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Percent Word Problems

Percent Applications and Word Problems

Percents are used everywhere in the real world, from calculating the final price of a pair of shoes on sale to figuring out the tip at a restaurant. Mastering percent applications means understanding how to find a part of a whole, and how to adjust original values based on percent increases or decreases.

The Percent Equation

At the core of all percent word problems is one simple equation:

Part=Whole×Percent\text{Part} = \text{Whole} \times \text{Percent}

Note: Always convert the percent to a decimal before multiplying. For example, 30%30\% becomes 0.300.30, 8%8\% becomes 0.080.08, and 150%150\% becomes 1.501.50.

Discounts and Sales

When an item is on sale, the discount is a percentage subtracted from the original price.

Example: A \80jacketisjacket is30%$ off. What is the sale price?

  1. Find the discount amount: Multiply the original price by the percent (as a decimal). Discount=$80×0.30=$24\text{Discount} = \$80 \times 0.30 = \$24
  2. Subtract from the original price: Sale Price=$80−$24=$56\text{Sale Price} = \$80 - \$24 = \$56

Shortcut: If the jacket is 30%30\% off, you are paying 70%70\% of the original price. \80 \times 0.70 = $56$.

Sales Tax and Tips

Sales tax and tips are extra amounts added to your original bill.

Example: A \50itemhasitem has8%$ sales tax. What is the total?

  1. Find the tax amount: Multiply the price by the tax rate. Tax=$50×0.08=$4\text{Tax} = \$50 \times 0.08 = \$4
  2. Add to the original price: Total=$50+$4=$54\text{Total} = \$50 + \$4 = \$54

Shortcut: You can multiply the original amount by 108%108\% (or 1.081.08) to find the total in one step: \50 \times 1.08 = $54$.

Percent Increase and Decrease

Sometimes you need to find the percentage by which a number has changed. To find the percent change, use this formula:

Percent Change=Amount of ChangeOriginal Amount×100\text{Percent Change} = \frac{\text{Amount of Change}}{\text{Original Amount}} \times 100

Example: A price went from \120toto$150$. What is the percent increase?

  1. Find the amount of change: $150−$120=$30\$150 - \$120 = \$30
  2. Divide by the original amount: 30120=0.25\frac{30}{120} = 0.25
  3. Convert to a percent: 0.25×100=25%0.25 \times 100 = 25\%

The price increased by 25%25\%.

Percents Greater Than 100% and Less Than 1%

Don't let unusual percents trick you; the rules stay exactly the same!

  • Greater than 100%: Represents more than the original whole. For example, a 150%150\% increase means the decimal is 1.501.50.
  • Less than 1%: Represents a very small fraction. For example, 0.5%0.5\% as a decimal is 0.0050.005. Always move the decimal point two places to the left!