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➗ Math · Numbers and Algebra · Lesson 6 of 8 · 6 min

Percent problems: part, whole, discount and percent change

🏷️ Sale prices, tax, raises and "what percent of the patients" questions all use one short equation.

💡 Key idea

Part = percent × whole, and percent change always divides by the original amount.

🧠 The idea

Almost every percent problem fits PART = PERCENT × WHOLE, with the percent written as a decimal. Find the missing piece. What is 15% of 80? Part = 0.15 × 80 = 12. 12 is what percent of 48? 12 ÷ 48 = 0.25 = 25%. 18 is 30% of what number? Whole = 18 ÷ 0.30 = 60. A DISCOUNT is subtracted from the original price and TAX or a tip is added to it; a 20% discount means you pay 80% of the price. PERCENT CHANGE = (new − original) ÷ original × 100. Always divide by the ORIGINAL amount, not the new one. A price that rises from $40 to $50 has increased by 10 ÷ 40 = 25%. Two percent changes in a row do not simply add: a 10% raise then a 10% cut leaves you at 0.9 × 1.1 = 99% of the start.

✏️ Worked example

Scrubs cost $36. They are 25% off, and then 5% sales tax is added to the sale price. What is the total?
Step 1: Discount: 25% of $36 = 0.25 × 36 = $9.
Step 2: Sale price: $36 − $9 = $27.
Step 3: Tax: 5% of $27 = 0.05 × 27 = $1.35.
Step 4: Total: $27 + $1.35 = $28.35.
Answer: $28.35.

📌 Takeaways

  • Part = percent × whole; write the percent as a decimal.
  • A p% discount means you pay (100 − p)% of the price.
  • Percent change = change ÷ original × 100.

✅ Check yourself

Question 1 of 3

What is 35% of 60?

  1. 25
  2. 210
  3. 21
  4. 2.1
Show the answer and rationale

Answer: C

Part = 0.35 × 60 = 21. The answers 2.1 and 210 slip the decimal point one place each way (0.035 × 60 and 3.5 × 60), and 25 subtracts 35 from 60 instead of taking a percent.

Question 2 of 3

A clinic saw 40 patients on Monday and 50 on Tuesday. What was the percent increase?

  1. 10%
  2. 20%
  3. 125%
  4. 25%
Show the answer and rationale

Answer: D

Change = 50 − 40 = 10. Divide by the original: 10 ÷ 40 = 0.25 = 25%. The answer 20% divides by the new amount (10 ÷ 50), 10% confuses the change in patients with a percent, and 125% is the new amount as a percent of the old (50 ÷ 40), not the increase.

Question 3 of 3

27 is 45% of what number?

  1. 12.15
  2. 60
  3. 72
  4. 45
Show the answer and rationale

Answer: B

Whole = part ÷ percent = 27 ÷ 0.45 = 60. Check: 0.45 × 60 = 27. The answer 12.15 multiplies 27 × 0.45 instead of dividing, and 72 adds 27 + 45.

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