Percentages — Discounts, Prices & Calculation

Percentages — Discounts, Prices & Calculation

Percentages are a handy way of saying "out of every hundred." Once you grasp the idea, calculating discounts in a store, a test score, or a restaurant tip becomes straightforward. In this page we will learn how to calculate a percent of a number and how to find a price after a discount.

Background and Basic Definitions

What is a percentage? A percentage (%) means parts out of one hundred. For example, \( 25\% \) means 25 out of every 100.

Formula — percent of a number:

\[ \text{Part} = \frac{\text{Percent}}{100} \times \text{Whole} \]

For example: \( 20\% \) of 80 is \( \frac{20}{100} \times 80 = 16 \).

Calculating a price after a discount:

\[ \text{Price after discount} = \text{Original price} - \frac{\text{Discount}}{100} \times \text{Original price} \]

Or in short: \[ \text{Price after discount} = \text{Original price} \times \left(1 - \frac{\text{Discount}}{100}\right) \]

PercentAs a fractionAs a decimal
\( 10\% \)\( \frac{1}{10} \)0.1
\( 25\% \)\( \frac{1}{4} \)0.25
\( 50\% \)\( \frac{1}{2} \)0.5
\( 75\% \)\( \frac{3}{4} \)0.75

Solution Steps

  1. Step 1 — Identify the "whole" (the number you are taking the percent of) and the required percent.
  2. Step 2 — Write the percent as a fraction with denominator 100: e.g. \( 30\% = \frac{30}{100} \).
  3. Step 3 — Multiply the fraction by the whole: \( \frac{30}{100} \times 60 = 18 \).
  4. Step 4 — If the question asks about a discount: subtract the amount you calculated from the original price.
  5. Step 5 — Check: does the answer make sense? (10% of 60 is 6, so 30% should be 18 — correct!)

Worked Examples

Example 1: Percent of a Number — Test Score

Problem: A test has 40 questions. A student answered \( 75\% \) of them correctly. How many questions did she answer correctly?

Solution:

  1. The whole is 40 questions; the required percent is \( 75\% \).
  2. Write as a fraction: \( \frac{75}{100} \).
  3. Calculate: \( \frac{75}{100} \times 40 = 0.75 \times 40 = 30 \).

Answer: The student answered 30 questions correctly.

Example 2: Discount at a Clothing Store

Problem: A shirt costs $120. At the end of the season there is a \( 30\% \) discount. How much will you pay?

Solution:

  1. Calculate the discount amount: \( \frac{30}{100} \times 120 = 36 \).
  2. Price after discount: \( 120 - 36 = 84 \).
  3. You can also use the shortcut: \( 120 \times 0.70 = 84 \).

Answer: You will pay $84.

Example 3: Discount on a Toy

Problem: A toy robot costs $250. It is on sale with a \( 20\% \) discount. What is the new price?

Solution:

  1. Discount amount: \( \frac{20}{100} \times 250 = 50 \).
  2. New price: \( 250 - 50 = 200 \).

Answer: The new price is $200.

Example 4: Finding the Percentage from Two Numbers

Problem: Out of 80 students in the class, 20 live more than 2 km from school. What percentage is that?

Solution:

  1. Write as a fraction: \( \frac{20}{80} \).
  2. Divide: \( \frac{20}{80} = 0.25 \).
  3. Multiply by 100: \( 0.25 \times 100 = 25\% \).

Answer: 25% of the students live far from school.

Example 5: Adding a Percentage — Price After an Increase

Problem: A movie ticket costs $40. The price went up by \( 15\% \). What is the new price?

Solution:

  1. Calculate the extra amount: \( \frac{15}{100} \times 40 = 6 \).
  2. New price: \( 40 + 6 = 46 \).
  3. You can also use: \( 40 \times 1.15 = 46 \).

Answer: The new price is $46.

Common Mistakes

✗ Common mistake: Calculating \( 20\% \) of 50 and getting 20 — because you forget to divide by 100 and do \( 20 \times 50 \) instead.

✓ The correct way: Always divide the percent by 100 before multiplying: \( \frac{20}{100} \times 50 = 10 \). Check: 10% of 50 is 5, so 20% is 10 — that makes sense!

✗ Common mistake: Forgetting to subtract the discount from the original price and reporting the discount amount as the final price.

✓ The correct way: The discount is the amount you save, not the price you pay. Always: final price = original price minus the discount amount.

✗ Common mistake: In a "what percentage" problem, dividing the larger number by the smaller one instead of the other way around.

✓ The correct way: Always divide the "part" by the "whole": \( \frac{\text{part}}{\text{whole}} \times 100 \). For example: 20 out of 80 is \( \frac{20}{80} \times 100 = 25\% \).

Practice Tips

  • Quick tip: \( 10\% \) of a number = divide by 10. So \( 30\% \) = three times \( 10\% \).
  • \( 50\% \) is exactly half; \( 25\% \) is a quarter; \( 75\% \) is three-quarters — worth memorizing.
  • For a quick check: estimate roughly before calculating precisely — this helps you spot big errors.
  • If \( p\% \) of \( N \) gives a result larger than \( N \), there is definitely a calculation error.

Summary and Key Formulas

  • \( p\% \) of \( N \) = \( \frac{p}{100} \times N \)
  • Price after discount = original price \( \times \left(1 - \frac{p}{100}\right) \)
  • Price after increase = original price \( \times \left(1 + \frac{p}{100}\right) \)
  • What percentage: \( \frac{\text{part}}{\text{whole}} \times 100 \)
  • Useful shortcuts: 10%=÷10, 50%=÷2, 25%=÷4