Decimal to Fraction Conversion: Complete Notes for Students

Comprehensive notes on converting decimals to fractions with examples, exercises, and solutions for 4th grade students in PDF format.

Notes Examples: Problems 1 to 3
1 Converting Tenths
Example 1
Convert 0.4 to a fraction.
Definition:

Decimal to fraction conversion: The process of changing a decimal number into an equivalent fraction by using place value to determine the denominator.

Conversion Method Steps:
  1. Identify the decimal number
  2. Count the number of digits after the decimal point
  3. Write the decimal digits as the numerator
  4. Write the denominator based on place value
  5. Simplify if possible
Decimal
0.4
Digits After Point
1
Numerator
4
Denominator
10
Fraction
\(\frac{4}{10}\)
Simplified
\(\frac{2}{5}\)
Step 1: Identify the decimal

Decimal: 0.4

Step 2: Count decimal places

There is 1 digit after the decimal point

Step 3: Write the numerator

Take the digits after the decimal point: 4

Step 4: Write the denominator

For 1 decimal place, denominator is 10

Step 5: Write the fraction

\(0.4 = \frac{4}{10}\)

Step 6: Simplify the fraction

GCF of 4 and 10 is 2

\(\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5}\)

\(0.4 = \frac{2}{5}\)
Final answer:

\(0.4 = \frac{2}{5}\)

Applied rules:

• Place Value Rule: The number of decimal places determines the denominator (1 decimal → 10, 2 decimals → 100, etc.)

• Simplification: Divide both numerator and denominator by their greatest common factor

• Verification: Cross multiplication confirms equivalence

2 Converting Hundredths
Example 2
Convert 0.65 to a fraction.
Definition:

Hundredths place: The second digit to the right of the decimal point, representing \(\frac{1}{100}\) of the whole.

Decimal
0.65
Decimal Places
2
Numerator
65
Denominator
100
Fraction
\(\frac{65}{100}\)
Simplified
\(\frac{13}{20}\)
Step 1: Identify the decimal

Decimal: 0.65

Step 2: Count decimal places

There are 2 digits after the decimal point

Step 3: Write the numerator

Take the digits after the decimal point: 65

Step 4: Write the denominator

For 2 decimal places, denominator is 100

Step 5: Write the fraction

\(0.65 = \frac{65}{100}\)

Step 6: Simplify the fraction

Factors of 65: 1, 5, 13, 65

Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

GCF = 5

\(\frac{65}{100} = \frac{65 \div 5}{100 \div 5} = \frac{13}{20}\)

\(0.65 = \frac{13}{20}\)
Final answer:

\(0.65 = \frac{13}{20}\)

Applied rules:

• Place Value Rule: 2 decimal places → denominator is 100

• Factor Identification: Find GCF to simplify the fraction

• Verification: Cross multiplication confirms equivalence

3 Converting Mixed Numbers
Example 3
Convert 2.25 to a mixed number fraction.
Definition:

Mixed number: A number that combines a whole number and a proper fraction.

Decimal
2.25
Whole Part
2
Decimal Part
0.25
Convert Decimal
\(\frac{25}{100}\)
Simplify
\(\frac{1}{4}\)
Mixed Number
\(2\frac{1}{4}\)
Step 1: Separate whole and decimal parts

2.25 = 2 + 0.25

Step 2: Convert decimal part to fraction

0.25 has 2 decimal places → \(\frac{25}{100}\)

Step 3: Simplify the fraction

GCF of 25 and 100 is 25

\(\frac{25}{100} = \frac{1}{4}\)

Step 4: Combine with whole number

\(2.25 = 2\frac{1}{4}\)

Step 5: Verify the conversion

\(2\frac{1}{4} = 2 + 0.25 = 2.25\) ✓

\(2.25 = 2\frac{1}{4}\)
Final answer:

\(2.25 = 2\frac{1}{4}\)

Applied rules:

• Separation Method: Separate whole number from decimal part

• Decimal Conversion: Convert decimal part to fraction using place value

• Simplification: Reduce the fractional part to simplest form

Notes Summary: Decimal to Fraction Conversion
\(0.d_1d_2...d_n = \frac{d_1d_2...d_n}{10^n}\)
Core Conversion Formula
Tenths
\(0.d = \frac{d}{10}\)
One decimal place
Hundredths
\(0.dd = \frac{dd}{100}\)
Two decimal places
Thousandths
\(0.ddd = \frac{ddd}{1000}\)
Three decimal places
Key definitions:

Decimal: A number that uses a decimal point to separate the whole number part from the fractional part.

Fraction: A number that represents part of a whole, expressed as \(\frac{\text{numerator}}{\text{denominator}}\).

Place value: The value of a digit based on its position in a number (tenths, hundredths, thousandths, etc.).

Tenths place: The first digit to the right of the decimal point, representing \(\frac{1}{10}\) of the whole.

Hundredths place: The second digit to the right of the decimal point, representing \(\frac{1}{100}\) of the whole.

Thousandths place: The third digit to the right of the decimal point, representing \(\frac{1}{1000}\) of the whole.

Simplest form: A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

Decimal to Fraction Conversion Method:
  1. Identify the decimal number you want to convert
  2. Count the decimal places after the decimal point
  3. Write the decimal digits as the numerator
  4. Write the denominator as 1 followed by the same number of zeros as decimal places
  5. Simplify the fraction by dividing both numerator and denominator by their GCF
Tip 1: Count the number of digits after the decimal point to determine the denominator.
Tip 2: For 1 decimal place → denominator is 10, 2 decimal places → denominator is 100, 3 decimal places → denominator is 1000.
Tip 3: Always check if the fraction can be simplified to its lowest terms.
Tip 4: For mixed numbers, separate the whole number part from the decimal part before converting.
Common errors: Miscounting decimal places, forgetting to simplify, mixing up numerator and denominator.
Key insight: The denominator is always a power of 10 based on the number of decimal places.
Essential formulas:

• General formula: \(0.d_1d_2...d_n = \frac{d_1d_2...d_n}{10^n}\) where n is the number of decimal places

• Tenths: \(0.d = \frac{d}{10}\)

• Hundredths: \(0.dd = \frac{dd}{100}\)

• Thousandths: \(0.ddd = \frac{ddd}{1000}\)

Notes Practice: Problems 4 to 5
4 Converting Multiple Decimals
Example 4
Convert 0.8 and 0.75 to fractions and compare them.
Definition:

Decimal comparison: Converting decimals to fractions allows for easier comparison of their values.

First Decimal
0.8
Convert
\(\frac{8}{10}\)
Simplify
\(\frac{4}{5}\)
Second Decimal
0.75
Convert
\(\frac{75}{100}\)
Simplify
\(\frac{3}{4}\)
Step 1: Convert 0.8 to a fraction

0.8 has 1 decimal place → tenths place

\(0.8 = \frac{8}{10} = \frac{4}{5}\)

Step 2: Convert 0.75 to a fraction

0.75 has 2 decimal places → hundredths place

\(0.75 = \frac{75}{100} = \frac{3}{4}\)

Step 3: Compare the fractions

To compare \(\frac{4}{5}\) and \(\frac{3}{4}\), find common denominator

\(\frac{4}{5} = \frac{16}{20}\) and \(\frac{3}{4} = \frac{15}{20}\)

Since \(\frac{16}{20} > \frac{15}{20}\), we have \(0.8 > 0.75\)

\(0.8 = \frac{4}{5}\) and \(0.75 = \frac{3}{4}\), so \(0.8 > 0.75\)
Final answer:

\(0.8 = \frac{4}{5}\), \(0.75 = \frac{3}{4}\), and \(0.8 > 0.75\)

Applied rules:

• Decimal to Fraction: Use place value to determine denominator

• Comparison: Convert to common denominator to compare fractions

• Simplification: Always reduce to simplest form for easier comparison

5 Word Problem Application
Example 5
A recipe calls for 0.6 cups of milk. Express this as a fraction in simplest form.
Definition:

Real-world application: Converting decimals to fractions is useful in everyday situations like cooking, measuring, and shopping.

Decimal
0.6
Place Value
Tenths
Numerator
6
Denominator
10
Fraction
\(\frac{6}{10}\)
Simplified
\(\frac{3}{5}\)
Step 1: Identify the decimal

Amount needed: 0.6 cups

Step 2: Identify the place value

0.6 has 1 digit after decimal → tenths place

Step 3: Write the fraction

\(0.6 = \frac{6}{10}\)

Step 4: Simplify the fraction

Find GCF of 6 and 10:

Factors of 6: 1, 2, 3, 6

Factors of 10: 1, 2, 5, 10

GCF = 2

\(\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}\)

Step 5: Interpret in context

The recipe calls for \(\frac{3}{5}\) cup of milk

\(0.6 = \frac{3}{5}\)
Final answer:

The recipe calls for \(\frac{3}{5}\) cup of milk

Applied rules:

• Real-world Context: Convert decimal measurements to fractions for practical use

• Decimal Conversion: Use place value to determine numerator and denominator

• Simplification: Reduce to simplest form for easier understanding

Detailed Summary: Decimal to Fraction Conversion
\(0.d_1d_2...d_n = \frac{d_1d_2...d_n}{10^n}\)
Core Conversion Principle
Key Definitions:

Decimal: A number that uses a decimal point to separate the whole number part from the fractional part.

Fraction: A number that represents part of a whole, expressed as \(\frac{\text{numerator}}{\text{denominator}}\).

Place value: The value of a digit based on its position in a number (tenths, hundredths, thousandths, etc.).

Tenths place: The first digit to the right of the decimal point, representing \(\frac{1}{10}\) of the whole.

Hundredths place: The second digit to the right of the decimal point, representing \(\frac{1}{100}\) of the whole.

Thousandths place: The third digit to the right of the decimal point, representing \(\frac{1}{1000}\) of the whole.

Simplest form: A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

Decimal to Fraction Conversion Steps:
  1. Identify the decimal number you want to convert
  2. Count the decimal places after the decimal point
  3. Write the decimal digits as the numerator
  4. Write the denominator as 1 followed by the same number of zeros as decimal places
  5. Simplify the fraction by dividing both numerator and denominator by their GCF
Tip 1: Count the number of digits after the decimal point to determine the denominator.
Tip 2: For 1 decimal place → denominator is 10, 2 decimal places → denominator is 100, 3 decimal places → denominator is 1000.
Tip 3: Always check if the fraction can be simplified to its lowest terms.
Tip 4: For mixed numbers, separate the whole number part from the decimal part before converting.
Core Rule: The fundamental principle is that the denominator is always a power of 10 based on the number of decimal places: \(0.d_1d_2...d_n = \frac{d_1d_2...d_n}{10^n}\).
Practical Application: Converting decimals to fractions is essential for precise measurements, comparisons, and mathematical operations.
Essential Rules and Formulas:

• General Formula: \(0.d_1d_2...d_n = \frac{d_1d_2...d_n}{10^n}\) where n is the number of decimal places

• Tenths Conversion: \(0.d = \frac{d}{10}\)

• Hundredths Conversion: \(0.dd = \frac{dd}{100}\)

• Thousandths Conversion: \(0.ddd = \frac{ddd}{1000}\)

• Simplification: \(\frac{a}{b} = \frac{a \div \text{GCF}}{b \div \text{GCF}}\)

Simple Example

Convert 0.3 to a fraction:

0.3 has 1 decimal place → tenths place

Numerator: 3

Denominator: 10

\(0.3 = \frac{3}{10}\)

No common factors between 3 and 10, so \(\frac{3}{10}\) is in simplest form

Intermediate Example

Convert 0.45 to a fraction:

0.45 has 2 decimal places → hundredths place

Numerator: 45

Denominator: 100

\(0.45 = \frac{45}{100}\)

Find GCF of 45 and 100: GCF = 5

\(\frac{45}{100} = \frac{9}{20}\)

Advanced Example

Convert 3.75 to a mixed number:

Separate: 3 + 0.75

Convert 0.75: 2 decimal places → \(\frac{75}{100}\)

Simplify: GCF of 75 and 100 is 25

\(\frac{75}{100} = \frac{3}{4}\)

\(3.75 = 3\frac{3}{4}\)

Key Takeaways:
  • Count decimal places to determine the denominator (powers of 10)
  • The numerator is the decimal number without the decimal point
  • Always simplify fractions to their lowest terms
  • For mixed numbers, convert the decimal part separately
  • Use divisibility rules to simplify fractions efficiently
  • Decimal to fraction conversion is essential for precise calculations
Practice Problems
Practice Problems

Convert the following decimals to fractions in simplest form:

  1. 0.7
  2. 0.25
  3. 0.85
  4. 1.4
  5. 2.75
Answer Key:

1. \(0.7 = \frac{7}{10}\) (already in simplest form)

2. \(0.25 = \frac{25}{100} = \frac{1}{4}\) (divide by GCF=25)

3. \(0.85 = \frac{85}{100} = \frac{17}{20}\) (divide by GCF=5)

4. \(1.4 = 1\frac{4}{10} = 1\frac{2}{5}\) (divide by GCF=2)

5. \(2.75 = 2\frac{75}{100} = 2\frac{3}{4}\) (divide by GCF=25)