Calculating Percent Increase

A pair of sneakers was 80 dollars last year. This year it costs 100 dollars. What is the percent increase?

Identify the values: Original = 80 dollars, New = 100 dollars = Original: 80, New: 100

Find the change: $100 - 80 = 20$ dollars = Change = 20

Divide by original: $\frac{20}{80} = 0.25$ = 0.25

Convert to percent: $0.25 \times 100\% = 25\%$ = 25% increase

Answer: The price increased by **25%**.

Calculating Percent Decrease

A laptop was 800 euros. It's now on sale for 680 euros. What is the percent decrease?

Identify the values: Original = 800 euros, New = 680 euros = Original: 800, New: 680

Find the change: $680 - 800 = -120$ euros = Change = -120

Divide by original: $\frac{-120}{800} = -0.15$ = -0.15

Convert to percent: $-0.15 \times 100\% = -15\%$ = 15% decrease

Answer: The price decreased by **15%**.

Finding the New Value After Increase

A restaurant bill is 60 euros. You want to leave a 20% tip. What is the total amount including tip?

Identify the increase: 20% increase means multiplier is $1 + 0.20 = 1.20$ = Multiplier = 1.20

Calculate the total: $60 \times 1.20 = 72$ euros = 72 euros

Verify: Tip: $60 \times 0.20 = 12$ euros. Total: $60 + 12 = 72$ euros = Confirmed!

Answer: The total bill including tip is **72 euros**.

Finding the New Value After Decrease

A jacket costs 120 dollars. It's 35% off. What is the sale price?

Identify the decrease: 35% decrease means multiplier is $1 - 0.35 = 0.65$ = Multiplier = 0.65

Calculate sale price: $120 \times 0.65 = 78$ dollars = 78 dollars

Verify: Discount: $120 \times 0.35 = 42$ dollars. Sale price: $120 - 42 = 78$ dollars = Confirmed!

Answer: The sale price is **78 dollars**.

Finding the Original Value

After a 25% increase, a membership now costs 75 euros. What was the original price?

Set up the equation: Original $\times$ 1.25 = 75 euros = Original $\times$ 1.25 = 75

Solve for original: Original $= \frac{75}{1.25}$ = $\frac{75}{1.25}$

Calculate: $\frac{75}{1.25} = 60$ euros = 60 euros

Verify: $60 \times 1.25 = 75$ euros = Confirmed!

Answer: The original price was **60 euros**.

Mistake: Using the new value instead of the original value in the denominator

Why: Percent change is always relative to where you started, not where you ended up.

Correct: Always divide by the ORIGINAL value: $\frac{\text{Change}}{\text{Original}} \times 100\%$

Mistake: Thinking a 50% increase followed by a 50% decrease returns to the original

Why: After a 50% increase, the new base is larger. A 50% decrease of a larger number takes away more.

Correct: Example: 100 increases by 50% to 150. Then 150 decreases by 50% to 75, not back to 100!

Mistake: Confusing percent increase with the new total

Why: A 20% increase doesn't mean the new value is 20% - it means you ADD 20% to 100%.

Correct: For 20% increase, multiply by 1.20 (not 0.20). The new value is 120% of the original.

Mistake: Forgetting to convert the decimal to a percent

Why: 0.15 and 15% look different but represent the same amount. Forgetting to multiply by 100 gives the wrong answer format.

Correct: After dividing, multiply by 100 to express as a percent: $0.15 \times 100 = 15\%$

Shopping and Discounts

Stores use percent decrease to show how much you save during sales.

A 40% off sale on a 90 dollar item means you pay $90 \times 0.60 = 54$ dollars.

Salary and Wages

Employers and employees use percent increase to discuss raises and cost-of-living adjustments.

A 5% raise on a 50,000 dollar salary means a new salary of $50,000 \times 1.05 = 52,500$ dollars.

Population Growth

Demographers use percent change to track how populations grow or shrink over time.

A city with 500,000 people that grows by 4% will have $500,000 \times 1.04 = 520,000$ people.

Q: What's the difference between percent increase and percent of?

A: Percent OF finds a part of something (e.g., 20% of 50 = 10). Percent INCREASE adds that amount to the original (e.g., 50 increased by 20% = 50 + 10 = 60).

Q: Can percent decrease be more than 100%?

A: No, a percent decrease cannot exceed 100% because you cannot lose more than you have. A 100% decrease means the value becomes zero.

Q: Why doesn't a 25% increase followed by a 25% decrease give back the original?

A: Because the 25% decrease applies to the INCREASED value, not the original. Example: 100 + 25% = 125. Then 125 - 25% = 93.75, not 100.