Percents - Grade 8

1. Convert Between Percents, Fractions, and Decimals

Percent to Decimal:

Divide by 100 (move decimal point 2 places left)

Formula: $ \text{Decimal} = \frac{\text{Percent}}{100} $

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

Decimal to Percent:

Multiply by 100 (move decimal point 2 places right)

Formula: $ \text{Percent} = \text{Decimal} \times 100 $

Example: $ 0.75 = 0.75 \times 100 = 75\% $

Percent to Fraction:

Write as fraction over 100, then simplify

Formula: $ \text{Percent}\% = \frac{\text{Percent}}{100} $ (simplify)

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

Fraction to Percent:

Divide numerator by denominator, then multiply by 100

Formula: $ \frac{a}{b} = \frac{a}{b} \times 100\% $

Example: $ \frac{3}{4} = 0.75 \times 100 = 75\% $

2. Compare Percents to Fractions and Decimals

Key Strategy: Convert all values to the same form (all percents, all decimals, or all fractions)

Example: Compare $ 0.6 $, $ 55\% $, and $ \frac{3}{5} $

Convert to percents: $ 0.6 = 60\% $, $ 55\% = 55\% $, $ \frac{3}{5} = 60\% $

Order: $ 55\% < 0.6 = \frac{3}{5} $

3. Find What Percent One Number is of Another

Main Formula:

$ \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100 $

Example: What percent is 15 of 60?

$ \frac{15}{60} \times 100 = 0.25 \times 100 = 25\% $

Word Problem Strategy:

Identify the part (smaller amount)
Identify the whole (total amount)
Apply the formula and solve

4. Estimate Percents of Numbers

Quick Estimation Techniques:

10%: Move decimal one place left → $ 10\% \text{ of } 80 = 8 $
1%: Move decimal two places left → $ 1\% \text{ of } 80 = 0.8 $
50%: Divide by 2 → $ 50\% \text{ of } 80 = 40 $
25%: Divide by 4 → $ 25\% \text{ of } 80 = 20 $
5%: Find 10% and divide by 2 → $ 5\% \text{ of } 80 = 4 $

Tip: Round numbers to friendly values for easier mental math

5. Percents of Numbers and Money Amounts

Basic Formula:

$ \text{Part} = \frac{\text{Percent}}{100} \times \text{Whole} $

or $ \text{Part} = \text{Decimal} \times \text{Whole} $

Example 1: Find 35% of 200

$ 0.35 \times 200 = 70 $

Example 2: Find 15% of $80

$ 0.15 \times 80 = \$12 $

Word Problem Tips:

Look for the word "of" (indicates multiplication)
Convert percent to decimal first
For money, round to 2 decimal places

6. Compare Percents of Numbers

Strategy: Calculate each percent, then compare results

Example: Which is greater: 40% of 50 or 25% of 80?

$ 40\% \text{ of } 50 = 0.40 \times 50 = 20 $

$ 25\% \text{ of } 80 = 0.25 \times 80 = 20 $

Answer: They are equal

7. Solve Percent Equations

Standard Percent Equation:

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

or $ P = r \times W $

Finding the Part:

$ P = r \times W $

Example: What is 30% of 90? → $ P = 0.30 \times 90 = 27 $

Finding the Whole:

$ W = \frac{P}{r} $

Example: 20 is 25% of what? → $ W = \frac{20}{0.25} = 80 $

Finding the Percent:

$ r = \frac{P}{W} \times 100 $

Example: 15 is what percent of 60? → $ r = \frac{15}{60} \times 100 = 25\% $

8. Percent of Change

Main Formula:

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

or $ \text{Percent Change} = \frac{\text{Amount of Change}}{\text{Original}} \times 100 $

Percent Increase:

When new value > original value (positive result)

Example: From 50 to 75

$ \frac{75-50}{50} \times 100 = \frac{25}{50} \times 100 = 50\% \text{ increase} $

Percent Decrease:

When new value < original value (negative result, report as decrease)

Example: From 80 to 60

$ \frac{60-80}{80} \times 100 = \frac{-20}{80} \times 100 = -25\% $ → 25% decrease

9. Percent of Change: Word Problems

Step-by-Step Process:

Identify original value (starting amount)
Identify new value (ending amount)
Find the difference (amount of change)
Divide by original value
Multiply by 100 to get percent
Determine increase or decrease

Example: A shirt costs $40. It goes on sale for $32. What is the percent of change?

Original = $40, New = $32

$ \frac{32-40}{40} \times 100 = \frac{-8}{40} \times 100 = -20\% $

Answer: 20% decrease

10. Find the Original Amount (Percent of Change)

When Percent Increase is Given:

$ \text{Original} = \frac{\text{New Value}}{1 + \frac{r}{100}} $

where $ r $ is the percent increase

Example: After a 20% increase, a price is $60. Find the original price.

$ \text{Original} = \frac{60}{1 + 0.20} = \frac{60}{1.20} = \$50 $

When Percent Decrease is Given:

$ \text{Original} = \frac{\text{New Value}}{1 - \frac{r}{100}} $

where $ r $ is the percent decrease

Example: After a 25% decrease, a price is $45. Find the original price.

$ \text{Original} = \frac{45}{1 - 0.25} = \frac{45}{0.75} = \$60 $

Quick Reference Formulas

What to Find	Formula
Percent to Decimal	$ \text{Decimal} = \frac{\text{Percent}}{100} $
Decimal to Percent	$ \text{Percent} = \text{Decimal} \times 100 $
Part	$ P = r \times W $
Whole	$ W = \frac{P}{r} $
Percent	$ r = \frac{P}{W} \times 100 $
Percent of Change	$ \frac{\text{New} - \text{Original}}{\text{Original}} \times 100 $
Original (after increase)	$ \text{Original} = \frac{\text{New}}{1 + \frac{r}{100}} $
Original (after decrease)	$ \text{Original} = \frac{\text{New}}{1 - \frac{r}{100}} $