Decimal Fraction Practice

Complete practice on converting between fractions and decimals, comparing decimal fractions, and mathematical reasoning with detailed solutions.

Practice Exercises 1-5
1 Fraction to decimal conversion
Exercise 1
Convert 3/10 to a decimal
Definition:

Fraction to decimal: Divide the numerator by the denominator to convert a fraction to a decimal

Conversion method:
  1. Divide the numerator by the denominator
  2. Write the result as a decimal
  3. For tenths, the decimal has one digit after the decimal point
Fraction
3/10
Division
3 ÷ 10
Decimal
0.3
3
10
→
0.3
Step 1: Understand the denominator

Denominator is 10, so we have tenths

Step 2: Divide numerator by denominator

3 ÷ 10 = 0.3

Step 3: Write the decimal

3/10 = 0.3

3/10 = 0.3
Final answer:

3/10 = 0.3

Applied rules:

• Denominator 10: One decimal place

• Division: Numerator ÷ Denominator

• Tenths: 1/10 = 0.1

2 Fraction to decimal conversion
Exercise 2
Convert 7/100 to a decimal
Definition:

Fraction to decimal: Divide the numerator by the denominator to convert a fraction to a decimal

Fraction
7/100
Division
7 ÷ 100
Decimal
0.07
7
100
→
0.07
Step 1: Understand the denominator

Denominator is 100, so we have hundredths

Step 2: Divide numerator by denominator

7 ÷ 100 = 0.07

Step 3: Write the decimal

7/100 = 0.07

7/100 = 0.07
Final answer:

7/100 = 0.07

Applied rules:

• Denominator 100: Two decimal places

• Division: Numerator ÷ Denominator

• Hundredths: 1/100 = 0.01

3 Decimal to fraction conversion
Exercise 3
Convert 0.4 to a fraction
Definition:

Decimal to fraction: Write the decimal as a fraction with denominator based on place value

Decimal
0.4
Place Value
4 tenths
Fraction
4/10
0.4
→
4
10
Step 1: Identify the place value

0.4 has one digit after decimal point, so it's tenths

Step 2: Write the numerator

The digit after the decimal is 4

Step 3: Write the denominator

One decimal place means denominator is 10

Step 4: Write the fraction

0.4 = 4/10

0.4 = 4/10
Final answer:

0.4 = 4/10

Applied rules:

• Decimal Place Count: Number of decimal places determines denominator

• Denominator: 1 followed by zeros equal to decimal places

• Numerator: Digits after decimal point

4 Decimal to fraction conversion
Exercise 4
Convert 0.25 to a fraction
Definition:

Decimal to fraction: Write the decimal as a fraction with denominator based on place value

Decimal
0.25
Place Value
25 hundredths
Fraction
25/100
0.25
→
25
100
Step 1: Identify the place value

0.25 has two digits after decimal point, so it's hundredths

Step 2: Write the numerator

The digits after the decimal are 25

Step 3: Write the denominator

Two decimal places means denominator is 100

Step 4: Write the fraction

0.25 = 25/100

0.25 = 25/100
Final answer:

0.25 = 25/100

Applied rules:

• Decimal Place Count: Number of decimal places determines denominator

• Denominator: 1 followed by zeros equal to decimal places

• Numerator: Digits after decimal point

5 Comparing decimal fractions
Exercise 5
Compare 3/10 and 0.3 using <, >, or =
Definition:

Comparing decimal fractions: Convert both to the same form (both fractions or both decimals) before comparing

Convert
3/10 = 0.3
Compare
0.3 = 0.3
3
10
→
0.3
Step 1: Convert fraction to decimal

3/10 = 0.3

Step 2: Compare the decimals

0.3 and 0.3 are the same

Step 3: Write the comparison

3/10 = 0.3

3/10 = 0.3
Final answer:

3/10 = 0.3

Applied rules:

• Conversion: Convert to same form before comparing

• Equivalence: 3/10 and 0.3 represent the same value

• Decimal Conversion: 3 ÷ 10 = 0.3

Practice Exercises 6-10
6 Fraction to decimal conversion
Exercise 6
Convert 45/100 to a decimal
Definition:

Fraction to decimal: Divide the numerator by the denominator to convert a fraction to a decimal

Conversion method:
  1. Divide the numerator by the denominator
  2. Write the result as a decimal
  3. For hundredths, the decimal has two digits after the decimal point
Fraction
45/100
Division
45 ÷ 100
Decimal
0.45
45
100
→
0.45
Step 1: Understand the denominator

Denominator is 100, so we have hundredths

Step 2: Divide numerator by denominator

45 ÷ 100 = 0.45

Step 3: Write the decimal

45/100 = 0.45

45/100 = 0.45
Final answer:

45/100 = 0.45

Applied rules:

• Denominator 100: Two decimal places

• Division: Numerator ÷ Denominator

• Hundredths: 1/100 = 0.01

7 Fraction to decimal conversion
Exercise 7
Convert 7/1000 to a decimal
Definition:

Fraction to decimal: Divide the numerator by the denominator to convert a fraction to a decimal

Fraction
7/1000
Division
7 ÷ 1000
Decimal
0.007
7
1000
→
0.007
Step 1: Understand the denominator

Denominator is 1000, so we have thousandths

Step 2: Divide numerator by denominator

7 ÷ 1000 = 0.007

Step 3: Write the decimal

7/1000 = 0.007

7/1000 = 0.007
Final answer:

7/1000 = 0.007

Applied rules:

• Denominator 1000: Three decimal places

• Division: Numerator ÷ Denominator

• Thousandths: 1/1000 = 0.001

8 Decimal to fraction conversion
Exercise 8
Convert 0.08 to a fraction
Definition:

Decimal to fraction: Write the decimal as a fraction with denominator based on place value

Decimal
0.08
Place Value
8 hundredths
Fraction
8/100
0.08
→
8
100
Step 1: Identify the place value

0.08 has two digits after decimal point, so it's hundredths

Step 2: Write the numerator

The digits after the decimal are 08 or 8

Step 3: Write the denominator

Two decimal places means denominator is 100

Step 4: Write the fraction

0.08 = 8/100

0.08 = 8/100
Final answer:

0.08 = 8/100

Applied rules:

• Decimal Place Count: Number of decimal places determines denominator

• Denominator: 1 followed by zeros equal to decimal places

• Numerator: Digits after decimal point

9 Decimal to fraction conversion
Exercise 9
Convert 0.625 to a fraction
Definition:

Decimal to fraction: Write the decimal as a fraction with denominator based on place value

Decimal
0.625
Place Value
625 thousandths
Fraction
625/1000
0.625
→
625
1000
Step 1: Identify the place value

0.625 has three digits after decimal point, so it's thousandths

Step 2: Write the numerator

The digits after the decimal are 625

Step 3: Write the denominator

Three decimal places means denominator is 1000

Step 4: Write the fraction

0.625 = 625/1000

0.625 = 625/1000
Final answer:

0.625 = 625/1000

Applied rules:

• Decimal Place Count: Number of decimal places determines denominator

• Denominator: 1 followed by zeros equal to decimal places

• Numerator: Digits after decimal point

10 Comparing decimal fractions
Exercise 10
Compare 25/100 and 0.2 using <, >, or =
Definition:

Comparing decimal fractions: Convert both to the same form (both fractions or both decimals) before comparing

Convert
25/100 = 0.25
Compare
0.25 > 0.2
25
100
→
0.25
Step 1: Convert fraction to decimal

25/100 = 0.25

Step 2: Compare the decimals

0.25 and 0.2

Step 3: Compare place values

0.25 has 2 tenths and 5 hundredths

0.2 has 2 tenths and 0 hundredths

Since 5 hundredths > 0 hundredths, then 0.25 > 0.2

Step 4: Write the comparison

25/100 > 0.2

25/100 > 0.2
Final answer:

25/100 > 0.2

Applied rules:

• Conversion: Convert to same form before comparing

• Place Value Comparison: Compare from left to right

• Decimal Conversion: 25 ÷ 100 = 0.25

Additional Practice Problems
Fraction to Decimal
3/10 = ___
47/100 = ___
9/1000 = ___
15/100 = ___
8/10 = ___
Decimal to Fraction
0.7 = ___
0.03 = ___
0.56 = ___
0.125 = ___
0.9 = ___
Comparisons
4/10 ___ 0.4
35/100 ___ 0.3
7/100 ___ 0.07
0.8 ___ 80/100
0.15 ___ 15/1000
Mixed Practice
Convert 0.09 to fraction
Convert 123/1000 to decimal
Compare: 0.6 and 6/10
Compare: 0.25 and 24/100
Order: 0.3, 3/10, 0.30
\frac{a}{10} = 0.a
Fraction to Decimal Rule
Tenths
1/10 = 0.1
Decimal Conversion
Hundredths
1/100 = 0.01
Decimal Conversion
Thousandths
1/1000 = 0.001
Decimal Conversion
Key definitions:

Decimal Fraction: A fraction where the denominator is a power of 10 (10, 100, 1000, etc.)

Decimal: A number with a decimal point representing fractional parts

Numerator: The top number in a fraction

Denominator: The bottom number in a fraction

Place Value: The value of a digit based on its position

Conversion methods:
  1. Fraction to Decimal: Divide numerator by denominator
  2. Decimal to Fraction: Write digits after decimal as numerator, denominator based on decimal places
  3. Comparing: Convert to same form (both fractions or both decimals)
Tip 1: For tenths, denominator is 10 and decimal has 1 place.
Tip 2: For hundredths, denominator is 100 and decimal has 2 places.
Tip 3: For thousandths, denominator is 1000 and decimal has 3 places.
Tip 4: Always convert to the same form before comparing.
Practice Summary: Decimal Fractions
\frac{a}{10^n} = 0.\underbrace{00...a}_{n \text{ digits}}
General Rule
Key definitions:

Decimal Fraction: A fraction where the denominator is a power of 10 (10, 100, 1000, etc.).

Decimal: A number that represents a fraction with denominators of 10, 100, 1000, etc.

Numerator: The top number in a fraction that tells how many parts are being considered.

Denominator: The bottom number in a fraction that tells how many equal parts the whole is divided into.

Place Value: The value of a digit based on its position relative to the decimal point.

Conversion methods:
  1. Fraction to Decimal: Divide the numerator by the denominator.
  2. Decimal to Fraction: Write the digits after the decimal point as the numerator, with the denominator being 1 followed by as many zeros as there are decimal places.
  3. Comparing: Convert both numbers to the same form (both fractions or both decimals) before comparing.
  4. Simplifying: Reduce fractions to lowest terms when possible.
Tip 1: When converting fractions with denominators of 10, 100, or 1000, the decimal has 1, 2, or 3 places respectively.
Tip 2: The decimal point separates the whole number part from the fractional part.
Tip 3: Leading zeros after the decimal point indicate smaller place values.
Tip 4: Equivalent fractions and decimals represent the same amount.
Tip 5: Always check your work by converting back to the original form.
Decimal Patterns: 1/10 = 0.1, 1/100 = 0.01, 1/1000 = 0.001. Each denominator adds one zero and moves the decimal point one place left.
Place Value Hierarchy: Ones > Tenths > Hundredths > Thousandths. Each position to the right is 1/10 the value of the position to its left.
Equivalence Principle: Different forms can represent the same value: 1/2 = 0.5 = 5/10 = 50/100.
Key formulas to know:

• Fraction to Decimal: a/b = a ÷ b

• Decimal to Fraction: 0.ab = ab/100 (where a and b are digits)

• Place Value: nth decimal place = 1/10^n

• Equivalence: a/10 = 0.a, a/100 = 0.0a, a/1000 = 0.00a

• Comparison: Convert to same form before comparing

Questions & Answers

Question: Why does 3/10 equal 0.3? How do I know where to put the decimal point?

Answer: Great question! When the denominator is 10, we're dealing with tenths:

  • 3/10 means 3 parts out of 10 equal parts
  • This is the same as 3 tenths
  • 3 tenths written as a decimal is 0.3

The denominator tells you how many decimal places: 10 has 1 zero, so 1 decimal place; 100 has 2 zeros, so 2 decimal places; 1000 has 3 zeros, so 3 decimal places.

Example: 7/100 = 0.07 (2 decimal places because denominator has 2 zeros)

Question: How do I explain to students that 0.5 is the same as 5/10?

Answer: Explain it this way:

  • 0.5 means 5 tenths
  • 5 tenths written as a fraction is 5/10
  • Both represent the same amount: half of a whole

Use visual aids like fraction bars or circles divided into 10 equal parts. Shade 5 parts to show both 5/10 and 0.5 represent the same quantity.

Real-world example: $0.50 is the same as 50 cents out of 100 cents, which is 50/100 = 1/2.

Question: My child struggles with converting decimals to fractions. Any strategies to help them remember the process?

Answer: Here's a helpful strategy to teach your child:

  1. Count decimal places: 1 place = denominator of 10, 2 places = 100, 3 places = 1000
  2. Write digits as numerator: Use the digits after the decimal point
  3. Write denominator: 1 followed by as many zeros as decimal places

Memory trick: "Count the dots, write the lots! Count decimal places, that's how many zeros."

Example: 0.25 → 2 decimal places → denominator is 100 → numerator is 25 → 25/100

Practice with real-world examples like money (cents out of 100) to make it more concrete.

Complete Summary: Decimal Fractions

Key Concepts & Definitions

Decimal Fraction: A fraction where the denominator is a power of 10 (10, 100, 1000, etc.).

Decimal: A number that represents a fraction with denominators of 10, 100, 1000, etc.

Numerator: The top number in a fraction that tells how many parts are being considered.

Denominator: The bottom number in a fraction that tells how many equal parts the whole is divided into.

Place Value: The value of a digit based on its position relative to the decimal point.

Equivalent Forms: Different representations of the same value (e.g., 1/2 = 0.5).

Core Rules & Laws

Decimal Place Count: Number of decimal places determines the denominator (1 place = 10, 2 places = 100, 3 places = 1000).

Fraction to Decimal: Divide numerator by denominator.

Decimal to Fraction: Numerator = digits after decimal, denominator = 1 followed by zeros equal to decimal places.

Comparison Rule: Convert to same form before comparing.

Equivalence Principle: Different forms can represent the same value.

Place Value Hierarchy: Ones > Tenths > Hundredths > Thousandths.

Step-by-Step Methods
  1. Fraction to Decimal: Divide numerator by denominator
  2. Decimal to Fraction: Write digits after decimal as numerator, denominator based on decimal places
  3. Comparing: Convert both to same form (fractions or decimals)
  4. Place Value: Identify position of each digit after decimal
  5. Verification: Convert back to original form to check accuracy
Examples (Simple to Advanced)
  • Simple: 1/10 = 0.1 → One tenth
  • Intermediate: 45/100 = 0.45 → Forty-five hundredths
  • Advanced: 123/1000 = 0.123 → One hundred twenty-three thousandths
  • Decimal to Fraction: 0.7 = 7/10 → Seven tenths
  • Comparison: 0.25 = 25/100 → Both represent same value
  • Complex: 0.625 = 625/1000 → Six hundred twenty-five thousandths
Tips & Common Pitfalls
Count decimal places: 1 place = tenths, 2 places = hundredths, 3 places = thousandths.
Leading zeros matter: 0.05 ≠ 0.5 (five hundredths vs five tenths).
Same form for comparison: Convert both to fractions or decimals before comparing.
Check your work: Convert back to verify your answer is correct.
Place value matters: Position determines the value of each digit.
Key Notes & Memory Aids

Memory Trick: "Count the dots, write the lots! Count decimal places, that's how many zeros."

Decimal Pattern: 1/10 = 0.1, 1/100 = 0.01, 1/1000 = 0.001

Place Value Chart: Ones.Tenths.Hundredths.Thousandths

Conversion Logic: Denominator tells you decimal places, numerator tells you digits.

Equivalence: 0.5 = 5/10 = 50/100 = 0.50

Real-World Connection: Think of money: $0.25 = 25 cents = 25/100 dollars