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Percents can be seen in numerous daily situations all over the world because they are an easy way to express and relate data. This lesson will focus on converting different forms of numbers into percents and vice versa. The lesson will also show how percents appear in everyday life.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

A sudden loud sound coming from Paulina's phone wakes her up. It iss $6:00AM$ and time to start getting ready for school. When she looks at her phone, Paulina realizes that she forgot to charge it the night before. The battery is at only $14%!$
She connects her phone to a charger and starts getting ready.

External credits: @jannoon028

a What fraction in its simplest form expresses the current battery level?

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b Paulina disconnects her phone as soon as she is ready to go to school. The battery is charged up to $65 $ of its capacity. What percent does Paulina see on the screen?

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">%<\/span><\/span><\/span><\/span>","answer":{"text":["83"]}}

Discussion

A percent is the ratio of a number to $100.$ Percents are usually denoted by the symbol $%.$

$p%=100p $

$5%=1005 $

Percentages are often used in real-life situations. For example, saline is a mixture of salt and water often used to help with dry eyes or to clean and store contact lenses, as it does not irritate the eyes. Saline contains $0.9%$ salt. This means that every $100$ grams of saline contains $0.9$ grams of salt.
$0.9%=1000.9 $

Another situation in which percentages can be used is sales. For example, if a jacket is on sale, a store may advertise that it is $20%$ off. Then, the price has been lowered by $20%.$ If the jacket originally cost $$100,$ then the price was lowered by $$20.$ $20%=10020 $

Discussion

It is possible to convert a fraction into a percent and vice versa. Consider the following fraction.
*expand_more*
*expand_more*
Note that the order of these two steps can be reversed. In other words, the fraction can be multiplied by $100$ first, before calculating the quotient of the numerator by the denominator. ### Extra

Converting Different Fractions

$8714 $

This fraction can be rewritten as a percent by following two steps.
1

Divide the Numerator by the Denominator

First, divide the numerator by the denominator. In this case, the numerator is $14$ and the denominator is $87.$ The procedure of long division can be used to divide up to four decimal places.

The quotient is $0.1609.$ This is the decimal equivalent of the fraction.

2

Multiply by $100$

Next, multiply the result by $100$ to convert it to percent. Note that multiplying a decimal number by $100$ comes down to moving the decimal point two places to the right.

The result is $16.09,$ which can be rounded to $16.1.$ Finally, add a percent sign after the number.

$8714 ≈16.1% $

Therefore, the fraction represents about $16.1%.$ Input a fraction whose numerator and denominator are less than $100.$ Then, click

Submit and Convertto see how it can be converted into a percent. In this applet, the fraction will be multiplied by $100$ first.

Example

Paulina's first class is math. The teacher gives everyone back their graded papers from last Thursday's test. Paulina is thrilled to see that she got $23$ answers correctly.

There were $25$ questions in total. Paulina wonders if she managed to answer $90%$ of the questions correctly. Find the fraction that describes Paulina's score and then express it as a percent.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">%<\/span><\/span><\/span><\/span>","answer":{"text":["92"]}}

Which number is the total and which is the part of the total? Use the definition of a percent.

It is given that Paulina answered $23$ out of $25$ questions correctly. This information will be used to first write a fraction. Since the total number of questions is $25,$ this is the denominator. The number of answers Paulina got correct is $23,$ which represents a part of $25,$ so it is the numerator.

$2523 $

Now this fraction can be expressed as a percent. Start by writing it as a fraction with a denominator of $100.$ Note that $25$ is a factor of $100.$
$25⋅4=100 $

This means that multiplying the numerator and denominator by $4$ will result in an equivalent fraction with a denominator of $100.$
Recall the definition of a percent — the ratio of a number to $100.$
$n%=100n $

This means that the fraction represents $92%.$
$10092 =92% $

Therefore, Paulina did score more than $90%$ on the math test. Great job!
Fraction $2523 $ represents the number of correct answers Paulina got on the test. This number can be rewritten as a percent by following two steps. First, divide the numerator by the denominator by using long division.

The quotient is $0.92.$ Next, multiply this decimal number by $100$ to write it as a percent. This can be done by moving the decimal point two places to the right.

The product is $92.0,$ or $92.$ By adding a percent sign, it can be concluded that Paulina got $92%$ of the answers correct on her math test.

Discussion

It is possible to convert a percent into a fraction and the other way around. Consider the following percent.
*expand_more*
*expand_more*
### Extra

Converting Different Percents Between $1$ and $100$

$48% $

This percent can be written as a fraction by following two steps.
1

Write as a Fraction Whose Denominator Is $100$

Since a percent is the ratio of a number to $100,$ write it as a fraction whose numerator is the number part of the percent and whose denominator is $100.$

$48%=10048 $

2

Simplify the Fraction

Next, check whether the fraction can be simplified. First, split the numerator and denominator into prime factors. This process is shown below for $10048 .$

$48100 =2⋅2⋅2⋅2⋅3=2⋅2⋅5⋅5 $

The numbers share two common factors. Their product is the greatest common factor (GCF) of $48$ and $100.$
$GCF(48,100)=2⋅2=4 $

Finally, divide both the numerator and denominator by $4.$
The fractions $2512 $ and $10048 $ are equivalent and they both correspond to $48%.$ Input a percent that is an integer between $1$ and $100.$ The process of converting this percent into a fraction will be demonstrated.

Example

It is lunch time! Paulina and her friends go to the school cafeteria. She is trying to decide between juice and a sandwich or soda and a taco. What a hard decision! Her good know-it-all friend Davontay interrupts her thoughts with an interesting fact.

External credits: @pch.vector

Paulina is curious about how many students out of $25$ would choose different options, if this statistic were in fact true.

a What fraction in simplest form corresponds to the percent of students choosing Option $2?$

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b How many students out of $25$ would choose Option $1?$

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a Calculate what percent corresponds to the number of students choosing Option $2.$ Use the definition of a percent.

b Think about what the numerator and denominator of the fraction represent.

a Davontay said that $68%$ of students choose Option $1$ for lunch. Let $100%$ represent all students in their school. Subtract $68%$ from $100%$ to find how many students choose Option $2.$

$Option2100%−68%=32% $

Next, this percent should be written as a fraction. Start by recalling that a percent is the ratio of a number to $100.$
$n%=100n $

Apply this formula to $32%.$
$32%=10032 $

This fraction should also be simplified. First, split the numerator and denominator into prime factors.
$32100 =2⋅2⋅2⋅2⋅2=2⋅2⋅5⋅5 $

The numbers share two common factors, $2$ and $2.$ Their product is the greates common factor (GCF) of $32$ and $100.$
$GCF(32,100)=2⋅2=4 $

Finally, divide the numerator and denominator by their GCF, $4.$
Therefore, $32%$ corresponds to $258 .$
b In Part A, it was found that $258 $ students prefer Option $2$ for lunch. This fraction means that $8$ students out of $25$ would choose the soda and taco. This leaves $25−8=17$ students who prefer Option $1.$

External credits: @pch.vector

After realizing this, Paulina decides to go for Option $2.$ Gotta show some love to the soda and taco! 😄

Pop Quiz

Discussion

It is possible to convert a mixed number into a percent and the other way around. Consider the following mixed number.
*expand_more*
*expand_more*
*expand_more*

$792 $

This mixed number can be rewritten into a percent by following three steps.
1

Convert the Fraction Part to a Decimal

The fraction part of $792 $ is $92 .$ To rewrite it as a decimal, divide the numerator $2$ by the denominator $9$ by using long division.

The quotient is approximately $0.22.$ Note that it is often enough to divide to two decimal places.

2

Add the Decimal to the Integer Part

The integer part of the mixed number is $7.$ In the previous step, the fraction part $92 $ was rewritten as the decimal number $0.22.$ Add the integer part and the decimal part.

$7+0.22=7.22 $

The mixed number can be written as the decimal number $7.22.$ 3

Convert the Decimal to a Percent

Rewrite the decimal as a percent by multiplying it by $100$ and adding a percent sign.

$7.22⋅100=722% $

Therefore, $792 $ corresponds to $722%.$ Discussion

It is possible to convert a percent into a mixed number and the other way around. Keep in mind that if a percent less than $100%$ is converted into a fraction, then the result is a proper fraction. This means that it only makes sense to convert percents greater than $100%$ into mixed numbers. Consider the following percent.
*expand_more*
*expand_more*
*expand_more*

$234% $

This percent can be rewritten into a mixed number by following three steps.
1

Write the Percent as a Fraction With $100$ as Denominator

Recall that a percent is the ratio of a number to $100.$ Start by writing the percent as a fraction whose numerator is the number part of the percent and whose denominator is $100.$

$234%=100234 $

2

Divide the Numerator by the Denominator

Divide the numerator by the denominator to rewrite the improper fraction $100234 $ as a mixed number. Use long division to find the quotient and remainder.

The quotient is $2$ and the remainder is $34.$ Write the quotient $2$ as the integer part of the mixed number and the remainder $34$ as the numerator of the fraction part. Remember that the denominator of the fraction part is the same as the denominator of the improper fraction, $100.$

$100234 =210034 $

3

Simplify the Fraction Part

The final step is to simplify the fraction part. Find the gretest common factor (GCF) of the numerator and denominator by splitting $34$ and $100$ into prime factors.

$34100 =2⋅17=2⋅2⋅5⋅5 $

The numbers share only one common factor, $2.$ This means that $2$ is the GCF of $34$ and $100.$ Divide the numerator and the denominator by $2.$
Finally, the mixed number that corresponds to $234%$ can be written.
$234%=25017 $

Example

After school, Paulina goes to her dance class. She and her partner learn a couple of new moves that keep them on their toes!
### Hint

a Rewrite the fraction as a mixed number by dividing the numerator by the denominator by using long division.
### Solution

At the end of the lesson, Paulina checks her fitness tracker and discovers that she completed $235%$ of her daily activity goal.

a What mixed number corresponds to this percent?

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b Paulina's partner has completed $1107 $ of his daily activity goal. What percent corresponds to this mixed number?

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">%<\/span><\/span><\/span><\/span>","answer":{"text":["170"]}}

a In order to rewrite $235%$ as a mixed number, start by recalling the definition of a percent. A percent is the ratio of a number to $100.$

$n%=100n $

This formula can be used to rewrite $235%$ as a fraction.
$235%=100235 $

Next, divide the numerator by the denominator to rewrite the fraction as a mixed number. Recall that a mixed number consists of an integer part and a fraction part. Use long division to find the integer quotient and remainder.
The quotient is $2,$ so this is the integer part of the mixed number. The remainder is $35,$ which makes it the numerator of the fraction part. The denominator of the fraction part of the mixed number is the same as the denominator of the fraction.

$100235 =210035 $

Finally, simplify the fraction part. Begin by splitting the numerator and denominator into prime factors.
$35100 =5⋅7=2⋅2⋅5⋅5 $

The numbers share only one common factor, $5.$ This means that this is their greatest common factor (GCF). Simplify the fraction by dividing its numerator and denominator by their GCF.
The percent of $235$ corresponds to the mixed number of $2207 .$
b This time the mixed number $1107 $ should be written as a percent. The fraction part is $107 .$ Divide the numerator by the denominator to write it as a decimal. Since the denominator is $10,$ move the decimal point one place to the left.

Now, add the integer part $1$ and the decimal part $0.7.$ This gives the decimal number $1.7.$

$1107 =1.7 $

Finally, multiply the number by $100$ to rewrite it as a percent. Remember to include a percent sign!
$1.7⋅100=170% $

Therefore, the mixed number $1107 $ is equivalent to $170%.$
Pop Quiz

Convert the given percent into its equivalent mixed number or the given mixed number into a percent. Write the fraction part of a mixed number in simplest form. Round the percent to the nearest integer.

Discussion

It is possible to convert a decimal number into a percent and vice versa. Consider the following decimal number.

$5.71 $

This decimal can be rewritten as a percent by multiplying it by $100.$ The multiplication can be done by moving the decimal point two places to the right.
Then add a percent sign.

Therefore, the decimal number $5.71$ is equivalent to $571%.$

Example

After dance class — what a workout! — Paulina decides to stop by her granddad's house. She finds him in the garden collecting honey from his bee hives.
### Hint

### Solution

Paulina likes watching how hard the bees work to make this delicacy. Her grandpa proudly tells her that he has already collected $1.58$ times more honey than the same month last year.

a Express $1.58$ in percent form.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">%<\/span><\/span><\/span><\/span>","answer":{"text":["158"]}}

b When the year comes to an end, grandpa calculates that in total he collected $2.8$ times more honey this year compared to the last year. What percent does this number correspond to?

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">%<\/span><\/span><\/span><\/span>","answer":{"text":["280"]}}

a Multiply the decimal number by $100$ and add a percent sign.

b Multiply the decimal number by $100$ and add a percent sign.

a To rewrite the decimal number $1.58$ as a percent, multiply the number by $100$ and add a percent sign.

The decimal number $1.58$ corresponds to $158%.$ This number implies that Paulina's grandpa collected $158%$ of the amount of honey he harvested in the same month last year.

b Now the decimal number $2.8$ will be rewritten as a percent. To do so, multiply by $100$ and add a percent sign afterwards. An easy way to do this is by moving the decimal point two places to the right.

Therefore, the product is $280.0,$ or just $280.$

As shown, $2$