Percentages: From Sales Discounts to Test Scores

Q: What does percent mean?

Percent means 'per hundred.' The symbol % means you are dividing a number by 100. So 45% means 45 out of every 100.

2026-05-16 · A2Z Lessons

Learning Objectives:

Understand what a percentage means and how to convert between percent, decimal, and fraction forms
Calculate the sale price after a percentage discount
Find the percentage score on a test or quiz
Compute percentage increase and percentage decrease
Apply percentage calculations to real-world scenarios including tax and tips

Prerequisites

Before this lesson you should be comfortable with basic multiplication and division, and understand that a fraction like 3/4 means "3 divided by 4." If those feel shaky, review our Decimal Arithmetic lesson first.

What Is a Percentage?

The word percent comes from the Latin per centum, meaning "per hundred." The percent symbol (%) is simply a shorthand for dividing by 100. This single idea unlocks everything else in this lesson.

Think of a large square divided into 100 equal tiles. If 45 tiles are shaded, exactly 45% of the square is shaded — because 45 out of 100 equals 45/100 = 0.45.

The Three Forms: Percent, Decimal, Fraction

Every percentage has an equivalent decimal and fraction form. Being able to switch between them quickly is the foundation of all percentage work.

Percent → Decimal: divide by 100 (move decimal left 2 places)
45% = 45 ÷ 100 = 0.45
7% = 7 ÷ 100 = 0.07
120% = 120 ÷ 100 = 1.20

Decimal → Percent: multiply by 100 (move decimal right 2 places)
0.38 = 0.38 × 100 = 38%
0.06 = 0.06 × 100 = 6%
1.75 = 1.75 × 100 = 175%

The Core Percentage Formula

Almost every percentage problem is a variation of this one formula:

Part = Percent × Whole

Rearrangements:
Percent = Part ÷ Whole (then × 100 to express as %)
Whole = Part ÷ Percent

Label each number in the problem as Part, Percent, or Whole, decide which is unknown, and apply the correct rearrangement.

Calculating a Test Score Percentage

You answered 36 questions correctly out of a possible 40. What is your percentage score?

Known: Part = 36 correct answers, Whole = 40 total questions
Unknown: Percent

Step 1 — Apply the formula:
Percent = Part ÷ Whole = 36 ÷ 40 = 0.90

Step 2 — Convert decimal to percent:
0.90 × 100 = 90%

Answer: You scored 90% on the test.

Another example: 17 out of 25 questions correct.

Percent = 17 ÷ 25 = 0.68
0.68 × 100 = 68%

Answer: 68%. A passing score!

Calculating a Sale Price (Discount)

Stores mark down prices using percentage discounts. There are two ways to reach the sale price. Both give the same answer — choose whichever is faster for you.

Method A: Find the Discount Amount, Then Subtract

A jacket costs $80. It is on sale for 25% off. What is the sale price?

Step 1 — Find the discount amount:
Discount = 25% × $80 = 0.25 × 80 = $20

Step 2 — Subtract from original price:
Sale Price = $80 − $20 = $60

Answer: The jacket costs $60 on sale.

Method B: Multiply by the Remaining Percentage (Faster)

If the discount is 25%, the buyer pays (100% − 25%) = 75% of the price.

Same jacket: $80 at 25% off

Sale Price = 75% × $80 = 0.75 × 80 = $60

Answer: $60. Same result, one step fewer.

A video game originally costs $60 and is discounted 40%.

Buyer pays: 100% − 40% = 60% of $60
Sale Price = 0.60 × $60 = $36

Answer: $36.

Calculating Tax and Tips

Tax and tips work in reverse — you add a percentage rather than subtract it.

A meal costs $45. Sales tax is 8%. What is the total bill?

Tax Amount = 8% × $45 = 0.08 × 45 = $3.60
Total Bill = $45.00 + $3.60 = $48.60

Or in one step: Total = 108% × $45 = 1.08 × 45 = $48.60

You want to leave a 20% tip on a $35 restaurant bill.

Tip = 20% × $35 = 0.20 × 35 = $7.00
Total = $35 + $7 = $42.00

Percentage Increase and Percentage Decrease

These formulas measure how much something changed relative to where it started.

Percentage Change = ((New Value − Old Value) ÷ Old Value) × 100

If the result is positive → percentage increase
If the result is negative → percentage decrease

A shirt was $40 last month and costs $52 this month. What is the percentage increase?

Change = $52 − $40 = $12
Percentage = ($12 ÷ $40) × 100 = 0.30 × 100 = 30%

Answer: The price increased by 30%.

A stock was valued at $200 and dropped to $150. What is the percentage decrease?

Change = $150 − $200 = −$50
Percentage = (−$50 ÷ $200) × 100 = −0.25 × 100 = −25%

Answer: The stock decreased by 25%.

Finding the Original Price After a Discount

Sometimes a problem gives you the sale price and asks for the original. This requires working backwards.

A pair of shoes is on sale for $63 after a 30% discount. What was the original price?

The sale price is 70% of the original (100% − 30% = 70%).

Part = $63
Percent = 70% = 0.70
Unknown: Whole (Original Price)

Whole = Part ÷ Percent = $63 ÷ 0.70 = $90

Answer: The original price was $90.

Practice Problems

You got 22 questions correct out of 25. What is your percentage score?
Solution: (22 ÷ 25) × 100 = 88%
A $120 coat is discounted 35%. What is the sale price?
Solution: 100% − 35% = 65%. Sale price = 0.65 × $120 = $78.
A town's population grew from 15,000 to 18,000. What is the percentage increase?
Solution: ((18,000 − 15,000) ÷ 15,000) × 100 = (3,000 ÷ 15,000) × 100 = 20%.
A laptop is on sale for $680 after a 15% discount. What was the original price?
Solution: Sale price = 85% of original. Original = $680 ÷ 0.85 = $800.
A dinner bill is $55. You add 18% tax and want to leave a 20% tip on the pre-tax amount. What is the total you pay?
Solution: Tax = 0.18 × $55 = $9.90. Tip = 0.20 × $55 = $11.00. Total = $55 + $9.90 + $11.00 = $75.90.

Common Mistakes to Avoid

Forgetting to convert percent to decimal. Writing "Part = 25 × 80" instead of "Part = 0.25 × 80" is the single most common error. Always divide by 100 first.
Confusing "percent of" with "percent more than." "10% more than $50" means $50 + (10% × $50) = $55, not just 10% × $50 = $5.
Applying a discount twice. A 20% discount then another 20% discount is not 40% off. The second discount applies to the already-reduced price: 0.80 × 0.80 = 0.64, so the total is 36% off.
Using the wrong base for percentage change. Always divide by the original (old) value, not the new one.

Frequently Asked Questions

What does percent mean?

Percent means "per hundred." The symbol % means you are dividing a number by 100. So 45% means 45 out of every 100.

How do I convert a percentage to a decimal?

Divide the percentage by 100 — or equivalently, move the decimal point two places to the left. For example, 35% = 0.35.

How do I find what percent one number is of another?

Divide the part by the whole, then multiply by 100. For example, 18 out of 24 is (18 ÷ 24) × 100 = 75%.

What is the formula for a percentage discount?

Sale Price = Original Price × (1 − Discount Rate). For a 30% discount on a $50 item: $50 × 0.70 = $35.

How do I calculate a percentage increase?

Percentage Increase = ((New Value − Old Value) ÷ Old Value) × 100. If a price goes from $40 to $50, the increase is ((50 − 40) ÷ 40) × 100 = 25%.