Converting a Percent to a Fraction

Write $75\%$ as a fraction in lowest terms.

Write as a fraction over 100: $75\% = \frac{75}{100}$ = $\frac{75}{100}$

Find the GCD of 75 and 100: Factors of 75: 1, 3, 5, 15, 25, 75\nFactors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100\nGCD = 25 = GCD = 25

Divide both by GCD: $\frac{75 \div 25}{100 \div 25} = \frac{3}{4}$ = $\frac{3}{4}$

Answer: $75\% = \frac{3}{4}$

Converting a Percent to a Decimal

Write $42\%$ as a decimal.

Remember: percent means 'per 100': $42\% = \frac{42}{100}$ = $\frac{42}{100}$

Divide by 100 (move decimal 2 places left): $42 \div 100 = 0.42$ = $0.42$

Answer: $42\% = 0.42$

Converting a Fraction to a Percent

Write $\frac{3}{5}$ as a percentage.

Convert to equivalent fraction with 100 as denominator: $\frac{3}{5} = \frac{3 \times 20}{5 \times 20} = \frac{60}{100}$ = $\frac{60}{100}$

Write as a percent: $\frac{60}{100} = 60\%$ = $60\%$

Answer: $\frac{3}{5} = 60\%$

Finding a Percentage of a Number

What is $20\%$ of 80?

Convert percent to decimal: $20\% = 0.20$ = $0.20$

Multiply by the number: $0.20 \times 80 = 16$ = $16$

Answer: $20\%$ of 80 is $16$

Real-World Discount Problem

A jacket costs 60 dollars. It's on sale for 30% off. What is the sale price?

Calculate the discount amount: $30\% \text{ of } 60 = 0.30 \times 60 = 18$ = 18 dollars off

Subtract from original price: $60 - 18 = 42$ = 42 dollars

Answer: The sale price is 42 dollars

Mistake: Confusing $0.5$ with $5\%$

Why: When converting, students sometimes forget to multiply by 100. $0.5 = 50\%$, not $5\%$.

Correct: To convert decimal to percent, multiply by 100: $0.5 \times 100 = 50\%$

Mistake: Forgetting that $100\%$ means the whole thing

Why: Students sometimes think percentages can't reach 100 or go beyond.

Correct: $100\% = 1$ (all of it). You can even have more than $100\%$: $150\%$ means 1.5 times the original.

Mistake: Moving the decimal the wrong direction

Why: Confusion about which way to move when converting between decimals and percents.

Correct: Percent to decimal: divide by 100 (move left 2 places). Decimal to percent: multiply by 100 (move right 2 places).

Shopping Discounts

Stores use percentages to advertise sales and help customers calculate savings.

A 25% discount on a 40 dollar item saves you 10 dollars.

Test Scores

Teachers use percentages to show how well you performed on tests and assignments.

Getting 17 out of 20 questions correct means you scored $\frac{17}{20} = 85\%$.

Nutrition Labels

Food packages show the percent of daily nutrients in each serving.

If a snack provides 20% of your daily fiber, eating 5 servings would give you 100% of your daily fiber needs.

Q: Can a percentage be more than 100%?

A: Yes! $150\%$ means 1.5 times the original amount. For example, if a price increased by $150\%$, it's now 2.5 times what it was (original + 1.5 times original).

Q: What's the difference between percent and percentage point?

A: If an interest rate goes from $5\%$ to $8\%$, it increased by 3 percentage points. But it increased by $60\%$ (because $\frac{3}{5} = 0.6 = 60\%$).

Q: Why do we use percentages instead of fractions?

A: Percentages make comparisons easier. It's simpler to compare $75\%$ vs $80\%$ than to compare $\frac{3}{4}$ vs $\frac{4}{5}$. Having a common denominator (100) helps!