- 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
- Frequently Asked Questions

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.

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.

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.

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`.

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.

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`.

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.

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.

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.

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 = 2/5`Hence,

`40%` as a percentage is equal to `2/5` as a fraction.

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%`.

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.

**Q`1`. 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%`?**

- `25%`
- `45%`
- `60%`
- `15%`

**Answer: **a

**Q`2`. If a shirt originally costs `$40` and is on sale for `20%` off, what is the sale price?**

- `$28`
- `$32`
- `$36`
- `$44`

**Answer: **b

**Q`3`. If `25%` of a number is `50`, what is the number?**

- `100`
- `150`
- `200`
- `250`

**Answer: **c

**Q`4`. A car's value depreciated by `10%` over the year. If its original value was `$20,000`, what is its current value?**

- `$18,000`
- `$18,500`
- `$19,000`
- `$19,500`

**Answer: **a

**Q`5`. If you score `80%` on a test with `50` questions, how many questions did you answer correctly?**

- `30`
- `35`
- `40`
- `45`

**Answer: **c

**Q`1`. What is the percentage formula?**

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

**Q`2`. How do you convert a decimal to a percentage?**

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

**Q`3`. 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.

**Q`4`. 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

**Q`5`. 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`.