Finding a Percentage of a Number

For Elementary Students

What Does "Percentage OF" Mean?

When you see "20% OF 80," it means "What is 20% times 80?"

Think about it like this: If you have 80 cookies and you want to give away 20% of them, how many cookies is that?

80 cookies total
20% of them = ?

That's what we're finding!

The Basic Rule: Convert and Multiply!

Step 1: Change the percentage to a decimal Step 2: Multiply by the number

Percentage → Decimal → Multiply!

Example 1: 20% of 80

Step 1: Convert 20% to decimal

20% = 20 ÷ 100 = 0.20
(Move decimal 2 places left!)

Step 2: Multiply

0.20 × 80 = 16

Answer: 16!

Example 2: 30% of 90

Step 1: 30% = 0.30

Step 2: 0.30 × 90 = 27

Answer: 27!

Example 3: 5% of 200

Step 1: 5% = 0.05

Step 2: 0.05 × 200 = 10

Answer: 10!

Super Easy Shortcuts!

Finding 10% (Super Fast!)

Just move the decimal point LEFT one place!

10% of 250 = 25.0 (move decimal left once)
10% of 43 = 4.3
10% of 7 = 0.7

Or think: Divide by 10!

10% of 250 = 250 ÷ 10 = 25

Finding 1% (Even Easier!)

Move the decimal point LEFT two places!

1% of 600 = 6.00 (move left twice)
1% of 50 = 0.50
1% of 8 = 0.08

Or think: Divide by 100!

1% of 600 = 600 ÷ 100 = 6

Finding 50% (EASIEST!)

Just divide by 2! (50% means HALF!)

50% of 36 = 36 ÷ 2 = 18
50% of 100 = 50
50% of 7 = 3.5

Building Trick: Use 10% and 1%!

You can find ANY percentage by combining 10% and 1%!

Example: Find 15% of 200

Step 1: Find 10% of 200 = 20

Step 2: Find 5% of 200 = half of 10% = 10

Step 3: Add them!
15% = 10% + 5% = 20 + 10 = 30

Answer: 30!

Example: Find 12% of 50

Step 1: 10% of 50 = 5

Step 2: 1% of 50 = 0.5

Step 3: 12% = 10% + 1% + 1%
        = 5 + 0.5 + 0.5
        = 6

Answer: 6!

Example: Find 25% of 80

Trick: 25% is the same as 1/4!

80 ÷ 4 = 20

Answer: 20!

Real-Life Example: Sale Prices!

Problem: A jacket costs $60 and is 25% off. How much do you save?

Solution:

Step 1: Find 25% of 60

25% = 0.25
0.25 × 60 = 15

Step 2: You save $15!

Step 3: Sale price = $60 − $15 = $45

Answer: You save $15, pay $45!

Real-Life Example: Tips!

Problem: Your meal costs $40. You want to leave a 15% tip. How much?

Method 1: Convert and multiply

15% = 0.15
0.15 × 40 = 6

Tip: $6!

Method 2: Use the 10% trick

10% of 40 = 4
5% of 40 = 2 (half of 10%)
15% = 10% + 5% = 4 + 2 = 6

Tip: $6!

Real-Life Example: Tax!

Problem: A toy costs $20. Tax is 8%. How much is the tax?

8% = 0.08

0.08 × 20 = 1.60

Tax: $1.60
Total cost: $20 + $1.60 = $21.60

Common Percentages to Remember

10% = 0.10  (one tenth)
20% = 0.20  (one fifth)
25% = 0.25  (one quarter)
50% = 0.50  (one half)
75% = 0.75  (three quarters)
100% = 1.00 (the whole thing!)

Visual Example

50% of 10:

Total: ■■■■■■■■■■ (10)

Half:  ■■■■■ (5)

50% of 10 = 5

25% of 8:

Total: ■■■■■■■■ (8)

Quarter: ■■ (2)

25% of 8 = 2

Memory Trick

"Percent to decimal, then multiply real!"

Quick Tips

Tip 1: To convert % to decimal: divide by 100 (or move decimal 2 left)

Tip 2: 10% → divide by 10

Tip 3: 50% → divide by 2

Tip 4: Use 10% and 1% to build other percentages!

Tip 5: Check if the answer makes sense (20% of 100 should be less than 100!)

For Junior High Students

Understanding "Percentage Of"

Finding a percentage of a number is a fundamental operation expressing a fractional part of a quantity relative to 100.

Mathematical interpretation: "p% of n" means (p/100) × n

Algebraic notation:

p% of n = (p/100) × n

Rationale: "Percent" means "per hundred," so p% represents the fraction p/100.

The Standard Algorithm

Method: Convert percentage to decimal, then multiply.

Procedure:

1. Convert percentage to decimal: p% → p/100 or p ÷ 100
2. Multiply by the number: (p/100) × n

Example 1: Calculate 35% of 120

Step 1: 35% = 35/100 = 0.35
Step 2: 0.35 × 120 = 42

Result: 42

Example 2: Calculate 8% of 250

Step 1: 8% = 0.08
Step 2: 0.08 × 250 = 20

Result: 20

Example 3: Calculate 125% of 40

Step 1: 125% = 1.25
Step 2: 1.25 × 40 = 50

Note: Percentages can exceed 100%, yielding results greater than the original number.

Alternative: Fraction Method

Procedure: Express percentage as fraction, then multiply.

Example: Find 60% of 80

Method 1 (decimal):
60% = 0.60
0.60 × 80 = 48

Method 2 (fraction):
60% = 60/100 = 3/5
(3/5) × 80 = (3 × 80)/5 = 240/5 = 48

Advantage of fraction method: Exact arithmetic with no rounding for certain percentages (e.g., 33⅓%, 66⅔%)

Mental Calculation Strategies

Strategy 1: Powers of 10

Finding 10%: Divide by 10 (shift decimal one place left)

10% of 470 = 47
10% of 6.5 = 0.65

Finding 1%: Divide by 100 (shift decimal two places left)

1% of 850 = 8.5
1% of 42 = 0.42

Finding 100%: The number itself

100% of any number n = n

Strategy 2: Common fractions

Example: 25% of 84

25% = 1/4
84 ÷ 4 = 21

Strategy 3: Building from benchmarks

Decompose complex percentages into sums of simpler ones.

Example: Find 15% of 200

15% = 10% + 5%

10% of 200 = 20
5% of 200 = 10 (half of 10%)

15% of 200 = 20 + 10 = 30

Example: Find 37% of 100

37% = 30% + 5% + 2%

30% of 100 = 30
5% of 100 = 5
2% of 100 = 2

37% of 100 = 30 + 5 + 2 = 37

Strategy 4: Scaling

For percentages like 20%, 30%, 40%, etc., use the relationship to 10%.

Example: 40% of 75

40% = 4 × 10%

10% of 75 = 7.5
40% of 75 = 4 × 7.5 = 30

Applications

Finance: Discounts

Problem: Item priced at $180 with 30% discount. Find discount amount.

30% of $180 = 0.30 × 180 = $54

Discount: $54
Sale price: $180 − $54 = $126

Finance: Tips and gratuities

Problem: Restaurant bill is $85. Calculate 18% tip.

Method 1 (standard):
18% of $85 = 0.18 × 85 = $15.30

Method 2 (building):
10% of 85 = 8.50
8% of 85 = 6.80 (using 10% of 85 × 0.8)
18% = 10% + 8% = 8.50 + 6.80 = $15.30

Finance: Sales tax

Problem: Purchase totals $250 before 7% sales tax. Find tax amount.

7% of $250 = 0.07 × 250 = $17.50

Tax: $17.50
Total: $250 + $17.50 = $267.50

Statistics: Sample proportions

Problem: In a survey of 500 people, 65% support a proposal. Find number of supporters.

65% of 500 = 0.65 × 500 = 325

Supporters: 325 people

Science: Concentrations

Problem: Solution contains 15% salt by mass. In 80 g of solution, find mass of salt.

15% of 80 g = 0.15 × 80 = 12 g

Salt mass: 12 g

Relationship to Proportion

"p% of n" is equivalent to setting up a proportion:

p/100 = x/n

Solving for x:
x = (p/100) × n

Example: What is 40% of 75?

Proportion: 40/100 = x/75

Cross-multiply: 100x = 40 × 75 = 3000
Solve: x = 3000/100 = 30

This demonstrates that percentage problems are fundamentally proportion problems.

Inverse Operation: Finding What Percentage

Given: A number a is p% of number b Find: p

Formula: p = (a/b) × 100

Example: 15 is what percentage of 60?

p = (15/60) × 100
  = 0.25 × 100
  = 25%

Verification: 25% of 60 = 0.25 × 60 = 15 ✓

Common Errors

Error 1: Forgetting to convert percentage to decimal

❌ 20% of 50 = 20 × 50 = 1000 ✓ 20% of 50 = 0.20 × 50 = 10

Error 2: Incorrect decimal conversion

❌ 5% = 0.5 ✓ 5% = 0.05

Error 3: Confusing "of" with "increase by"

"20% of 100" = 20 "100 increased by 20%" = 120 (different!)

Error 4: Misplacing decimal in mental math

When finding 10% of 43: ❌ 10% of 43 = 0.43 ✓ 10% of 43 = 4.3

Tips for Success

Tip 1: Always convert percentage to decimal first (or fraction)

Tip 2: Use mental math shortcuts for 10%, 1%, 50%, 25%

Tip 3: Build complex percentages from simple ones (15% = 10% + 5%)

Tip 4: Verify reasonableness: result should be less than original for percentages < 100%

Tip 5: For multiples of 10%, scale from 10% (e.g., 30% = 3 × 10%)

Tip 6: Remember: "of" means multiply

Tip 7: Practice common percentages to develop fluency

Extensions: Percentage Chains

Successive percentages:

Finding p% of q% of n requires two multiplications.

Example: Find 20% of 50% of 200

Step 1: 50% of 200 = 0.50 × 200 = 100
Step 2: 20% of 100 = 0.20 × 100 = 20

Result: 20

Alternative (combined):

(0.20 × 0.50) × 200 = 0.10 × 200 = 20

Note: 20% of 50% = 10% (multiplicative, not additive)

Summary

Key formula:

p% of n = (p/100) × n

Standard procedure:

1. Convert percentage to decimal: p% → p/100
2. Multiply by number: (p/100) × n

Mental shortcuts:

• 10%: divide by 10
• 1%: divide by 100
• 50%: divide by 2
• 25%: divide by 4
• Build complex percentages from simple ones

Applications:

• Discounts and sales prices
• Tips and gratuities
• Sales tax
• Statistical proportions
• Concentrations

Practice