Decimals and Percents

How do we convert between decimals and percents? Understanding this relationship is essential for working with proportions, discounts, and real-world calculations!

What Are Decimals?

Decimals represent parts of a whole using place value.

Place values to the right of decimal point:

• Tenths (0.1)
• Hundredths (0.01)
• Thousandths (0.001)
• And so on...

Examples:

• 0.5 = five tenths
• 0.25 = twenty-five hundredths
• 0.375 = three hundred seventy-five thousandths

Decimals show parts using powers of 10!

What Are Percents?

Percent means "per hundred" or "out of 100."

Symbol: %

Examples:

• 50% = 50 per hundred = 50/100
• 25% = 25 per hundred = 25/100
• 100% = 100 per hundred = whole thing

Percent compares to 100!

Think: If something was divided into 100 equal parts, how many parts?

The Relationship

Key concept:

• Percent = parts per 100
• Decimal = parts per 1

They represent the same value, different forms!

Example:

• 0.5 (decimal) = 50% (percent)
• Both mean "half"
• Just written differently

Converting Percent to Decimal

To convert percent to decimal:

Divide by 100 (or move decimal point 2 places left)

Method 1: Divide 50% = 50 ÷ 100 = 0.5

Method 2: Move decimal 50% = 50.% → move 2 left → 0.50 = 0.5

Remove the % sign when converting!

Examples: Percent to Decimal

Example 1: 75% 75 ÷ 100 = 0.75 Or: 75.% → 0.75

Example 2: 8% 8 ÷ 100 = 0.08 Or: 8.% → 0.08 (needs zero placeholder!)

Example 3: 100% 100 ÷ 100 = 1.00 = 1 Or: 100.% → 1.00

Example 4: 150% 150 ÷ 100 = 1.50 = 1.5 Or: 150.% → 1.50

Works for any percent!

Converting Decimal to Percent

To convert decimal to percent:

Multiply by 100 (or move decimal point 2 places right)

Add the % sign!

Method 1: Multiply 0.6 = 0.6 × 100 = 60%

Method 2: Move decimal 0.6 = 0.60 → move 2 right → 60.% = 60%

Examples: Decimal to Percent

Example 1: 0.25 0.25 × 100 = 25% Or: 0.25 → 25.% = 25%

Example 2: 0.08 0.08 × 100 = 8% Or: 0.08 → 08.% = 8%

Example 3: 1.5 1.5 × 100 = 150% Or: 1.50 → 150.% = 150%

Example 4: 0.375 0.375 × 100 = 37.5% Or: 0.375 → 37.5% (decimal in percent!)

Percents Greater Than 100%

Percents can be more than 100%!

100% = the whole thing More than 100% = more than the whole

Example: 250% = 250 ÷ 100 = 2.5 = 2.5 times the original = 2½ times as much

Common in:

• Growth rates (population increased 150%)
• Price increases
• Comparisons ("sales are 200% of last year")

Percents Less Than 1%

Percents can be less than 1%!

Example 1: 0.5% = 0.5 ÷ 100 = 0.005

Example 2: 0.25% = 0.25 ÷ 100 = 0.0025

Very small amounts:

• Interest rates (0.5% APR)
• Chemical concentrations
• Statistical probabilities

Common Conversions to Memorize

Essential conversions:

Halves:

• 50% = 0.5 = 1/2

Fourths:

• 25% = 0.25 = 1/4
• 75% = 0.75 = 3/4

Tenths:

• 10% = 0.1 = 1/10
• 20% = 0.2 = 1/5
• 30% = 0.3 = 3/10
• And so on...

Whole:

• 100% = 1.0 = 1

Knowing these speeds up calculations!

Moving the Decimal Point

Key pattern:

Percent → Decimal: Move 2 places LEFT

• 45% → 0.45
• 8% → 0.08
• 150% → 1.50

Decimal → Percent: Move 2 places RIGHT

• 0.35 → 35%
• 0.08 → 8%
• 1.2 → 120%

Remember the direction!

Why Does This Work?

Percent means per 100:

Converting TO decimal divides by 100:

• Moving decimal 2 left = dividing by 100
• 50% = 50/100 = 0.50

Converting FROM decimal multiplies by 100:

• Moving decimal 2 right = multiplying by 100
• 0.50 = 0.50 × 100 = 50%

It's all about that relationship with 100!

Adding Zeros When Needed

Sometimes need placeholder zeros:

Example 1: 5% to decimal 5% = 05% = 0.05 (need zero in tenths place)

Example 2: 0.3 to percent 0.3 = 0.30 = 30% (add zero to move decimal)

Don't be afraid to add zeros for clarity!

Decimal Percents

Percents can have decimals:

Example 1: 12.5% = 12.5 ÷ 100 = 0.125

Example 2: 6.25% = 6.25 ÷ 100 = 0.0625

Example 3: 0.5% = 0.5 ÷ 100 = 0.005

Still move decimal 2 places left!

Real-World: Sales Tax

Sales tax is a percent:

Example: 8% sales tax Convert to decimal: 8% = 0.08

**To calculate tax on $50:**$50 × 0.08 = $4.00 tax

Total: $50 +$4 = $54

Or use shortcut: $50 × 1.08 =$54 (1.08 represents 100% + 8%)

Real-World: Discounts

Discounts are percents:

Example: 25% off $80 item

Method 1: Find discount amount 25% = 0.25 Discount: $80 × 0.25 =$20 Sale price: $80 -$20 = $60

Method 2: Direct calculation Pay 75% (100% - 25% = 75%) 0.75 × $80 =$60

Both work!

Real-World: Test Scores

Scores often as percents:

Example: 18 out of 20 correct

As decimal: 18/20 = 0.9

As percent: 0.9 = 90%

Scored 90% on the test!

Convert: fraction → decimal → percent

Comparing Decimals and Percents

Which is larger?

Example: 0.45 or 40%

Convert to same form: 0.45 = 45%

Compare: 45% > 40%

So 0.45 > 40%

Convert to same form to compare!

Using Percents in Calculations

To find percent of a number:

Step 1: Convert percent to decimal Step 2: Multiply

Example: Find 35% of 60 35% = 0.35 0.35 × 60 = 21

35% of 60 is 21

Fractions, Decimals, and Percents

All three forms connected:

Example: One half

• Fraction: 1/2
• Decimal: 0.5
• Percent: 50%

Convert between any two:

• Fraction → Decimal: Divide
• Decimal → Percent: × 100
• Percent → Decimal: ÷ 100
• Decimal → Fraction: Use place value

Percent Increase and Decrease

Percent change uses decimals:

Increase: Original increases by percent

• 20% increase on $50
• Increase: $50 × 0.20 =$10
• New value: $50 +$10 = $60
• Or: $50 × 1.20 =$60

Decrease: Original decreases by percent

• 15% decrease on $80
• Decrease: $80 × 0.15 =$12
• New value: $80 -$12 = $68
• Or: $80 × 0.85 =$68

Finding What Percent One Number Is of Another

Example: 15 is what % of 60?

Step 1: Write as fraction 15/60

Step 2: Convert to decimal 15/60 = 0.25

Step 3: Convert to percent 0.25 = 25%

Answer: 15 is 25% of 60

Benchmark Percents

Useful percents to know:

10% = 0.1 (one tenth)

• Easy to find: move decimal one left
• 10% of $50 =$5

50% = 0.5 (half)

• Divide by 2
• 50% of $80 =$40

25% = 0.25 (one quarter)

• Divide by 4
• 25% of $80 =$20

Use these to estimate!

Mental Math Tricks

Finding 10%: Move decimal one place left

• 10% of 45 = 4.5

Finding 1%: Move decimal two places left

• 1% of 45 = 0.45

Finding 5%: Find 10%, then divide by 2

• 10% of 60 = 6
• 5% of 60 = 3

Finding 20%: Find 10%, then double

• 10% of 40 = 4
• 20% of 40 = 8

Common Mistakes to Avoid

❌ Mistake 1: Moving decimal wrong direction

• Percent → Decimal: LEFT 2 places
• Decimal → Percent: RIGHT 2 places

❌ Mistake 2: Forgetting the % sign

• 0.5 ≠ 5 (that's 0.5 and 500%!)
• Always include % when writing percent

❌ Mistake 3: Not adding placeholder zeros

• 5% = 0.05 (not 0.5!)
• Need that zero in tenths place

❌ Mistake 4: Confusing percent with decimal

• 0.15 is NOT 0.15%
• 0.15 = 15%

❌ Mistake 5: Wrong calculation

• Don't multiply when should divide
• Percent to decimal: ÷ 100

Problem-Solving Strategy

For conversion problems:

Percent → Decimal:

1. Divide by 100 (or move decimal 2 left)
2. Remove % sign
3. Add zeros if needed

Decimal → Percent:

1. Multiply by 100 (or move decimal 2 right)
2. Add % sign
3. Add zeros if needed

For applications:

1. Identify what you're finding
2. Convert percent to decimal
3. Multiply or calculate as needed
4. Interpret in context

Quick Reference

Conversions:

• Percent → Decimal: ÷ 100 (move 2 left)
• Decimal → Percent: × 100 (move 2 right)

Common values:

• 100% = 1.0
• 50% = 0.5
• 25% = 0.25
• 10% = 0.1
• 1% = 0.01

Remember:

• Percent means "per 100"
• Always check which direction
• Add zeros when needed
• Include % sign for percents

Applications:

• Tax: multiply price by tax rate
• Discount: multiply price by discount rate
• Change: new - old, then convert

Practice Tips

Tip 1: Practice both directions

• Percent to decimal AND decimal to percent
• Builds flexibility

Tip 2: Memorize common conversions

• 10%, 25%, 50%, 75%, 100%
• Makes problems faster

Tip 3: Visualize the movement

• Picture decimal point jumping 2 places
• Remember which direction

Tip 4: Check reasonableness

• 50% should be 0.5, not 5 or 0.05
• Does answer make sense?

Tip 5: Use real-world examples

• Store discounts
• Test scores
• Sports statistics
• Makes it concrete!

Summary

Decimals and percents are two ways to express the same value:

Percent:

• Per hundred (out of 100)
• Uses % symbol
• Examples: 25%, 50%, 100%

Decimal:

• Place value based
• Uses decimal point
• Examples: 0.25, 0.5, 1.0

Converting Percent to Decimal:

• Divide by 100
• Move decimal 2 places LEFT
• Remove % sign
• Example: 45% = 0.45

Converting Decimal to Percent:

• Multiply by 100
• Move decimal 2 places RIGHT
• Add % sign
• Example: 0.45 = 45%

Key concepts:

• Both represent parts of a whole
• Can convert between them easily
• Percents greater than 100% possible
• Percents less than 1% possible
• Add placeholder zeros when needed

Applications:

• Sales tax and discounts
• Test scores and grades
• Interest rates
• Statistical data
• Comparisons and proportions

Skills needed:

• Understanding place value
• Moving decimal points
• Multiplying and dividing by 100
• Recognizing equivalent forms

Mastering decimal-percent conversions is essential for working with real-world percentages and proportions!