Percents

Understand the language of 'out of 100'

If you have ever felt confused by percentages, you are in good company. Between sales tags shouting “40% OFF!”, credit card statements showing “24.99% APR,” and news headlines declaring that “73% of people prefer…” it can feel like percent is some secret code that everyone else understands.

Here is the good news: you already understand percents more than you realize. Every time you have said “I’m 100% sure” or “That’s only half,” you have been using the core idea behind percentages. Let’s pull back the curtain and make this concept crystal clear.

Core Concepts

What Does “Percent” Actually Mean?

The word percent comes from the Latin per centum, which literally means “per hundred” or “out of 100.” That is it. When you see a percent, you are looking at a number out of 100.

Think of it this way: imagine slicing a pizza into exactly 100 tiny, equal pieces. If you eat 25 of those pieces, you have eaten 25 percent of the pizza. Written as $25%$, this simply means “25 out of 100.”

This “out of 100” system is incredibly useful because it gives us a universal way to compare things. Whether you are talking about a test score, a discount at a store, or the battery level on your phone, percent lets you express “how much of the whole” in a consistent way.

Why 100?

You might wonder why we use 100 as our reference point. The answer is practical: 100 is easy to work with mentally. It is big enough to allow for precision (you can talk about 47% or 83.5%) but small enough to visualize. When someone says they got 85% on a test, you instantly have a feel for how they did, without needing to know if the test had 20 questions or 200.

Notation and Terminology

Percent symbol (%): This symbol means “out of 100” or “divided by 100.”

$50%$ means $\frac{50}{100}$ or $0.50$

Base (or Whole): The total amount you are taking a percentage of.

Part: The portion of the whole that corresponds to the percentage.

Rate: The percent itself, expressed as a number.

These three quantities are related by a simple formula:

$$\text{Part} = \text{Rate} \times \text{Base}$$

Or, rearranged:

$$\text{Rate} = \frac{\text{Part}}{\text{Base}} \quad \text{and} \quad \text{Base} = \frac{\text{Part}}{\text{Rate}}$$

Converting Between Percents, Decimals, and Fractions

Here is something important: percents, decimals, and fractions are just three different ways of writing the same value. Learning to convert between them fluently will make working with percents much easier.

Percent to Decimal

Divide by 100 (or move the decimal point two places to the left):

$$45% = \frac{45}{100} = 0.45$$

$$7% = \frac{7}{100} = 0.07$$

$$125% = \frac{125}{100} = 1.25$$

Decimal to Percent

Multiply by 100 (or move the decimal point two places to the right):

$$0.72 = 72%$$

$$0.08 = 8%$$

$$1.5 = 150%$$

Percent to Fraction

Write the percent over 100, then simplify:

$$25% = \frac{25}{100} = \frac{1}{4}$$

$$60% = \frac{60}{100} = \frac{3}{5}$$

Fraction to Percent

Divide the numerator by the denominator, then multiply by 100:

$$\frac{3}{4} = 0.75 = 75%$$

$$\frac{1}{8} = 0.125 = 12.5%$$

Examples

Example 1: Finding a Percent of a Number

You want to leave an 18% tip on a $45 restaurant bill. How much should you tip?

Solution:

Convert 18% to a decimal: $18% = 0.18$

Multiply by the bill amount:

$$\text{Tip} = 0.18 \times 45 = 8.10$$

You should leave an $8.10 tip.

Quick check: 18% is close to 20%, and 20% of $45 is $9. Our answer of $8.10 is a bit less, which makes sense.

Example 2: Finding What Percent One Number Is of Another

You scored 42 points out of 50 on a quiz. What is your percentage score?

Solution:

Use the formula: $\text{Rate} = \frac{\text{Part}}{\text{Base}}$

$$\text{Rate} = \frac{42}{50} = 0.84$$

Convert to percent: $0.84 = 84%$

Your score is 84%.

Example 3: Finding the Whole When Given a Part and Percent

A jacket is on sale for $63, which is 70% of the original price. What was the original price?

Solution:

Here, $63 is the part and 70% is the rate. We need to find the base (original price).

$$\text{Base} = \frac{\text{Part}}{\text{Rate}} = \frac{63}{0.70} = 90$$

The original price was $90.

Check: 70% of $90 = $0.70 \times 90 = $63. Correct!

Example 4: Percent Increase and Decrease

A streaming service raised its monthly price from $12 to $15. What is the percent increase?

Solution:

First, find the amount of change:

$$\text{Change} = 15 - 12 = 3$$

Then divide by the original amount (this is important!):

$$\text{Percent Change} = \frac{\text{Change}}{\text{Original}} = \frac{3}{12} = 0.25 = 25%$$

The price increased by 25%.

Note: We divide by the original price ($12), not the new price. This is a common mistake to watch out for.

Example 5: Combining Discounts and Tax

A $120 pair of sneakers is on sale for 30% off. After the discount, you must pay 8.5% sales tax. What is the final price?

Solution:

Step 1: Calculate the discount amount.

$$\text{Discount} = 0.30 \times 120 = 36$$

Step 2: Subtract the discount from the original price.

$$\text{Sale Price} = 120 - 36 = 84$$

Step 3: Calculate the sales tax on the sale price.

$$\text{Tax} = 0.085 \times 84 = 7.14$$

Step 4: Add the tax to the sale price.

$$\text{Final Price} = 84 + 7.14 = 91.14$$

The final price is $91.14.

Alternative approach: You can also think of “30% off” as paying 70% of the price, and then paying 108.5% of that (to include tax):

$$120 \times 0.70 \times 1.085 = 91.14$$

Key Properties and Rules

The Percent Equation

$$\text{Part} = \text{Percent} \times \text{Whole}$$

This single equation, rearranged as needed, solves almost every basic percent problem.

Percent Change Formula

$$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100%$$

If the result is positive, it is an increase.
If the result is negative, it is a decrease.

Simple Interest Formula

$$I = P \times r \times t$$

Where:

$I$ = Interest earned (or owed)
$P$ = Principal (starting amount)
$r$ = Annual interest rate (as a decimal)
$t$ = Time in years

Common Percent Equivalents Worth Memorizing

Percent	Decimal	Fraction
10%	0.1	1/10
20%	0.2	1/5
25%	0.25	1/4
33.3%	0.333	1/3
50%	0.5	1/2
75%	0.75	3/4
100%	1	1

Quick Mental Math Tricks

To find 10%: Move the decimal one place left. (10% of 80 = 8)
To find 5%: Find 10%, then halve it. (5% of 80 = 4)
To find 15%: Find 10% + 5%. (15% of 80 = 8 + 4 = 12)
To find 20%: Find 10%, then double it. (20% of 80 = 16)
To find 1%: Move the decimal two places left. (1% of 80 = 0.80)

Real-World Applications

Shopping and Discounts

When a store advertises “Buy One, Get One 50% Off,” what does that really mean? If shirts cost $30 each:

First shirt: $30
Second shirt: 50% off = $15
Total: $45 for two shirts
Effective discount: You saved $15 on $60 worth of shirts = 25% off

Understanding percents helps you see through marketing language and know what you are actually paying.

Tipping at Restaurants

Many people tip between 15% and 20% at restaurants. A quick method:

Find 10% of your bill (move the decimal)
For 15%: add half of that 10%
For 20%: double the 10%

On a $73 bill:

10% = $7.30
15% = $7.30 + $3.65 = $10.95
20% = $7.30 + $7.30 = $14.60

Simple Interest on Savings and Loans

If you deposit $2,000 in a savings account earning 4% annual simple interest, how much interest do you earn in 3 years?

$$I = P \times r \times t = 2000 \times 0.04 \times 3 = 240$$

You would earn $240 in interest over 3 years.

Note: Most real-world interest is compound interest, which grows faster. But simple interest gives you a baseline understanding.

Grade Calculations

Your final grade might be weighted:

Homework: 20%
Quizzes: 30%
Final Exam: 50%

If you scored 90% on homework, 80% on quizzes, and 75% on the final:

$$\text{Final Grade} = (0.20 \times 90) + (0.30 \times 80) + (0.50 \times 75)$$ $$= 18 + 24 + 37.5 = 79.5%$$

Nutrition Labels

When a nutrition label says a serving has “15% of your daily value of iron,” it means that one serving provides 15 out of the 100 units of iron recommended for a day.

Self-Test Problems

Problem 1 (Easy): Convert 0.045 to a percent.

Show Answer

$0.045 = 4.5%$

Move the decimal point two places to the right.

Problem 2 (Easy): What is 35% of 80?

Show Answer

$$35% \times 80 = 0.35 \times 80 = 28$$

Problem 3 (Medium): A phone that originally cost $800 is now selling for $680. What is the percent discount?

Show Answer

First, find the discount amount: $800 - 680 = 120$

Then find the percent: $\frac{120}{800} = 0.15 = 15%$

The discount is 15%.

Problem 4 (Medium): After a 20% price increase, a video game costs $72. What was the original price?

Show Answer

If the price increased by 20%, the new price is 120% of the original.

$$\text{Original} = \frac{72}{1.20} = 60$$

The original price was $60.

Check: 20% of $60 = $12, and $60 + $12 = $72. Correct!

Problem 5 (Hard): You borrow $5,000 at 6% simple annual interest for 2.5 years. What is the total amount you must repay?

Show Answer

First, calculate the interest:

$$I = P \times r \times t = 5000 \times 0.06 \times 2.5 = 750$$

Then add interest to principal:

$$\text{Total} = 5000 + 750 = 5750$$

You must repay $5,750.

Summary

Percent means “per hundred.” The symbol % is shorthand for “out of 100.”
Converting is straightforward:
- Percent to decimal: divide by 100 (move decimal left 2 places)
- Decimal to percent: multiply by 100 (move decimal right 2 places)
- Percent to fraction: put over 100, then simplify
The core equation $\text{Part} = \text{Percent} \times \text{Whole}$ solves most basic percent problems. Rearrange it to find whichever value you need.
Percent change is always calculated relative to the original value, not the new value.
Simple interest uses the formula $I = Prt$, where rate is expressed as a decimal and time is in years.
Mental math shortcuts (finding 10% first, then adjusting) make everyday calculations faster.
Percents are everywhere: shopping, tipping, grades, loans, statistics, and more. Mastering them gives you a practical skill you will use for the rest of your life.

You have now built a solid foundation in percents. With practice, these calculations will become second nature, and you will find yourself quickly estimating tips, comparing discounts, and understanding statistics with confidence.