Relating Fractions and Decimals: Complete Educational Exercises

Master the relationship between fractions and decimals through these 5 detailed exercises with step-by-step solutions.

Solution: Exercises 1 to 3
1 Converting fractions to decimals
Exercise 1
Convert the fraction \(\frac{3}{4}\) to a decimal.
Definition:

Fraction: A number representing part of a whole, written as \(\frac{numerator}{denominator}\)

Decimal: A number with a decimal point representing parts of a whole

Conversion method:
  1. Divide the numerator by the denominator
  2. Perform the division until remainder is zero or a pattern emerges
  3. Write the quotient as the decimal equivalent
Fraction
\(\frac{3}{4}\)
Division
\(3 ÷ 4\)
Decimal
\(0.75\)
Step 1: Set up the division

We need to divide 3 by 4: \(3 ÷ 4\)

Step 2: Add decimal point and zeros

\(3.000 ÷ 4\)

Step 3: Perform the division

\(4\) goes into \(30\) seven times (since \(4 × 7 = 28\)), remainder \(2\)

Bring down the next \(0\), making it \(20\)

\(4\) goes into \(20\) five times (since \(4 × 5 = 20\)), remainder \(0\)

Step 4: Write the result

So, \(\frac{3}{4} = 0.75\)

\(\frac{3}{4} = 0.75\)
Final answer:

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

Applied rules:

• Division method: Numerator divided by denominator gives the decimal equivalent

• Place value: Each digit after decimal point represents tenths, hundredths, thousandths, etc.

• Terminating decimals: Division ends when remainder is zero

Tip: For denominators that are powers of 10 (10, 100, 1000), conversion is easier: \(\frac{3}{10} = 0.3\), \(\frac{7}{100} = 0.07\)
2 Converting decimals to fractions
Exercise 2
Convert the decimal \(0.6\) to a fraction in simplest form.
Definition:

Simplest form: A fraction where numerator and denominator have no common factors except 1

Decimal
\(0.6\)
Fraction
\(\frac{6}{10}\)
Simplified
\(\frac{3}{5}\)
Step 1: Identify place value

The decimal \(0.6\) has one digit after the decimal point, so it's in the tenths place

Step 2: Write as fraction

Write the digits after the decimal point over the appropriate power of 10

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

Step 3: Simplify the fraction

Find the greatest common divisor (GCD) of 6 and 10

GCD of 6 and 10 is 2

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

Step 4: Check if simplified

GCD of 3 and 5 is 1, so \(\frac{3}{5}\) is in simplest form

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

\(0.6 = \frac{3}{5}\)

Applied rules:

• Place value rule: 1 decimal place → denominator of 10, 2 decimal places → denominator of 100, etc.

• Simplification: Divide both numerator and denominator by their GCD

• Simplest form: Fraction is in simplest form when GCD of numerator and denominator is 1

Tip: To find GCD, list the factors of each number and identify the largest common factor.
3 Comparing fractions and decimals
Exercise 3
Which is greater: \(\frac{2}{5}\) or \(0.45\)?
Definition:

Comparing rational numbers: Numbers must be in the same form (both fractions or both decimals) to compare them accurately

Fraction
\(\frac{2}{5}\)
Convert to decimal
\(0.4\)
Compare
\(0.4 < 0.45\)
Step 1: Convert to same form

Convert \(\frac{2}{5}\) to decimal: \(2 ÷ 5 = 0.4\)

Step 2: Compare decimals

Now compare \(0.4\) and \(0.45\)

Step 3: Analyze place values

Both have 0 in the ones place

Both have 4 in the tenths place

But \(0.4\) has 0 in the hundredths place, while \(0.45\) has 5 in the hundredths place

Step 4: Make comparison

Since \(0 < 5\) in the hundredths place, \(0.4 < 0.45\)

\(\frac{2}{5} < 0.45\)
Final answer:

\(0.45\) is greater than \(\frac{2}{5}\)

Applied rules:

• Conversion rule: Convert to the same form before comparing

• Place value comparison: Compare digits starting from leftmost place value

• Equality rule: \(\frac{a}{b} = \frac{a×n}{b×n}\) for any non-zero n

Alternative method: Convert both to fractions with common denominator or both to decimals with same number of places.
Key Rules, Laws, and Methods
\(\frac{a}{b} = a ÷ b\)
Fraction to Decimal Conversion
Rule 1
\(\frac{a}{10^n} = 0.\underbrace{0...0}_{n-1}a\)
Fraction with power of 10 denominator
Rule 2
\(0.\underbrace{0...0}_{n}a = \frac{a}{10^{n+1}}\)
Decimal to fraction conversion
Rule 3
\(\frac{a}{b} = \frac{a×n}{b×n}\)
Equivalent fractions
Key definitions:

Numerator: The top number in a fraction representing how many parts we have

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

Decimal: A number system based on powers of 10, with a decimal point separating whole and fractional parts

Equivalent: Two numbers that represent the same value, even though they may look different

Complete methodology:
  1. Identify the operation needed: Are you converting fraction to decimal, decimal to fraction, or comparing?
  2. Choose the method: Long division, place value recognition, or conversion to same form
  3. Apply the method: Follow the steps systematically
  4. Verify the result: Check if the conversion makes sense
Tip 1: Remember common fraction-to-decimal equivalents: \(\frac{1}{2}=0.5\), \(\frac{1}{4}=0.25\), \(\frac{3}{4}=0.75\), \(\frac{1}{5}=0.2\), \(\frac{2}{5}=0.4\)
Tip 2: When converting decimals to fractions, count the decimal places to determine the denominator (10, 100, 1000, etc.)
Tip 3: To simplify fractions, divide both numerator and denominator by their greatest common divisor (GCD)
Tip 4: When comparing, always convert to the same form first (either both fractions or both decimals)
Common errors: Forgetting to simplify fractions, miscounting decimal places, comparing different forms directly.
Exam preparation: Practice converting between forms, memorize common equivalences, work on comparison problems.
Formulas to know by heart:

• Conversion: \(\frac{a}{b} = a ÷ b\) (fraction to decimal)

• Place value: 1 decimal place → tenths (\(\frac{1}{10}\)), 2 decimal places → hundredths (\(\frac{1}{100}\))

• Equivalence: \(\frac{a}{b} = \frac{a×n}{b×n}\) for any non-zero n

• Simplification: Divide numerator and denominator by GCD

Visual Understanding: Fraction-Decimal Equivalents
Exercise 6: Equivalent Values
Visual representation of common fraction-decimal equivalents:
\(\frac{1}{2} = 0.5\), \(\frac{1}{4} = 0.25\), \(\frac{3}{4} = 0.75\), \(\frac{1}{5} = 0.2\), \(\frac{2}{5} = 0.4\)

Analysis: The chart shows how fractions and decimals represent the same quantities visually.

  • \(\frac{1}{2} = 0.5\) (half of a whole)
  • \(\frac{1}{4} = 0.25\) (quarter of a whole)
  • \(\frac{3}{4} = 0.75\) (three-quarters of a whole)
  • \(\frac{1}{5} = 0.2\) (one-fifth of a whole)
  • \(\frac{2}{5} = 0.4\) (two-fifths of a whole)

Questions & Answers

Question: Why does \(\frac{1}{4} = 0.25\)? It seems like it should be 0.4 since the numerator is 1 and the denominator is 4.

Answer: Great question! The fraction \(\frac{1}{4}\) means "1 divided by 4", not "1 and 4".

  • To convert \(\frac{1}{4}\) to a decimal, we perform the division: \(1 ÷ 4\)
  • Since 1 is smaller than 4, we add a decimal point and zeros: \(1.000 ÷ 4\)
  • 4 goes into 10 two times (since \(4 × 2 = 8\)), remainder 2
  • Bring down the next 0, making it 20
  • 4 goes into 20 five times (since \(4 × 5 = 20\)), remainder 0

So, \(\frac{1}{4} = 0.25\), not 0.4.

Think of it this way: if you have 1 pizza and want to share it equally among 4 people, each person gets 0.25 (or 25%) of the pizza.

Question: How do I know when a fraction will have a terminating decimal versus a repeating decimal?

Answer: This is an advanced concept, but here's the rule:

  • A fraction in its simplest form has a terminating decimal if its denominator has only 2 and/or 5 as prime factors
  • If the denominator has any other prime factors, the decimal will repeat

Examples of terminating decimals:

  • \(\frac{3}{4}\) → denominator is 4 = \(2^2\), so it terminates as 0.75
  • \(\frac{7}{20}\) → denominator is 20 = \(2^2 × 5\), so it terminates as 0.35

Examples of repeating decimals:

  • \(\frac{1}{3}\) → denominator is 3 (not 2 or 5), so it repeats as 0.333...
  • \(\frac{2}{7}\) → denominator is 7 (not 2 or 5), so it repeats as 0.285714285714...

For 4th grade, just focus on fractions with denominators like 2, 4, 5, 10, 20, 25, 50, 100 which typically result in terminating decimals.

Question: My child struggles with simplifying fractions after converting from decimals. Any advice for practice?

Answer: Here are some strategies to help with fraction simplification:

  1. Learn divisibility rules: Numbers divisible by 2 end in even digits, by 3 if the sum of digits is divisible by 3, by 5 if ending in 0 or 5
  2. Start with small primes: Try dividing by 2, then 3, then 5, etc., until you can't divide evenly anymore
  3. Use factor trees: Break down numerator and denominator into prime factors to easily see common factors
  4. Practice with common denominators: Focus on denominators like 10, 100, 1000 first, since these come from decimal conversions

Example: Convert 0.6 to fraction

  • 0.6 = \(\frac{6}{10}\) (since 6 is in the tenths place)
  • GCF of 6 and 10 is 2
  • \(\frac{6 ÷ 2}{10 ÷ 2} = \frac{3}{5}\)

Practice with simple examples first, then gradually increase complexity.