Understanding Percentages: Definition, Methods, and Examples

What are percentages?

Percentages act as a universal scale for comparing quantities. Because they always use a base of $100$, they make it easy to evaluate proportions that might otherwise be difficult to compare.

Learning how to work with fractions decimals and percentages allows you to express the same mathematical relationship in the format best suited to the problem. Whether analyzing a discount, a test score, or a probability, a percentage tells you exactly how many parts out of one hundred are being considered.

Understand per hundred

The term "percent" translates directly to "for every hundred" or "out of one hundred."

Whenever you see a percentage, you can rewrite it as a fraction with a denominator of $100$. This makes calculating with percentages straightforward. If you have $7$ parts out of $100$, you have $7$ percent. If you have a complete whole, you have all $100$ parts out of $100$, which is $100$ percent.

The foundation of every percentage is the ratio of a value to one hundred.

Use the percent symbol

The symbol $\%$ is placed immediately after a number to indicate a percentage.

Writing $43\%$ is a compact way of writing the fraction $\dfrac{43}{100}$ or the decimal $0.43$. The percent symbol mathematically instructs you to divide the preceding number by $100$.

• $1\% = \dfrac{1}{100} = 0.01$
• $43\% = \dfrac{43}{100} = 0.43$
• $99\% = \dfrac{99}{100} = 0.99$

Whenever you need to use a percentage in a calculation, replace the $\%$ symbol by either placing the number over $100$ or moving the decimal point two places to the left.

Represent percent with grids and fractions

Visual models such as hundred grids and bar models are powerful tools for understanding percentages.

A hundred grid contains exactly $100$ small squares. The number of shaded squares directly matches the percentage. You can summarize these relationships in an equivalence table to help with fractions decimals and percents conversion.

Recognise percentages above 100

A percentage is not capped at $100\%$. While $100\%$ represents exactly one complete whole, percentages can exceed $100$ when describing a value larger than the original amount.

For example, if an item's cost increases, its new price might be $125\%$ of its original price. This means the new price includes the full original amount ($100\%$) plus an additional portion ($25\%$).

Worked examples

Applying these methods involves structured steps. Finding a percent of a number is the most common application, followed by calculating increases or working backwards to find original values.

Example 1: Finding the percent of a number

Question: What is $35\%$ of $400$?

Method:

1. Write the percentage as a decimal by dividing by $100$ to get $0.35$.
2. Multiply the decimal by the total amount: $0.35 \times 400 = 140$.

Answer: $140$.

Check: Since $10\%$ of $400$ is $40$, $30\%$ is $120$. Half of $10\%$ is $5\%$, which is $20$. Adding $120$ and $20$ gives $140$.

Example 2: Calculating a percent increase

Question: A plant measuring $60\text{ cm}$ grows by $15\%$. What is its new height?

Method:

1. Find $15\%$ of the original height: $0.15 \times 60 = 9$.
2. Add the increase to the original height: $60 + 9 = 69$.

Answer: The new height is $69\text{ cm}$.

Check: You can also multiply the original height by $1.15$, which represents $115\%$. Calculating $60 \times 1.15$ gives $69$.

Example 3: Finding the original amount (reverse percent)

Question: A jacket is on sale for $48\text{ dollars}$, which is a $20\%$ discount from its original price. What was the original price?

Method:

1. Determine the percentage of the original price that the sale price represents. A $20\%$ discount means the jacket costs $80\%$ of its original price.
2. Set up the relationship: $80\%$ of the original price equals $48$.
3. Find $1\%$ of the original price by dividing by $80$: $48 \div 80 = 0.60$.
4. Multiply by $100$ to find the full $100\%$ original price: $0.60 \times 100 = 60$.

Answer: The original price was $60\text{ dollars}$.

Check: A $20\%$ discount on $60\text{ dollars}$ is $12\text{ dollars}$. Subtracting $12$ from $60$ gives the sale price of $48\text{ dollars}$.

Common mistakes

When working with percentages, a few errors appear frequently. Being aware of them helps you avoid incorrect calculations.

• Confusing decimals and percentages: It is common to incorrectly convert $0.5$ to $5\%$. Because percent means "out of one hundred," you must multiply the decimal by $100$. The correct conversion for $0.5$ is $50\%$.
• Assuming percentages cannot exceed $100\%:$While a probability cannot exceed $100\%$, growth and price increases often result in values above $100\%$. An increase of $150\%$ means more than doubling the original amount.
• Forgetting to add or subtract for changes: When a problem asks for a percentage increase, finding the percentage is only the first step. You must add that value to the original amount to find the final total.

Frequently asked questions

What is a percentage?

A percentage is a standardized mathematical way of showing a part of a whole by expressing it as a fraction out of $100$.

How do you find the percent of a number?

Convert the percentage into a decimal by dividing by $100$, or write it as a fraction over $100$. Then, multiply that decimal or fraction by the given number.

What does $0\%$ mean?

A value of $0\%$ means zero parts out of one hundred, which is mathematically equal to $0$. It represents none of the whole amount.