Training Percents Percent Change and Applications
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Percent Change and Applications

24 min Percents

Percent Change

Percent change measures how much a quantity has increased or decreased relative to its original value. When a stock rises from 50 dollars to 65 dollars, the percent increase is 30 percent — a single number that captures the magnitude of the change regardless of the original size.

The formula for percent change is straightforward: subtract the original from the new value, divide by the original, and multiply by 100. This lesson covers both percent increase and percent decrease, along with common applications like markups, discounts, and population growth.

Formula

$$\text{Percent Change} = \frac{|\text{New} - \text{Original}|}{\text{Original}} \times 100\%$$

  • New $>$ Original → percent increase
  • New $<$ Original → percent decrease
Example 1

A shirt's price drops from $\$40$ to $\$30$. Find the percent decrease.

$$\frac{|30 - 40|}{40} \times 100\% = \frac{10}{40} \times 100\% = 25\% \text{ decrease}$$

Example 2

A stock rises from $\$25$ to $\$31$. Percent increase?

$$\frac{31 - 25}{25} \times 100\% = \frac{6}{25} \times 100\% = 24\%$$

Sales Tax

Formula

$$\text{Total Price} = \text{Original} + (\text{Tax Rate} \times \text{Original}) = \text{Original} \times (1 + r)$$

Example 3

An item costs $\$85$, tax rate $7\%$. Find the total.

Tax $= 0.07 \times 85 = \$5.95$. Total $= \$85 + \$5.95 = \$90.95$.

Or: $85 \times 1.07 = \$90.95$.

Discount

Example 4

A $\$120$ jacket is $30\%$ off. Find the sale price.

Discount $= 0.30 \times 120 = \$36$. Sale price $= 120 - 36 = \$84$.

Or: $120 \times 0.70 = \$84$.

Tip

Example 5

Dinner bill is $\$65$. You leave an $18\%$ tip. Total?

Tip $= 0.18 \times 65 = \$11.70$. Total $= \$76.70$.

Simple Interest

Formula

$$I = P \cdot r \cdot t$$

$P$ = principal, $r$ = annual rate (decimal), $t$ = time in years.

Example 6

$P = \$2{,}000$, $r = 5\%$, $t = 3$ years. Find the interest and total amount.

$I = 2000 \times 0.05 \times 3 = \$300$. Total $= \$2{,}300$.

Practice Problems

1. A phone goes from $\$600$ to $\$720$. Percent increase?
2. Gas drops from $\$3.50$ to $\$2.80$ per gallon. Percent decrease?
3. Shoes cost $\$95$, tax $6\%$. Total?
4. TV originally $\$800$, on sale for $25\%$ off. Sale price?
5. Lunch is $\$38$. Leave a $20\%$ tip. Total?
6. Invest $\$5{,}000$ at $4\%$ simple interest for 2 years. Total?
7. Population grows from $15{,}000$ to $18{,}600$. Percent increase?
8. A $\$250$ coat is $40\%$ off, then $8\%$ tax on sale price. Final cost?
9. You earned $\$450$ interest on a $\$3{,}000$ investment over 3 years. Rate?
10. Rent increases from $\$1{,}200$ to $\$1{,}350$. Percent increase?
11. Item marked $\$64$ after a $20\%$ discount. Original price?
12. A car depreciates $15\%$ per year. Value after 1 year if bought at $\$22{,}000$?
Show Answer Key

1. $20\%$

2. $20\%$

3. $\$100.70$

4. $\$600$

5. $\$45.60$

6. $\$5{,}400$

7. $24\%$

8. $\$150 \times 1.08 = \$162$

9. $5\%$

10. $12.5\%$

11. $\$80$ (since $\$80 \times 0.80 = \$64$)

12. $\$18{,}700$

Solving Percent Problems Percent Word Problems