Introduction to Percentages

Fractions, decimals, and percentages are just three different ways of expressing the same idea: a part of a whole. The key is understanding how they relate.

Start with a fraction like $\frac{3}{4}$. This means "3 divided by 4," which gives 0.75. That decimal tells you the size of the part. To turn it into a percentage, multiply by 100 because you're asking, "how many parts out of 100 is this?" So 0.75 becomes 75%.

Going the other way, 40% means 40 out of 100, or $\frac{40}{100}$. Simplifying gives $\frac{2}{5}$.

A very common mistake is forgetting why we multiply or divide by 100. It's not random—it's because percentages are defined relative to 100.

Compare these:

• $\frac{1}{2} = 0.5 = 50\%$ (half of the whole)
• $\frac{1}{5} = 0.2 = 20\%$ (a smaller portion)

Notice how all three forms describe the same quantity, just in different languages.

Mental model: Think of fractions, decimals, and percentages as translations of the same idea—like saying the same sentence in different languages.

When you find a percentage of a quantity, you're finding a part of that total.

For example, 20% of 50 means: "what is 20 out of every 100, applied to 50?" To make this calculation easy, we convert 20% into a decimal (0.2). Why? Because decimals are designed for multiplication—they directly scale a number.

So instead of thinking "20 out of 100 of 50," we think "0.2 times 50," which gives 10.

A common mistake is forgetting to convert the percentage to a decimal before multiplying. For example, doing 20 × 50 instead of 0.2 × 50 gives a completely wrong answer.

Compare:

• Correct: $0.2 \times 50 = 10$
• Incorrect: $20 \times 50 = 1000$ (this ignores the "out of 100" idea)

Mental model: A percentage is a "scaling factor." Converting to a decimal turns it into a multiplier that shrinks or grows the original number.

Percentage change measures how much something has increased or decreased relative to where it started. That last part is crucial.

Suppose a price goes from 50 to 60. The change is 10—but is that a big change? It depends on the starting point. Compared to 50, an increase of 10 is 20%.

The process works like this:

1. Find the change (new − original)
2. Compare that change to the original
3. Convert to a percentage

A very common mistake is dividing by the new value instead of the original. This gives the wrong interpretation.

Compare:

• Correct: $10 \div 50 = 0.2 = 20\%$
• Incorrect: $10 \div 60 \approx 16.7\%$ (this misrepresents the change)

Why do we use the original? Because it represents the baseline we are measuring from.

Mental model: Percentage change asks, "How big is the change compared to where we started?"

Percentages are useful because they let you quickly judge and compare situations without needing full details.

For example:

• A 30% discount tells you how much you save relative to the original price
• A survey showing 70% agreement tells you the majority opinion
• Nutrition labels show percentages so you can compare intake across foods

A common mistake is treating percentages as absolute amounts. For example, 50% off a $10 item and 50$100 item are very different in actual savings.

Compare:

• 50% of $10 =$5 saved
• 50% of $100 =$50 saved

Same percentage, very different impact.

Mental model: Percentages are like "relative measurements"—they tell you how big something is compared to its whole, not its actual size.