# Percentage

• Introduction
• What is Percentage?
• What is Percentage Formula
• How to Calculate Percentage?
• Find the Part Given a Whole and a Percent
• Find the Whole Given a Part and a Percent
• Decimal to Percent
• Percent to Decimal
• Fraction to Percent
• Percent to Fraction
• How to Find Percentage Increase?
• How to Find Percentage Decrease?
• Practice Problems

## Introduction

The term "percentage" comes from the Latin word "per centum," which translates to "by the hundred." In simpler terms, percentages are just fractions where the denominator is always 100. It's a way of expressing the relationship between a part and a whole, with the whole always considered as 100.

Understanding percentages is essential in various contexts, helping you analyze data and make comparisons easily. It's also valuable in converting percentages into fractions and decimals. By mastering this concept, you can navigate through various mathematical scenarios with confidence.

## What is Percentage?

Percentages represent a way of expressing a fraction or ratio with the total value always set at 100. Take, for instance, if Emma achieved a 40% score on her science quiz, it means she answered correctly for 40 out of every 100 questions. This fraction is written as \frac{40}{100} or, in terms of a ratio, 40:100. The "%" symbol, representing percentage, is pronounced as "percent" or "percentage." You can interchange this symbol with "divided by 100" to convert the percentage into an equivalent fraction or decimal.

## What is Percentage Formula

The percentage formula is used to figure out a part of something about the whole, expressed as a fraction of 100. It's a way of representing numbers on a scale of 0 to 100. The formula is pretty straightforward:

\text{Percentage} = (\frac{\color{#38761d}\text{Part}}{\color{#6F2DBD}\text{Whole}}) \times 100

For example, let's say you have a class with 40 students, and you want to find out the percentage of girls. If there are 10 girls in the class, you can apply the percentage formula:

Percentage of girls = (\color{#38761d}10/\color{#6F2DBD}40) \times 100 = 25%

In this case, 25% represents the proportion of girls in the class.

Remember, whether you're dealing with test scores, populations, or any other situation where you want to express a part of a whole, the percentage formula can come in handy. It's a simple and effective way to understand and communicate proportions.

## How to Calculate Percentage?

Let's explore how to find percentage with some examples.

Example 1: Discounts at a Store

Let's say you found a shirt with an original price of $50, and it's on sale for $35. Find the discount percentage.

Solution:

To find out the discount percentage, you can use the formula:
\text{Percentage} = \frac{\color{#6F2DBD}\text{Original Price} - \color{#fb8500}\text{Sale Price}}{\color{#6F2DBD}\text{Original Price}} \times 100
\text{Percentage} = \frac{\color{#6F2DBD}50 - \color{#fb8500}35}{\color{#6F2DBD}50} \times  100 = 30%
So, the shirt is discounted by 30%.

Example 2: Exam Scores

What percentage did you score if you achieved 75 out of 100 on a math test?

Solution:

You can use the formula:
\text{Percentage} = \frac{\color{#38761d}\text{Score}}{\color{#6F2DBD}\text{Total Possible Score}} \times 100
\text{Percentage} = \frac{\color{#38761d}75}{\color{#6F2DBD}100} \times 100 = 75%
Therefore, your score is 75% on the math test.

Example 3: Population Growth

Suppose a town had a population of 5,000 in 2020 and grew to 6,500 in 2022. What’s the percentage of growth?

Solution:

To find the percentage growth, you can use the formula:
\text{Percentage} = \frac{\color{#fb8500}\text{New Population} - \color{#6F2DBD}\text{Old Population}}{\color{#6F2DBD}\text{Old Population}} \times 100
\text{Percentage} = \frac{\color{#fb8500}(6,500) - \color{#6F2DBD}(5,000)}{\color{#6F2DBD}(5,000)} \times 100 = 30%
Thus, the town's population increased by 30% from 2020 to 2022.

## Find the Part Given a Whole and a Percent

To find the part given a whole and the percent, you can use the following formula:

\color{#38761d}\text{Part} = \frac{\text{Percentage}}{100} \times \color{#6F2DBD}\text{Whole}

Example:  Suppose you have a jar filled with 80 red and blue marbles and 20% of them are red. Find the number of red marbles.

Solution:

\color{#38761d}\text{Part} = \frac{20}{100} \times \color{#6F2DBD}80 = \color{#38761d}(16)
So, there are 16 red marbles in the jar.

## Find the Whole Given a Part and a Percent

To find the whole given a part and a percent, you can use the following formula:

\color{#6F2DBD}\text{Whole} = \frac{\color{#38761d}\text{Part}}{\text{Percentage}} \times 100

Example: You have 40% of a certain quantity, and it amounts to 120. Find the total quantity.

Solution:

\color{#6F2DBD}\text{Whole} = \frac{\color{#38761d}\text{Part}}{\text{Percentage}} \times 100
= (\color{#38761d}(120) / 40) \times 100
= 3 \times 100
= \color{#6F2DBD}300
So, the total quantity is \color{#6F2DBD}300.

## Decimal to Percent

To convert a decimal to a percent, multiply it by 100 and add the percentage sign.

\text{Percent} = \color{#fb8500}\text{Decimal} \times  100

Example: Convert 0.75 to a percentage.

Solution:

Percent = 0.75 \times 100 = 75%
Therefore, 0.75 as a decimal is equivalent to 75% as a percentage.

## Percent to Decimal

To convert a percent to a decimal, divide it by 100.

\color{#fb8500}\text{Decimal} = \frac{\text{Percent}}{100}

Example: Convert 25% to a decimal.

Solution:

\color{#fb8500}\text{Decimal} = 25 / 100 = 0.25
Thus, 25% as a percentage is equal to 0.25 as a decimal.

## Fraction to Percent

To convert a fraction to a percent, multiply the fraction by 100.

\text{Percent} = \color{#fb8500}\text{Fraction} \times 100

Example: Convert 3/4 to a percentage.

Solution:

Percent = (3/4) \times 100 = 75%
Therefore, 3/4 as a fraction is equivalent to 75% as a percentage.

## Percent to Fraction

To convert a percentage to a fraction, divide it by 100 and simplify if possible.

\color{#fb8500}\text{Fraction} = \frac{\text{Percent}}{100}

Example: Convert 40% to a fraction.

Solution:

\color{#fb8500}\text{Fraction} = 40 / 100 = 2/5Hence,
40% as a percentage is equal to 2/5 as a fraction.

## How to Find Percentage Increase?

Percentage increase tells us how much a value has gone up in percentage over a certain period. It's handy when we want to figure out things like population growth or the increase in bacteria on a surface. Calculating percentage increases is pretty straightforward. You just use this formula:

\text{Percentage Increase} = \frac{(\color{#fb8500}\text{Increased Value} - \color{#6F2DBD}\text{Original Value})}{\color{#6F2DBD}\text{Original Value}} \times 100

Let's say a jacket's price went up from $50 to $75. To find out the percentage increase, plug the numbers into the formula:

\text{Percentage Increase} = \frac{(\color{#fb8500}($75) - \color{#6F2DBD}($50))}{\color{#6F2DBD}($50)} \times 100 = 50% So, the jacket's price increased by 50%. ## How to Find Percentage Decrease? On the flip side, percentage decrease helps us understand how much a value has decreased over time. Whether it's less rainfall or a decrease in the number of COVID-19 patients, a percentage decrease comes to our rescue. The formula for percentage decrease is: \text{Percentage Decrease} = \frac{(\color{#6F2DBD}\text{Original Value} - \color{#fb8500}\text{Decreased Value})}{\color{#6F2DBD}\text{Original Value}} \times 100 Suppose the amount of rainfall drops from 100 mm to 80 mm. To find the percent decrease, plug these values into the formula. \text{Percentage Decrease} = \frac{(\color{#6F2DBD}100 - \color{#fb8500}(80))}{\color{#6F2DBD}100} \times 100 = 20% So, the rainfall decreased by 20%. Understanding these percentage changes can give us insights into various situations, helping us make sense of trends and variations. ## Practice Problems Q1. In a class of 200 students, 50 students scored above 90% in the recent exam. What's the percentage of students who scored above 90%? 1. 25% 2. 45% 3. 60% 4. 15% Answer: a Q2. If a shirt originally costs $40 and is on sale for 20% off, what is the sale price?

1. $28 2. $32
3. $36 4. $44

Q3. If 25% of a number is 50, what is the number?

1. 100
2. 150
3. 200
4. 250

Q4. A car's value depreciated by 10% over the year. If its original value was $20,000, what is its current value? 1. $18,000
2. $18,500 3. $19,000
4. \$19,500

Q5. If you score 80% on a test with 50 questions, how many questions did you answer correctly?

1. 30
2. 35
3. 40
4. 45

Q1. What is the percentage formula?

Answer: The percentage formula is \frac{\text{Part}}{\text{Whole}} \times 100, representing a part of the whole.

Q2. How do you convert a decimal to a percentage?

Answer: To convert a decimal to a percentage, multiply the decimal by 100.

Q3. What is the difference between percentage increase and percentage decrease?

Answer: Percentage increase is calculated as \frac{(\text{New Value} - \text{Original Value})}{\text{Original Value}} \times 100, while percentage decrease follows the same formula with a negative result.

Q4. How do you find the original value after a percentage increase or decrease?

Answer: To find the original value after a percentage change, use the formula

• \text{Original Value} = \frac{\text{New Value}}{1 + \frac{\text{Percentage Change}}{100}} for increase
• \text{Original Value} = \frac{\text{New Value}}{1 - \frac{\text{Percentage Change}}{100}} for decrease

Q5. What is the relationship between fractions and percentages?

Answer: Fractions and percentages are related, as a percentage can be expressed as a fraction by putting it over 100. For example, 25% is equivalent to the fraction 1/4.