6.2 Solve General Applications of Percent

Solving Percent Problems in Real-World Situations

Percentages show up constantly: shopping discounts, tax calculations, tips, interest rates. Solving percent problems comes down to one core equation and knowing how to rearrange it. This section covers that equation, then applies it to the real-world situations you'll actually encounter.

Solving Basic Percent Equations

Every percent problem involves three quantities: the percent, the whole (also called the base), and the part. If you know any two of them, you can find the third.

A percent is just a ratio out of 100. So 60% means $\frac{60}{100}$, which equals 0.60 as a decimal. To convert any percent to a decimal, divide by 100 (75% = 0.75). To go the other way, multiply by 100 (0.4 = 40%).

The fundamental equation is:

$\text{percent} \times \text{whole} = \text{part}$

You can rearrange this to solve for whichever value is missing:

• Finding the part: $\text{part} = \frac{\text{percent}}{100} \times \text{whole}$
  • Example: 15% of 60 is $0.15 \times 60 = 9$
• Finding the whole: $\text{whole} = \frac{\text{part}}{\text{percent as a decimal}}$
  • Example: If 25 is 10% of a number, the number is $\frac{25}{0.10} = 250$
• Finding the percent: $\text{percent} = \frac{\text{part}}{\text{whole}} \times 100$
  • Example: 15 red marbles out of 60 total means $\frac{15}{60} \times 100 = 25\%$ are red

Here's how to apply this in context: if 20% of 150 students are seniors, you're finding the part. Multiply $0.20 \times 150 = 30$ seniors.

Percent Calculations in Everyday Life

Discounts reduce a price. To find the sale price:

1. Multiply the original price by the discount percent (as a decimal) to get the discount amount.
2. Subtract the discount amount from the original price.

Example: A $50 item with a 20% discount → discount amount is $50 \times 0.20 = 10$, so the sale price is $50 - 10 = \$40$.

Watch out: a 20% discount followed by another 20% discount is not a 40% discount overall. The second 20% applies to the already-reduced price, so you end up with a 36% total discount.

Sales tax increases a price. To find the total cost:

1. Multiply the pre-tax price by the tax rate (as a decimal) to get the tax amount.
2. Add the tax amount to the pre-tax price.

Example: A $100 item with 7% tax → tax is $100 \times 0.07 = 7$, so the total is $100 + 7 = \$107$.

Tips work the same way as tax: multiply the bill by the tip percent, then add it to the bill.

Example: A $40 bill with a 15% tip → tip is $40 \times 0.15 = 6$, so the total is $40 + 6 = \$46$.

Markup is how retailers set prices above their cost. Markup percentage is calculated based on the cost (not the selling price): $\text{markup \%} = \frac{\text{selling price} - \text{cost}}{\text{cost}} \times 100$. If a store buys a shirt for $20 and sells it for $30, the markup is $\frac{30 - 20}{20} \times 100 = 50\%$.

Calculating Percent Changes

Percent change tells you how much a value increased or decreased relative to where it started. The formula is:

$\text{percent change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100$

A positive result means an increase; a negative result means a decrease.

Percent increase example: An item costs $200 and increases by 15%. $200 + (200 \times 0.15) = 200 + 30 = \$230$

Percent decrease example: An item costs $80 and decreases by 25%. $80 - (80 \times 0.25) = 80 - 20 = \$60$

A common trap: repeated percent changes compound. A 10% increase followed by another 10% increase is not a 20% increase. It's actually a 21% increase, because the second 10% applies to the already-increased value. Similarly, a 100% decrease means the value drops all the way to zero.

Percentage points vs. percent change is a distinction that trips people up. If an interest rate goes from 5% to 7%, that's an increase of 2 percentage points. But as a percent change, it's $\frac{7 - 5}{5} \times 100 = 40\%$. These are very different numbers describing the same situation.

Financial Applications of Percent

Commission is money earned as a percentage of sales. If a salesperson earns 8% commission and sells $5,000 worth of products, their commission is $5{,}000 \times 0.08 = \$400$.

Simple interest is calculated with this formula:

$I = P \times r \times t$

where $I$ = interest earned, $P$ = principal (the starting amount), $r$ = annual interest rate (as a decimal), and $t$ = time in years.

Example: You deposit $1,000 at 5% annual interest for 3 years. $I = 1{,}000 \times 0.05 \times 3 = \$150$

With simple interest, you earn the same amount each year because interest is only calculated on the original principal. Compound interest is different: it calculates interest on the principal plus any interest already earned, so the amount grows faster over time. For this course, focus on simple interest, but know that the distinction exists.

When solving financial problems, always check that your rate and time use matching units. If the rate is annual, time should be in years. If you're given months, convert to years by dividing by 12.