole.

Solving problems with percentages involves translating a word problem into a mathematical equation. There are three types of problems with percentages:

  1. Finding the part
  2. Finding the whole
  3. Finding the percent

We’ll discuss these three problem types together and work on some examples to help you understand how to solve them.


Let’s start with the first problem type, finding the part.

Finding the Part

The first problem type involves finding part of a given number that’s equal to a given percent.

Consider the following percentage problem: What is 60% of 5? To answer this question, we need to understand the information we are given in the problem. Let’s look at each part of the word problem and translate it into an algebraic equation. The first part of the sentence, “What is,” means “What number is equal to.” Since we don’t know the number in question, we use the variable \(n\) to represent this portion of the equation:

\(n=\)

The next portion of the question is “\(60%\).” Since \(60%\) equals \(0.60\), write \(0.60\) in the equation after the equal sign:

\(n=0.60\)

The next part of the question is “of.” When translating written equations into numerical equations, the word “of” indicates multiplication. Therefore, write a multiplication sign after \(0.60\) in the equation:

\(n=0.60 \times\)

The last portion of the question is 5. Write 5 after the multiplication sign in the equation:

\(n=0.60 \times 5\)

From here, solve the equation for \(n. 0.60\times 5=3\), so \(n=3\). Therefore, \(60%\) of \(5\) is \(3\).

Now it’s your turn. Consider the following percentage problem: 40% of 20 is what number?

Pause the video here, convert the written equation into an algebraic equation, and solve.

Now that you’ve tried this one on your own, let’s take a look at it together. The algebraic equation that represents this question is:

\(0.40\times 20=n\)

From here, solve the equation for \(n\). \(0.40\times20=8\), so \(n=8\). Therefore, 40% of \(20\) is \(8\).

Great job!


Finding the Whole

The next percentage problem type is finding the whole. In these problems, we are given the percent and its equivalent part. We need to identify the whole.

Consider the following percentage problem: 3 is 60% of what number? To answer this question, we need to understand the information we are given in the problem. Let’s look at each part of the word problem and translate it into an algebraic equation. The question starts with “\(3\) is,” which means “\(3\) is equal to.” Start the equation with \(3\) and an equal sign:

\(3=\)

The next part of the question is 60%. Since 60% equals \(0.60\), write \(0.60\) in the equation after the equal sign:

\(3=0.60\)

The next word in the question is “of.” When translating written equations to algebraic equations, the word “of” indicates multiplication. Therefore, write a multiplication sign after \(0.60\) in the equation:

\(3=0.60\times\)

The next part of the question is “what number.” Since we don’t know what number is being referred to, use the variable \(n\) to represent this number:

\(3=0.60\times n\)

When multiplying a number by a variable, remember that no multiplication sign is needed. We can write the equation with the coefficient \(0.60\) next to the variable \(n\):

\(3=0.60n\)

From here, solve the equation for \(n\).

To solve for \(n\), isolate the variable by doing inverse operations. The opposite of multiplying by \(0.60\) is dividing by \(0.60\), so divide both sides of the equation by \(0.60\).

\(\frac{3}{0.60}=\frac{0.60n}{0.60}\)

\(3\div 0.60-5\), so write \(5\) on the left side of the equation. And \(0.60\div 0.60=1\), so write \(1n\), or just \(n\), on the right side of the equation.

\(5=n\)

Therefore, \(3\) is 60% of \(5\).

Now it’s your turn. Consider the following percentage problem: 50 is 20% of what number?

Pause the video here, translate the written equation into an algebraic equation, and solve.

Now that you’ve tried this one on your own, let’s take a look at it together. The algebraic equation that represents this question is:

\(50=0.20n\)

From here, solve the equation for \(n\).

To solve for \(n\), isolate the variable by doing inverse operations. The opposite of multiplying by \(0.20\) is dividing by \(0.20\), so divide both sides of the equation by \(0.20\).

\(\frac{50}{0.20}=\frac{0.20n}{0.20}\)

\(50\div 0.20=250\), so write \(250\) on the left side of the equation. Then, if we divide the right side of the equation, we’ll get \(1\), so we’re left with \(1n\) on the right side of the equation. Since \(1n\) is the same thing as \(n\), \(250\) is equal to \(n\).

\(250=n\)

So \(50\) is 20% of \(250\).

Nice work!


Finding the Percent

The last percentage problem type involves finding the percent. In these scenarios, we are given the part and the whole and we need to find the percent.

Consider the following percentage problem: What percent of \(5\) is \(3\)? To answer the question, we need to understand the information we are given in the problem. Let’s look at each part of the word problem and translate it into an algebraic equation. The question starts with “What percent”. Since we don’t know the percent, we’ll use the variable \(p\) to represent this number:

\(p\)

The next word in the question is “of.” Remember, when translating written equations into algebraic equations, the word “of” indicates multiplication. Therefore, write a multiplication sign after \(p\) in the equation:

\(p\times\)

The next part of the question is the number \(5\). Write \(5\) in the equation after the multiplication sign:

\(p\times 5\)

The next part of the question is the word “is.” When translating written equations into algebraic equations, the word “is” means “is equal to.” Therefore, write an equal sign after \(5\) in the equation:

\(p\times 5=\)

The next part of the equation is the number \(3\). Write \(3\) in the equation after the equal sign:

\(p\times 5=3\)

When multiplying a number by a variable, remember that no multiplication sign is needed. We can write the equation with the coefficient \(5\) next to the variable \(p\), like this:

\(5p=3\)

From here, solve the equation for \(p\).

To solve for \(p\), isolate the variable by doing inverse operations. The opposite of multiplying by \(5\) is dividing by \(5\), so divide both sides of the equation by \(5\).

\(\frac{5p}{5}=\frac{3}{5}\)

\(5\div 5=1\), so write \(1p\), or just \(p\), on the left side of the equation. \(3\div 5=0.60\) so \(p=0.60\).

\(p = 0.60\)

Now that we know \(p=0.60\), we need to convert the decimal into a percentage to answer the original question. \(0.60\) is equal to \(\frac{60}{100}\), or 60%. Therefore, \(3\) is 60% of \(5\).

Now it’s your turn. Consider the following percentage problem: 11 is what percent of 20?

Pause the video here, convert the written equation into an algebraic equation, and solve.

Now that you’ve tried this one on your own, let’s take a look at it together. The algebraic equation that represents this question is:

\(11 = 20p\)

From here, solve the equation for \(p\).

To solve for \(p\), isolate the variable by doing inverse operations. Simply divide both sides of the equation by \(20\).

\(\frac{11}{20}=\frac{20p}{20}\)

\(20\div 20=1\), so write \(p\) on the right side of the equation. And \(11\div 20=0.55\), so \(p\) is equal to \(0.55\).

\(0.55=p\)

Now that we know that \(p=0.55\), we need to convert this decimal into a percentage to answer our original question. \(0.55\) is equal to \(\frac{55}{100}\), or 55%. Therefore, \(11\) is 55% of \(20\).

Great work!


Computations with percentages are used all the time in the real world. For instance, store discounts, interest rates, and test scores are often expressed as percentages. Percentages are also used when calculating the tip on a bill at a restaurant.

I have one more problem for you to try. This one is a little more challenging, but I know you can handle it! Consider the following percentage problem:

Rachael is taking her friend out to dinner. When she gets the check, the total is $56.75, not including the tip. Rachael wants to leave a 20% tip. How much money should Rachael leave for a tip?

Pause the video here, translate the written equation into an algebraic equation, and then solve.

Now that you’ve tried this one on your own, let’s take a look at it together.

To answer this question, we need to figure out what number is equal to 20% of $56.75. The algebraic equation that represents this question is:

\(n=0.20\times 56.75\)

From here, solve the equation for \(n. 0.20\times 56.75=11.35\), so \(n=11.35\). Therefore, for a 20% tip, Rachael should leave $11.35.

Great job!

I hope this video about computation with percentages was helpful. Thanks for watching, and happy studying!

Frequently Asked Questions

Q

How do you find the percentage of a number?

A

Find the percentage of a number by turning the percentage into a decimal and multiplying by the number.
Ex. What is 75% of 152?
0.75 × 152 = 144

Q

How do you calculate percentage increase?

A

Calculate percentage increase using this formula:
% increase=\(\frac{new-old}{old}×100\)
Ex. A student scored a 73 on his first test and a 92 on his second test. What is the percentage increase from his first test score to his second test score?
% increase=\(\frac{92-73}{73}×100=\frac{19}{73}×100=26\%\)

Q

How do you calculate percentage change?

A

Calculate percentage change using this formula:
% change=\(\frac{new-old}{old}×100\)
Ex. A house is originally priced at $350,000. After 6 months of not selling, the price is lowered to $315,000. What is the percent change from the original to the new price?
% change= \(\frac{315,000-350,000}{350,000}×100=-10\%\)

Q

How do you calculate percentage difference?

A

Calculate percentage difference using this formula:
% difference=\(|\frac{\text{Value }1-\text{Value }2}{\frac{\text{Value }1+\text{Value }2}{2}}|×100\%\)
Ex. If chocolate ice cream costs $4.25 and cookie dough ice cream costs $6.35, what is the percentage difference between the two ice creams?
% difference=\(|\frac{4.25-6.35}{\frac{(4.25+6.35)}{2}}|×100\%=|\frac{-2.1}{5.3}|×100\%=39.62\%\)

Q

How do you calculate a grade percentage?

A

To calculate a grade percentage, add each grade from one category and then divide by the number of grades in that category (take the average). Next, multiply by the weight (percentage) that category holds. Repeat for as many categories as there are and then add them together.
Ex. At my school, major grades are worth 60% of my final grade, daily grades are worth 30% of my final grade, and my final exam is worth 10% of my final grade. If my grades are as follows, what is my final grade?
Daily Grades: 95, 87, 73, 92, 98
Major Grades: 93, 88, 95
Final Exam: 94
Daily Grade Average=\(\frac{95+87+73+92+98}{5}=\frac{445}{5}=89\)
Daily Grade Portion=\(89×30\%=89×.3=26.7\%\)
Major Grade Average=\(\frac{93+88+95}{3}=\frac{276}{3}=92\)
Major Grade Portion=\(92×60\%=92×.6=55.2\%\)
Final Exam Portion=\(94×10\%=94×.1=9.4\%\)
Final Grade=\(26.7+55.2+9.4=91.3\%\)

Q

How do you calculate percentage with a calculator?

A

To calculate percentage with a calculator, divide the part by the whole and multiply by 100.
Ex. What percentage of 20 is 5?
\(\frac{5}{20}=.25×100=25\%\)

Q

How do you multiply percentages?

A

Multiply percentages by first converting them to decimals. Then, multiply the decimals. Finally, convert back to a percentage, if necessary.
Ex. \(13\%×21\%=.13×.21=.0273=2.73\%\)

Practice Questions

Question #1:

 
What is \(90\%\) of \(30\)?

\(29\)
\(24\)
\(27\)
\(26\)
Answer:

This problem is about finding the part. To solve, read the question carefully and form an equation from it. The word “what” is equivalent to the variable in the equation. Let’s use \(n\), for the number that will solve the problem. Next, the word “is” is much like the word “equals.” Write an equals sign after \(n\).
\(n=…\)

Moving along, “\(90\%\)” is equivalent to \(\frac{90}{100}\), or \(0.90\).
\(n=0.90…\)

Finally, we have “of \(30\).” Whenever you see the word “of” in a math problem, remember that it is equivalent to a multiplication sign! This means the equation from the problem statement will be
\(n=0.90\times30\)

To solve for \(n\), simply multiply.
\(n=27\)

The solution to this problem is \(27\), because \(27\) is \(90\%\) of \(30\).

Question #2:

 
\(20\) is \(40\%\) of what number?

\(50\)
\(40\)
\(45\)
\(80\)
Answer:

To answer this question, start by writing an equation from the problem statement. The first word is “\(20\)”. Since this is already in the language of math, go ahead and write it down in the equation.
\(20\)

Next, the word “is” is much like the word “equals.” Write an equals sign after \(20\).
\(20=\)

\(40\%\) is the same as \(\frac{40}{100}\), or equivalently, \(0.40\). Write this down next.
\(20=0.40…\)

Remember, the word “of” implies multiplication. Write a multiplication sign after \(0.40\).
\(20=0.40\times…\)

Finally, the problem statement has the words “what number.” In place of these words, write your variable. This can be any letter, but here we will use \(n\).
\(20=0.40\times n\)

To solve for the value of \(n\), you must get it all by itself on one side of the equals sign. Get rid of the \(0.40\) by dividing by \(0.40\) on both sides.
\(\frac{20}{0.40}=\frac{0.40\times n}{0.40}\)

\(50=n\)

This means that \(20\) is \(40\%\) of \(50\).

Question #3:

 
What percent of \(75\) is \(15\)?

\(20\%\)
\(15\%\)
\(75\%\)
\(25\%\)
Answer:

To answer this question, it is again important to convert the problem statement into an algebraic expression. First, we have the words “what percent.” Let’s use the variable \(p\) to represent this.

Next is the word “of.” Like in the other problems, this word is equivalent to a multiplication sign. Now we have \(p\times …\)

Next is the number \(75\), followed by “is \(15\).” The word “is” should be replaced with an equals sign, and, of course, \(75\) and \(15\) should remain as numbers. Putting all the pieces together, we have the equation \(p\times75=15\).

To solve for \(p\), divide both sides by \(75\).
\(\frac{p\times75}{75}=\frac{15}{75}\)

\(p=\frac{1}{5}=0.20\)

Converting this to a percent, \(p=20\%\).

Question #4:

 
Miss Turner’s chemistry students recently took an exam. Of the \(40\) students in her class, \(16\) of them made A’s. What percent of the class made A’s on the exam?

\(20\%\)
\(24\%\)
\(33\%\)
\(40\%\)
Answer:

Word problems are a little trickier to solve, but we can make an equation just like in the previous problems by studying the information closely. Notice that the question at the end of the problem asks “what percent of the class made A’s.” Let’s use this to start building the equation.

The words “what percent” indicate that we want to know a percentage, so let’s use the variable \(p\). Next, the words “of the class” indicate that we need a multiplication sign, followed by the number of students in the class. Now we have \(p\times40\).

The next part of the question specifies that we are interested in the students who made A’s. Since \(16\) students got an A on the exam, set the equation equal to \(16\).
\(p\times40=16\)

Now, solve for \(p\) by dividing both sides by \(40\).
\(\frac{p\times40}{40}=\frac{16}{40}\)
\(p=0.4\)

Converting this to a percentage, we see that \(p=40\%\). So \(40\%\) of Miss Turner’s students made an A on the chemistry exam.

Question #5:

 
George and his family are out for lunch, and their bill comes out to \($36.00\). If they want to leave a \(20\%\) tip for their waiter, how much will the tip be?

\($6.00\)
\($5.75\)
\($7.20\)
\($8.25\)
Answer:

To answer this question, first put the information into an equation. We know the cost of the bill and the percent of the tip. Recall that tips are a certain percent of the bill. Let’s use the variable t to represent the tip amount. Notice that the word “are” is another form of the word “is.” Write an equals sign after t.
\(t=\)

On the other side of the equation, we will write the tip percentage multiplied by the bill amount. This is because “tips are a certain percent of the bill amount,” and the word “of” indicates multiplication.
\(t=20\% \times 36.00\)

To do this multiplication, convert \(20\%\) to a decimal.
\(t=0.20\times36\)
\(t=7.2\)

The tip should be \($7.20\).

 

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by Mometrix Test Preparation | Last Updated: December 20, 2021