Percents

A percent is a ratio that compares a number to 100. The word percent means per hundred. For example, 35% means 35 out of 100, or 35/100. This can also be written as the decimal 0.35.

To convert a percent to a decimal, divide by 100 or move the decimal point two places to the left. For example, 62% = 0.62, 8% = 0.08, and 125% = 1.25. To convert a decimal to a percent, multiply by 100 or move the decimal point two places to the right. For example, 0.47 = 47%, 0.6 = 60%, and 1.35 = 135%.

To convert a fraction to a percent, divide the numerator by the denominator and multiply the result by 100. For example, 3/4 = 0.75 = 75%. To convert a percent to a fraction, write the percent over 100 and simplify. For example, 40% = 40/100 = 2/5.

Most percent problems involve a part, a whole, and a percent. They can be represented by the equation part = percent × whole, where the percent is written as a decimal. For example, to find 30% of 80, calculate 0.30 × 80 = 24.

To find what percent one number is of another, divide the part by the whole and multiply by 100. If 18 of 24 students completed an assignment, then 18 ÷ 24 = 0.75, or 75%.

To find the whole when the part and percent are known, divide the part by the percent written as a decimal. If 21 is 35% of a number, then the whole is 21 ÷ 0.35 = 60.

A percent proportion can also be used: part/whole = percent/100. Cross multiplication produces an equation that can be solved for the unknown value. The proportion 18/24 = p/100 gives 24p = 1,800, so p = 75.

A discount is an amount subtracted from an original price. First calculate the discount, and then subtract it from the original price. A $60 item discounted by 25% has a discount of 0.25 × 60 = $15. The sale price is $60 − $15 = $45. The sale price may also be found directly by multiplying by the percent remaining: 100% − 25% = 75%, so 0.75 × 60 = $45.

A markup is an amount added to an original cost. If a store buys an item for $40 and applies a 30% markup, the markup is 0.30 × 40 = $12. The selling price is $40 + $12 = $52. It may also be calculated directly using 130% of the original cost: 1.30 × 40 = $52.

Sales tax is added to the purchase price. If an item costs $80 and the tax rate is 7.5%, the tax is 0.075 × 80 = $6. The total cost is $86. Unless a problem states otherwise, calculate sales tax after applying any discount.

A tip is usually calculated as a percent of the bill before tax unless the situation states otherwise. For a $48 restaurant bill with an 18% tip, the tip is 0.18 × 48 = $8.64. The total before any tax is $56.64. A reasonable estimate can be used to check the answer.

Percent increase compares the amount of increase to the original value. Use percent increase = increase ÷ original × 100%. If a price rises from $50 to $65, the increase is $15. The percent increase is 15 ÷ 50 = 0.30, or 30%.

Percent decrease compares the amount of decrease to the original value. Use percent decrease = decrease ÷ original × 100%. If enrollment falls from 240 students to 198 students, the decrease is 42. The percent decrease is 42 ÷ 240 = 0.175, or 17.5%.

The original value is always the denominator when calculating percent change. Dividing by the new value is a common error. Identify the starting amount before calculating the percent change.

To find an original amount after a percent increase or decrease, write an equation using the percent multiplier. After a 20% increase, the new amount is 120% of the original, so new = 1.20 × original. After a 20% decrease, the new amount is 80% of the original, so new = 0.80 × original. If a sale price of $72 reflects a 20% discount, then 72 = 0.80x, and the original price is $90.

Multi-step percent problems must be completed in the correct order. Suppose an $80 jacket is discounted by 25% and then taxed at 8%. First calculate the sale price: 0.75 × 80 = $60. Then calculate tax on the sale price: 0.08 × 60 = $4.80. The final cost is $64.80.

Successive percent changes are not usually canceled by applying equal percentages. If a $100 price increases by 20%, it becomes $120. If the new price then decreases by 20%, it becomes 0.80 × 120 = $96, not $100, because the second percentage uses a different whole.

Common misconceptions and corrections: Convert the percent to a decimal before multiplying, identify the correct whole, use the original value when calculating percent change, subtract discounts, add markups and taxes, and perform multi-step calculations in the order described.