Percents

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Convert between percents, decimals, and fractions.
  • Compute percent of a number, percent increase/decrease, percent change, and reverse percent.
  • Solve word problems involving tax, discounts, interest, and mixtures.

Key Ideas

  • Percent means out of 100:

    \[ 1\% = \frac{1}{100} = 0.01 \]

  • Conversions:

    • percent → decimal: ÷100 (25% → 0.25)
    • decimal → percent: ×100 (0.07 → 7%)
    • percent → fraction: write over 100 and simplify (50% = 1/2)
Important

Percent increase and decrease use multiplicative factors:
Increase → multiply by \(1 + r\)
Decrease → multiply by \(1 - r\)

Common Problem Types

Percent of a Number

Example: What is 18% of 250?
\(0.18 \times 250 = 45\)

Percent Increase / Decrease

Example: A jacket costs $80 and is discounted 25%.
Discount = \(0.25 \times 80 = 20\)
Sale price = 80 − 20 = $60

Percent Change Formula

To find the percent a value increases/decreases relative to the original:

\[ \text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\% \]

Interpreting the Sign

The percent change formula can produce either a positive or negative result.

  • Positive result → percent increase
  • Negative result → percent decrease

Example:

A price drops from $50 to $40.

\[ \frac{40-50}{50}\times100\% = -20\% \]

The negative sign indicates a decrease.

So you may report the answer as either:

  • \(-20\%\) change
  • 20% decrease

Both describe the same situation.

Reverse Percent (Finding Original)

Example: After a 20% discount, the price is $64.
\(0.80x = 64 \Rightarrow x = 80\)

Consecutive Percent Changes

Multiply factors:

\[ (1.10)(0.90) = 0.99 \]

Net change = −1%.

Percents in Context

  • Tax/tip → multiply by \(1+r\)
  • Discounts → multiply by \(1-r\)
  • Multi-step → apply each percent in order

Strategies

  • Convert percents to decimals before multiplying.
  • Use percent change formula anytime the phrase “by what percent” appears.
  • Watch wording:
    • “increase by 20%” vs “increase to 20%”.
  • For reverse percent problems, divide by the growth/decay factor.

Worked Examples

Example 1

Question: What is 12% of 450?
Solution: \(0.12 \times 450 = 54\)

Example 2

Question: A score rises from 40 to 50. What is the percent increase?
Change = \(10\)
Percent increase:

\[ \frac{10}{40} = 0.25 = 25\% \]

Example 3

Question: A laptop marked $1200 has a 15% discount and then 8% tax.
Discount: \(1200 \times 0.85 = 1020\)
Tax: \(1020 \times 1.08 = 1101.60\)

WarningCommon Mistakes
  • Using additive thinking for percent increase/decrease instead of factors (e.g., adding 20 instead of multiplying by \(1.20\)).
  • Computing percent change with the wrong denominator (should be the original value).
  • Reversing percent problems incorrectly (e.g., dividing by 0.20 instead of 0.80 for a 20% discount).
  • Assuming consecutive percent changes add (e.g., +20% then −20% = 0 — this is false).
  • Mixing up “increase by 20%” vs “increase to 20%.”

Practice Problems

  1. What is 7.5% of 200?
  2. A coat is reduced 30% and sells for $140. What was the original price?
  3. A price rises 12% then 5%. Approximate the net percent change.

1.
\(7.5\% = 0.075\)
\(0.075 \times 200 = 15\)
Answer: 15

2.
30% discount → pay 70% of original.
\(0.70x = 140\)
\(x = 200\)
Answer: $200

3.
+12% → \(1.12\)
+5% → \(1.05\)
Product → \(1.12 \times 1.05 = 1.176\)
Net = \(17.6\%\) increase
Answer: +17.6%

Summary

  • Percent = out of 100.
  • Convert using ×100 / ÷100.
  • Increase → multiply by \(1 + r\); decrease → multiply by \(1 - r\).
  • Percent change uses
    \[ \frac{\text{new - original}}{\text{original}} \times 100\%. \]
  • Consecutive percents multiply (not add).
  • Reverse percent uses division by the percent factor.
  • Change percent to decimal immediately before multiplying.
  • Percent change → always divide by the original value.
  • For reverse percent problems, divide by \(1 \pm r\).
  • Consecutive percent changes multiply — never add.