Chapter 4: Percents
"Percent" means per hundred — a percent is simply a ratio with a denominator of 100. Percents are one of the most universally applied mathematical tools in everyday life. They appear in sales taxes, discounts, interest rates on loans and savings, nutrition labels, test scores, population statistics, and investment returns. This chapter builds a complete toolkit for working with percents in any real-world context.
4.1 Understanding Percent
Definition: Percent
A percent (symbol: %) is a ratio that compares a number to 100. $$P\% = \frac{P}{100}$$ For example, $73\% = \dfrac{73}{100} = 0.73$. A percent can be any non-negative value — values greater than 100% indicate more than the whole.
Less than 100%
A part of the whole
e.g., 40% of a pizza
Exactly 100%
The entire whole
e.g., 100% attendance
More than 100%
More than the whole
e.g., 150% of budget
4.2 Converting Between Percent, Decimal, and Fraction
Percent → Decimal
Divide by 100 (move decimal point 2 places to the left).
$45\% \to 0.45$
$7\% \to 0.07$
$125\% \to 1.25$
Decimal → Percent
Multiply by 100 (move decimal point 2 places to the right).
$0.38 \to 38\%$
$0.075 \to 7.5\%$
$1.6 \to 160\%$
Percent → Fraction
Write over 100 and simplify.
$60\% = \dfrac{60}{100} = \dfrac{3}{5}$
$12.5\% = \dfrac{12.5}{100} = \dfrac{1}{8}$
Fraction → Percent
Divide numerator by denominator, then multiply by 100.
$\dfrac{3}{8} = 0.375 = 37.5\%$
$\dfrac{7}{4} = 1.75 = 175\%$
Memorize These Common Conversions
| Fraction | Decimal | Percent |
|---|---|---|
| $\frac{1}{100}$ | $0.01$ | $1\%$ |
| $\frac{1}{10}$ | $0.1$ | $10\%$ |
| $\frac{1}{5}$ | $0.2$ | $20\%$ |
| $\frac{1}{4}$ | $0.25$ | $25\%$ |
| $\frac{1}{3}$ | $0.\overline{3}$ | $33.\overline{3}\%$ |
| $\frac{1}{2}$ | $0.5$ | $50\%$ |
| $\frac{2}{3}$ | $0.\overline{6}$ | $66.\overline{6}\%$ |
| $\frac{3}{4}$ | $0.75$ | $75\%$ |
| $1$ | $1.0$ | $100\%$ |
4.3 Finding a Percent of a Number
Percent of a Number
To find $P\%$ of a number $W$ (the whole/base):
$$\text{Part} = \frac{P}{100} \times W = P\% \times W$$Alternatively, convert the percent to a decimal first, then multiply.
Example 4.1 — Finding the Percent of a Number
(a) Find 40% of 75.
(b) A store has 240 items. 35% are on sale. How many items are on sale?
(c) Find 6.5% of $\$380$.
(d) Find 150% of 60.
4.4 Percent Change
Definition: Percent Change
Percent change measures how much a quantity has increased or decreased relative to its original value: $$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$$ A positive result is a percent increase; a negative result is a percent decrease.
Percent Change Formula
Example 4.2 — Percent Increase and Decrease
(a) Percent Increase. A worker earns $\$48{,}000$/year and receives a raise to $\$52{,}800$/year. Find the percent increase.
(b) Percent Decrease. A sweater originally priced at $\$85$ is now $\$59.50$. Find the percent decrease.
Finding the New Value Directly
Instead of computing percent change in two steps, multiply the original by a single multiplier:
- Increase by $r\%$: New value $= \text{Original} \times (1 + \frac{r}{100})$
Example: Increase $\$200$ by 15%: $200 \times 1.15 = \$230$ - Decrease by $r\%$: New value $= \text{Original} \times (1 - \frac{r}{100})$
Example: Decrease $\$200$ by 15%: $200 \times 0.85 = \$170$
4.5 Discount and Sales Tax
Discounts
Discount Formulas
$$\text{Discount Amount} = \text{Original Price} \times \text{Discount Rate}$$ $$\text{Sale Price} = \text{Original Price} - \text{Discount Amount}$$ $$\text{Sale Price} = \text{Original Price} \times (1 - \text{Discount Rate})$$Example 4.3 — Discount Calculation
A $\$120$ jacket is 25% off. What is the sale price?
Sales Tax
Sales Tax Formulas
$$\text{Tax Amount} = \text{Price} \times \text{Tax Rate}$$ $$\text{Total Cost} = \text{Price} + \text{Tax Amount} = \text{Price} \times (1 + \text{Tax Rate})$$Example 4.4 — Sales Tax
A television costs $\$650$. The sales tax rate is 8.25%. Find the total cost.
Example 4.5 — Discount Then Tax
A $\$180$ coat is 20% off. Sales tax is 7%. What is the final price?
Note: Tax is computed on the sale price, not the original price. The combined multiplier is $0.80 \times 1.07 = 0.856$, so $180 \times 0.856 = \$154.08$.
4.6 Tip Calculations
When dining out, a tip (gratuity) is a percent of the pre-tax bill paid to the server. Common tip amounts range from 15% to 20%, but any percent can be used.
Example 4.6 — Calculating Tips
A restaurant bill before tax is $\$56.40$. Calculate a 18% tip and find the total amount paid if the tax rate is 8%.
Quick tip estimation: For a 20% tip, take 10% (move decimal one place left: $\$5.64$) and double it: $\$11.28$. For 15%, take 10% ($\$5.64$) + half of that ($\$2.82$) = $\$8.46$.
4.7 Simple Interest
Definition: Simple Interest
Simple interest is computed only on the original amount deposited or borrowed (the principal), not on accumulated interest. The formula is: $$I = P \cdot r \cdot t$$ where $I$ is the interest earned or owed, $P$ is the principal, $r$ is the annual interest rate (as a decimal), and $t$ is the time in years. $$A = P + I = P(1 + rt)$$ where $A$ is the total amount (principal + interest) after time $t$.
Simple Interest Formula
Example 4.7 — Simple Interest: Savings Account
Ana deposits $\$2{,}500$ in a savings account with a 4.5% annual simple interest rate. How much interest does she earn after 3 years? What is her total balance?
Example 4.8 — Simple Interest: Car Loan
James borrows $\$8{,}000$ to buy a used car. The annual interest rate is 6%, and he will repay the loan in 30 months. How much total interest will he pay? What is his total repayment?
Example 4.9 — Finding the Rate
An investment of $\$5{,}000$ earned $\$750$ in interest over 2 years. What was the annual interest rate?
4.8 Real-World Multi-Step Percent Problems
Example 4.10 — Successive Percent Changes
A store increases the price of a $\$200$ item by 20%, then later decreases the new price by 20%. Is the final price $\$200$?
This illustrates that successive percent changes do not cancel. A 20% increase followed by a 20% decrease gives a net change of $1.20 \times 0.80 = 0.96$, a 4% decrease overall.
Example 4.11 — Finding the Original Price
After a 30% discount, a pair of shoes costs $\$84$. What was the original price?
A common mistake is to add 30% back: $84 \times 1.30 = \$109.20 \neq \$120$. Always divide by the remaining percent.
Example 4.12 — Commission
A real estate agent earns 3% commission on the sale of a house. If the house sells for $\$285{,}000$, what is the agent's commission? If the agent needs to earn $\$12{,}000$ in commissions per month, how many $\$285{,}000$ homes must be sold?
Percent Problem Strategy Guide
| Question Type | Known | Unknown | Formula |
|---|---|---|---|
| Find the part | $P\%$, Whole $W$ | Part $A$ | $A = \frac{P}{100} \times W$ |
| Find the whole | $P\%$, Part $A$ | Whole $W$ | $W = \frac{A \times 100}{P}$ |
| Find the percent | Part $A$, Whole $W$ | Percent $P$ | $P = \frac{A}{W} \times 100$ |
| Percent change | Old, New | $\%$ change | $\frac{\text{New}-\text{Old}}{\text{Old}} \times 100$ |
| New after change | Old, $r\%$ | New | Old $\times$ $(1 \pm r/100)$ |
| Simple interest | $P$, $r$, $t$ | $I$ or $A$ | $I = Prt$, $A = P(1+rt)$ |
Practice Problems
Chapter 4 Practice
Conversions
- Convert $0.085$ to a percent.
- Convert $\dfrac{5}{8}$ to a percent.
- Convert $130\%$ to a decimal and a fraction.
Percent of a Number
- What is 72% of 350?
- 18 is 24% of what number?
- $\$63$ is what percent of $\$420$?
Percent Change, Discount, and Tax
- A population grew from 45{,}000 to 54{,}000. Find the percent increase.
- A $\$240$ TV is on sale for 35% off. Find the sale price.
- The sale price is $\$156$ after a 40% discount. What was the original price?
- Find the total cost of a $\$92$ item with 6.5% sales tax.
Tips and Interest
- A dinner bill is $\$74.00$. Calculate a 20% tip.
- $\$3{,}600$ is invested at 5% annual simple interest. How much interest is earned in 4 years?
- A loan of $\$10{,}000$ at 8% simple interest for 18 months. Find the total repayment amount.
- After a 20% price increase, a textbook costs $\$96$. What was the original price?
Answers: 1) 8.5% 2) 62.5% 3) 1.3; $\frac{13}{10}$ 4) 252 5) 75 6) 15% 7) 20% 8) $\$156$ 9) $\$260$ 10) $\$97.98$ 11) $\$14.80$ 12) $\$720$ 13) $\$11{,}200$ 14) $\$80$
Chapter Summary
- A percent means "per hundred": $P\% = \dfrac{P}{100}$.
- Conversions: Percent ↔ Decimal: divide/multiply by 100. Percent ↔ Fraction: write over 100 and simplify.
- Percent of a number: Multiply the whole by the decimal form of the percent.
- Percent change: $\dfrac{\text{New} - \text{Original}}{\text{Original}} \times 100\%$. Positive = increase; negative = decrease.
- Multiplier method: Increase by $r\%$ → multiply by $(1 + r/100)$; Decrease by $r\%$ → multiply by $(1 - r/100)$.
- Successive percent changes are not additive; apply them one at a time.
- Simple interest: $I = Prt$, $A = P(1 + rt)$, where $t$ is in years.