Recurring Decimals
Subtopic Navigator
Lesson Objectives
- Understand what recurring decimals are and how to represent them using dot notation
- Convert fractions into recurring decimals using division
- Convert pure recurring decimals $(0.\dot{a})$ into fractions
- Convert mixed recurring decimals $(0.a\dot{b})$ into fractions
- Apply algebraic methods to convert recurring decimals to fractions
- Simplify fractions to their lowest terms after conversion
- Recognise the relationship between fractions and their decimal representations
Introduction to Recurring Decimals
A recurring decimal (also called a repeating decimal) is a decimal number in which a digit or a group of digits repeats infinitely. For example, $\frac{1}{3} = 0.33333...$ and $\frac{1}{7} = 0.142857142857...$. Recurring decimals occur when a fraction cannot be expressed as a terminating decimal — that is, when its denominator (in simplest form) has prime factors other than 2 and 5. Understanding how to convert between fractions and recurring decimals is essential for precise mathematical work.
Recurring Decimal Notation
$0.\dot{3} = 0.333333...$ (single digit repeats)
$0.\dot{1}4\dot{2} = 0.142142142...$ (group repeats)
$0.1\dot{6} = 0.166666...$ (mixed: non-repeating part followed by repeating part)
$0.\dot{3} = 0.333333...$ (single digit repeats)
$0.\dot{1}4\dot{2} = 0.142142142...$ (group repeats)
$0.1\dot{6} = 0.166666...$ (mixed: non-repeating part followed by repeating part)
Key Definitions:
• Recurring Decimal: A decimal where a digit or block of digits repeats indefinitely.
• Dot Notation: A dot placed over the first and last digit of the repeating block (e.g., $0.\dot{1}\dot{4}$).
• Pure Recurring Decimal: All digits after the decimal point repeat (e.g., $0.\dot{3}$, $0.\dot{1}2\dot{3}$).
• Mixed Recurring Decimal: Some non-repeating digits before the repeating block (e.g., $0.1\dot{6}$, $0.23\dot{4}\dot{5}$).
• Terminating Decimal: A decimal that ends (e.g., 0.5, 0.125).
• Recurring Decimal: A decimal where a digit or block of digits repeats indefinitely.
• Dot Notation: A dot placed over the first and last digit of the repeating block (e.g., $0.\dot{1}\dot{4}$).
• Pure Recurring Decimal: All digits after the decimal point repeat (e.g., $0.\dot{3}$, $0.\dot{1}2\dot{3}$).
• Mixed Recurring Decimal: Some non-repeating digits before the repeating block (e.g., $0.1\dot{6}$, $0.23\dot{4}\dot{5}$).
• Terminating Decimal: A decimal that ends (e.g., 0.5, 0.125).
Converting Fractions to Recurring Decimals
To convert a fraction to a recurring decimal, perform long division of the numerator by the denominator. If the division never terminates, a repeating pattern will emerge. The length of the repeating block is related to the denominator's factors.
Step-by-Step Method: Fraction to Recurring Decimal
1. Perform long division of numerator ÷ denominator.
2. If the remainder repeats a previous remainder, the decimal will repeat from that point.
3. Identify the repeating block of digits.
4. Write using dot notation: a dot over the first and last digit of the repeating block.
1. Perform long division of numerator ÷ denominator.
2. If the remainder repeats a previous remainder, the decimal will repeat from that point.
3. Identify the repeating block of digits.
4. Write using dot notation: a dot over the first and last digit of the repeating block.
Example 1: Simple Recurring Decimal
Problem: Convert $\frac{1}{3}$ to a decimal.
Solution:
1 ÷ 3 = 0 remainder 1, bring down 0 → 10 ÷ 3 = 3 remainder 1, bring down 0 → 10 ÷ 3 = 3 remainder 1...
The remainder 1 keeps repeating, so the decimal is $0.333333...$
Answer: $0.\dot{3}$
Problem: Convert $\frac{1}{3}$ to a decimal.
Solution:
1 ÷ 3 = 0 remainder 1, bring down 0 → 10 ÷ 3 = 3 remainder 1, bring down 0 → 10 ÷ 3 = 3 remainder 1...
The remainder 1 keeps repeating, so the decimal is $0.333333...$
Answer: $0.\dot{3}$
Example 2: Longer Repeating Block
Problem: Convert $\frac{1}{7}$ to a decimal.
Solution:
1 ÷ 7 = 0 remainder 1 → 10 ÷ 7 = 1 r3 → 30 ÷ 7 = 4 r2 → 20 ÷ 7 = 2 r6 → 60 ÷ 7 = 8 r4 → 40 ÷ 7 = 5 r5 → 50 ÷ 7 = 7 r1
Remainder 1 repeats after 6 steps. The repeating block is "142857".
Answer: $0.\dot{1}4285\dot{7}$ (often written as $0.\overline{142857}$)
Problem: Convert $\frac{1}{7}$ to a decimal.
Solution:
1 ÷ 7 = 0 remainder 1 → 10 ÷ 7 = 1 r3 → 30 ÷ 7 = 4 r2 → 20 ÷ 7 = 2 r6 → 60 ÷ 7 = 8 r4 → 40 ÷ 7 = 5 r5 → 50 ÷ 7 = 7 r1
Remainder 1 repeats after 6 steps. The repeating block is "142857".
Answer: $0.\dot{1}4285\dot{7}$ (often written as $0.\overline{142857}$)
Example 3: Mixed Recurring Decimal from Fraction
Problem: Convert $\frac{5}{6}$ to a decimal.
Solution:
5 ÷ 6 = 0 remainder 5 → 50 ÷ 6 = 8 r2 → 20 ÷ 6 = 3 r2 → 20 ÷ 6 = 3 r2...
After the first digit (8), the remainder 2 repeats, giving 3 repeating.
Answer: $0.8\dot{3}$ (0.833333...)
Problem: Convert $\frac{5}{6}$ to a decimal.
Solution:
5 ÷ 6 = 0 remainder 5 → 50 ÷ 6 = 8 r2 → 20 ÷ 6 = 3 r2 → 20 ÷ 6 = 3 r2...
After the first digit (8), the remainder 2 repeats, giving 3 repeating.
Answer: $0.8\dot{3}$ (0.833333...)
Watch Out!
Not all fractions produce recurring decimals. If the denominator (in simplest form) has only prime factors 2 and/or 5, the decimal terminates. For example, $\frac{3}{8} = 0.375$ terminates because $8 = 2^3$.
Not all fractions produce recurring decimals. If the denominator (in simplest form) has only prime factors 2 and/or 5, the decimal terminates. For example, $\frac{3}{8} = 0.375$ terminates because $8 = 2^3$.
Practice for Concept 1 (Fractions to Recurring Decimals)
- Convert $\frac{2}{3}$ to a recurring decimal.
- Convert $\frac{5}{11}$ to a recurring decimal.
- Convert $\frac{7}{9}$ to a recurring decimal.
- Convert $\frac{2}{7}$ to a recurring decimal.
- Convert $\frac{4}{15}$ to a recurring decimal.
Converting Pure Recurring Decimals to Fractions
A pure recurring decimal has all digits after the decimal point repeating. To convert, let $x$ equal the decimal, multiply by a power of 10 to shift one full repeating block to the left of the decimal, then subtract to eliminate the repeating part.
Step-by-Step Method for Pure Recurring Decimals:
1. Let $x$ equal the recurring decimal.
2. Multiply $x$ by $10^n$ where $n$ is the length of the repeating block.
3. Subtract the original equation from this new equation.
4. Solve for $x$ as a fraction.
5. Simplify the fraction to lowest terms.
1. Let $x$ equal the recurring decimal.
2. Multiply $x$ by $10^n$ where $n$ is the length of the repeating block.
3. Subtract the original equation from this new equation.
4. Solve for $x$ as a fraction.
5. Simplify the fraction to lowest terms.
Example 1: Single Digit Repeating
Problem: Convert $0.\dot{3}$ to a fraction.
Solution:
Let $x = 0.\dot{3} = 0.333333...$
Multiply by 10 (since 1 digit repeats): $10x = 3.333333...$
Subtract: $10x - x = 3.333333... - 0.333333...$
$9x = 3$
$x = \frac{3}{9} = \frac{1}{3}$
Answer: $\frac{1}{3}$
Problem: Convert $0.\dot{3}$ to a fraction.
Solution:
Let $x = 0.\dot{3} = 0.333333...$
Multiply by 10 (since 1 digit repeats): $10x = 3.333333...$
Subtract: $10x - x = 3.333333... - 0.333333...$
$9x = 3$
$x = \frac{3}{9} = \frac{1}{3}$
Answer: $\frac{1}{3}$
Example 2: Two Digits Repeating
Problem: Convert $0.\dot{1}\dot{4}$ to a fraction.
Solution:
Let $x = 0.\dot{1}\dot{4} = 0.14141414...$
Multiply by 100 (since 2 digits repeat): $100x = 14.14141414...$
Subtract: $100x - x = 14.141414... - 0.141414...$
$99x = 14$
$x = \frac{14}{99}$
Answer: $\frac{14}{99}$
Problem: Convert $0.\dot{1}\dot{4}$ to a fraction.
Solution:
Let $x = 0.\dot{1}\dot{4} = 0.14141414...$
Multiply by 100 (since 2 digits repeat): $100x = 14.14141414...$
Subtract: $100x - x = 14.141414... - 0.141414...$
$99x = 14$
$x = \frac{14}{99}$
Answer: $\frac{14}{99}$
Example 3: Three Digits Repeating
Problem: Convert $0.\dot{1}23\dot{5}$ to a fraction (repeating block "1235").
Solution:
Let $x = 0.\dot{1}235\dot{?}$ Wait, careful: $0.\dot{1}235$ means the repeating block is "1235" (4 digits).
Actually, $0.\dot{1}23\dot{5}$ means digits 1,2,3,5 repeat? The notation $\dot{1}23\dot{5}$ means the block from 1 to 5 repeats, so block "1235".
$x = 0.123512351235...$
Multiply by 10000 (4 digits repeat): $10000x = 1235.12351235...$
Subtract: $10000x - x = 1235$
$9999x = 1235$
$x = \frac{1235}{9999}$
Check simplification: 1235 = 5 × 247 = 5 × 13 × 19; 9999 = 9 × 1111 = 9 × 101 × 11? No common factors.
Answer: $\frac{1235}{9999}$
Problem: Convert $0.\dot{1}23\dot{5}$ to a fraction (repeating block "1235").
Solution:
Let $x = 0.\dot{1}235\dot{?}$ Wait, careful: $0.\dot{1}235$ means the repeating block is "1235" (4 digits).
Actually, $0.\dot{1}23\dot{5}$ means digits 1,2,3,5 repeat? The notation $\dot{1}23\dot{5}$ means the block from 1 to 5 repeats, so block "1235".
$x = 0.123512351235...$
Multiply by 10000 (4 digits repeat): $10000x = 1235.12351235...$
Subtract: $10000x - x = 1235$
$9999x = 1235$
$x = \frac{1235}{9999}$
Check simplification: 1235 = 5 × 247 = 5 × 13 × 19; 9999 = 9 × 1111 = 9 × 101 × 11? No common factors.
Answer: $\frac{1235}{9999}$
Shortcut for Pure Recurring Decimals:
$0.\dot{a} = \frac{a}{9}$
$0.\dot{a}\dot{b} = \frac{ab}{99}$
$0.\dot{a}bc\dot{d} = \frac{abcd}{9999}$ (where abcd is the repeating block)
$0.\dot{a} = \frac{a}{9}$
$0.\dot{a}\dot{b} = \frac{ab}{99}$
$0.\dot{a}bc\dot{d} = \frac{abcd}{9999}$ (where abcd is the repeating block)
Practice for Concept 2 (Pure Recurring to Fractions)
- Convert $0.\dot{7}$ to a fraction.
- Convert $0.\dot{2}\dot{7}$ to a fraction.
- Convert $0.\dot{1}2\dot{3}$ to a fraction.
- Convert $0.\dot{4}5\dot{6}$ to a fraction.
- Convert $0.\dot{9}$ to a fraction. What do you notice?
Converting Mixed Recurring Decimals to Fractions
A mixed recurring decimal has non-repeating digits before the repeating block. The method is similar but requires two multiplications: one to shift the non-repeating part and another to shift the repeating part.
Step-by-Step Method for Mixed Recurring Decimals:
1. Let $x$ equal the decimal.
2. Multiply by $10^m$ where $m$ is the number of non-repeating digits to get a pure recurring decimal.
3. Then multiply by $10^n$ where $n$ is the length of the repeating block to align the repeats.
4. Subtract to eliminate the recurring part.
5. Solve for $x$ and simplify.
1. Let $x$ equal the decimal.
2. Multiply by $10^m$ where $m$ is the number of non-repeating digits to get a pure recurring decimal.
3. Then multiply by $10^n$ where $n$ is the length of the repeating block to align the repeats.
4. Subtract to eliminate the recurring part.
5. Solve for $x$ and simplify.
Example 1: One Non-Repeating Digit
Problem: Convert $0.1\dot{6}$ to a fraction.
Solution:
Let $x = 0.1\dot{6} = 0.166666...$
Step 1: Multiply by 10 to move non-repeating digit: $10x = 1.\dot{6} = 1.666666...$
Step 2: Now $1.\dot{6}$ is pure recurring. Let $y = 1.\dot{6}$
For $0.\dot{6}$: $10y_2 = 6.666...$ where $y_2 = 0.\dot{6}$
Or directly: $1.\dot{6} = 1 + 0.\dot{6} = 1 + \frac{6}{9} = 1 + \frac{2}{3} = \frac{5}{3}$
So $10x = \frac{5}{3}$
$x = \frac{5}{30} = \frac{1}{6}$
Answer: $\frac{1}{6}$
Problem: Convert $0.1\dot{6}$ to a fraction.
Solution:
Let $x = 0.1\dot{6} = 0.166666...$
Step 1: Multiply by 10 to move non-repeating digit: $10x = 1.\dot{6} = 1.666666...$
Step 2: Now $1.\dot{6}$ is pure recurring. Let $y = 1.\dot{6}$
For $0.\dot{6}$: $10y_2 = 6.666...$ where $y_2 = 0.\dot{6}$
Or directly: $1.\dot{6} = 1 + 0.\dot{6} = 1 + \frac{6}{9} = 1 + \frac{2}{3} = \frac{5}{3}$
So $10x = \frac{5}{3}$
$x = \frac{5}{30} = \frac{1}{6}$
Answer: $\frac{1}{6}$
Example 2: Using the Subtraction Method
Problem: Convert $0.2\dot{4}5\dot{7}$ (repeating block "457", non-repeating digit "2")? Actually $0.2\dot{4}5\dot{7}$ means non-repeating "2", repeating "457".
Let's do $0.2\dot{4}5\dot{7} = 0.2457457457...$
Solution:
Let $x = 0.2457457457...$
Multiply by 10 (1 non-repeating digit): $10x = 2.457457457...$
Now $2.\dot{4}57\dot{?}$ has repeating block "457" (3 digits).
Multiply $10x$ by 1000: $10000x = 2457.457457...$
Subtract: $10000x - 10x = 2457.457... - 2.457... = 2455$
$9990x = 2455$
$x = \frac{2455}{9990} = \frac{491}{1998}$ (divide numerator and denominator by 5)
Answer: $\frac{491}{1998}$
Problem: Convert $0.2\dot{4}5\dot{7}$ (repeating block "457", non-repeating digit "2")? Actually $0.2\dot{4}5\dot{7}$ means non-repeating "2", repeating "457".
Let's do $0.2\dot{4}5\dot{7} = 0.2457457457...$
Solution:
Let $x = 0.2457457457...$
Multiply by 10 (1 non-repeating digit): $10x = 2.457457457...$
Now $2.\dot{4}57\dot{?}$ has repeating block "457" (3 digits).
Multiply $10x$ by 1000: $10000x = 2457.457457...$
Subtract: $10000x - 10x = 2457.457... - 2.457... = 2455$
$9990x = 2455$
$x = \frac{2455}{9990} = \frac{491}{1998}$ (divide numerator and denominator by 5)
Answer: $\frac{491}{1998}$
Example 3: Simpler Mixed Recurring
Problem: Convert $0.8\dot{3}$ to a fraction.
Solution:
Let $x = 0.8\dot{3} = 0.833333...$
Multiply by 10: $10x = 8.\dot{3} = 8.33333...$
$0.\dot{3} = \frac{1}{3}$, so $8.\dot{3} = 8 + \frac{1}{3} = \frac{24}{3} + \frac{1}{3} = \frac{25}{3}$
$10x = \frac{25}{3}$
$x = \frac{25}{30} = \frac{5}{6}$
Answer: $\frac{5}{6}$
Problem: Convert $0.8\dot{3}$ to a fraction.
Solution:
Let $x = 0.8\dot{3} = 0.833333...$
Multiply by 10: $10x = 8.\dot{3} = 8.33333...$
$0.\dot{3} = \frac{1}{3}$, so $8.\dot{3} = 8 + \frac{1}{3} = \frac{24}{3} + \frac{1}{3} = \frac{25}{3}$
$10x = \frac{25}{3}$
$x = \frac{25}{30} = \frac{5}{6}$
Answer: $\frac{5}{6}$
Example 4: Two Non-Repeating Digits
Problem: Convert $0.12\dot{3}$ to a fraction ($0.123333...$).
Solution:
Let $x = 0.12\dot{3} = 0.123333...$
Multiply by 100 (2 non-repeating digits): $100x = 12.\dot{3} = 12.33333...$
$12.\dot{3} = 12 + \frac{1}{3} = \frac{36}{3} + \frac{1}{3} = \frac{37}{3}$
$100x = \frac{37}{3}$
$x = \frac{37}{300}$
Answer: $\frac{37}{300}$
Problem: Convert $0.12\dot{3}$ to a fraction ($0.123333...$).
Solution:
Let $x = 0.12\dot{3} = 0.123333...$
Multiply by 100 (2 non-repeating digits): $100x = 12.\dot{3} = 12.33333...$
$12.\dot{3} = 12 + \frac{1}{3} = \frac{36}{3} + \frac{1}{3} = \frac{37}{3}$
$100x = \frac{37}{3}$
$x = \frac{37}{300}$
Answer: $\frac{37}{300}$
Shortcut Formula for Mixed Recurring:
$0.a\dot{b} = \frac{ab - a}{90}$ (where a is non-repeating digit(s), b is repeating digit(s))
More generally: $0.a_1a_2...a_m\dot{b_1}b_2...\dot{b_n} = \frac{\text{whole number formed by all digits} - \text{whole number formed by non-repeating digits}}{10^m(10^n - 1)}$
$0.a\dot{b} = \frac{ab - a}{90}$ (where a is non-repeating digit(s), b is repeating digit(s))
More generally: $0.a_1a_2...a_m\dot{b_1}b_2...\dot{b_n} = \frac{\text{whole number formed by all digits} - \text{whole number formed by non-repeating digits}}{10^m(10^n - 1)}$
Watch Out!
When converting mixed recurring decimals, be careful to count the number of non-repeating digits correctly. Multiply by the appropriate power of 10 to isolate the repeating part.
When converting mixed recurring decimals, be careful to count the number of non-repeating digits correctly. Multiply by the appropriate power of 10 to isolate the repeating part.
Practice for Concept 3 (Mixed Recurring to Fractions)
- Convert $0.2\dot{3}$ to a fraction.
- Convert $0.5\dot{7}$ to a fraction.
- Convert $0.34\dot{5}$ to a fraction.
- Convert $0.12\dot{3}\dot{4}$ to a fraction.
- Convert $0.01\dot{6}$ to a fraction.
Methods & Techniques
Mastering recurring decimals requires systematic algebraic techniques. Use these strategies to ensure accuracy.
Verification / Checking Strategy:
1. For fraction to decimal: Multiply the decimal back by the denominator to check if you get the numerator.
2. For decimal to fraction: Convert the fraction back to decimal using division to verify.
3. Use calculator check: Most scientific calculators can display recurring decimals as fractions.
4. Simplify fractions: Always check if the resulting fraction can be reduced by finding common factors.
1. For fraction to decimal: Multiply the decimal back by the denominator to check if you get the numerator.
2. For decimal to fraction: Convert the fraction back to decimal using division to verify.
3. Use calculator check: Most scientific calculators can display recurring decimals as fractions.
4. Simplify fractions: Always check if the resulting fraction can be reduced by finding common factors.
Example: Checking Conversion Work
Original problem: Convert $0.\dot{1}2\dot{3}$ to a fraction.
Your solution: $\frac{123}{999} = \frac{41}{333}$
Check:
Convert $\frac{41}{333}$ to decimal: 41 ÷ 333 = 0.123123123... ✓ matches $0.\dot{1}2\dot{3}$
Original problem: Convert $0.\dot{1}2\dot{3}$ to a fraction.
Your solution: $\frac{123}{999} = \frac{41}{333}$
Check:
Convert $\frac{41}{333}$ to decimal: 41 ÷ 333 = 0.123123123... ✓ matches $0.\dot{1}2\dot{3}$
Common Pitfalls & How to Avoid Them:
• Pitfall 1: Forgetting that $0.\dot{9} = 1$ exactly → Solution: Recognize that 0.999... is mathematically equal to 1.
• Pitfall 2: Incorrectly identifying the repeating block length → Solution: Perform division until a remainder repeats, or carefully examine the decimal pattern.
• Pitfall 3: Errors in subtraction when using the algebraic method → Solution: Align decimals carefully and double-check subtraction.
• Pitfall 4: Not simplifying fractions to lowest terms → Solution: Always divide numerator and denominator by their greatest common factor (GCF).
• Pitfall 5: Confusing mixed recurring with pure recurring → Solution: Count digits after decimal before the repeating block starts.
• Pitfall 1: Forgetting that $0.\dot{9} = 1$ exactly → Solution: Recognize that 0.999... is mathematically equal to 1.
• Pitfall 2: Incorrectly identifying the repeating block length → Solution: Perform division until a remainder repeats, or carefully examine the decimal pattern.
• Pitfall 3: Errors in subtraction when using the algebraic method → Solution: Align decimals carefully and double-check subtraction.
• Pitfall 4: Not simplifying fractions to lowest terms → Solution: Always divide numerator and denominator by their greatest common factor (GCF).
• Pitfall 5: Confusing mixed recurring with pure recurring → Solution: Count digits after decimal before the repeating block starts.
Summary Table: Converting Recurring Decimals to Fractions
| Decimal Type | Example | Method | Result |
|---|---|---|---|
| Pure (1 digit) | $0.\dot{7}$ | $\frac{7}{9}$ | $\frac{7}{9}$ |
| Pure (2 digits) | $0.\dot{3}\dot{6}$ | $\frac{36}{99}$ | $\frac{4}{11}$ |
| Mixed (1 non-repeating) | $0.1\dot{6}$ | $\frac{16-1}{90}$ | $\frac{15}{90}=\frac{1}{6}$ |
| Mixed (2 non-repeating) | $0.12\dot{3}$ | $\frac{123-12}{900}$ | $\frac{111}{900}=\frac{37}{300}$ |
Technique Practice
- Verify: Is $0.\dot{3} = \frac{1}{3}$? Check by multiplying $\frac{1}{3} \times 3 = 1$.
- Check the conversion: $0.2\dot{7} = \frac{5}{18}$. Verify by converting $\frac{5}{18}$ to decimal.
- Identify the error: A student wrote $0.\dot{9} = \frac{9}{10}$. What is wrong? Correct it.
- A student converted $0.1\dot{6}$ as $\frac{16}{99}$. Is this correct? Explain.
Real-World Applications
Recurring decimals appear naturally when dividing quantities that cannot be expressed as terminating decimals. They are important in precise calculations, measurement, and understanding rational numbers.
Application 1: Measurement and Construction
Scenario: A carpenter has a board of length $\frac{5}{6}$ metres. If this length is written as a decimal for a cutting machine, what decimal should be entered?
Problem: Convert $\frac{5}{6}$ to a decimal.
Solution:
$\frac{5}{6} = 0.833333... = 0.8\dot{3}$
For practical purposes, the carpenter might use 0.8333 m, but precise work requires understanding that the decimal repeats.
Scenario: A carpenter has a board of length $\frac{5}{6}$ metres. If this length is written as a decimal for a cutting machine, what decimal should be entered?
Problem: Convert $\frac{5}{6}$ to a decimal.
Solution:
$\frac{5}{6} = 0.833333... = 0.8\dot{3}$
For practical purposes, the carpenter might use 0.8333 m, but precise work requires understanding that the decimal repeats.
Application 2: Currency and Finance
Scenario: A recurring decimal appears when converting certain currencies. For example, $\frac{1}{3}$ dollar = $0.\dot{3}$ dollars. If a store sells 3 items for $1, what is the price per item?
Problem: Price per item = $1 ÷ 3 = \frac{1}{3} = 0.\dot{3}$ dollars.
Solution:
The exact price is $\frac{1}{3}$ dollar ≈ $0.3333, which rounds to $0.33 for cash transactions but is exactly $\frac{1}{3}$ for accounting.
Scenario: A recurring decimal appears when converting certain currencies. For example, $\frac{1}{3}$ dollar = $0.\dot{3}$ dollars. If a store sells 3 items for $1, what is the price per item?
Problem: Price per item = $1 ÷ 3 = \frac{1}{3} = 0.\dot{3}$ dollars.
Solution:
The exact price is $\frac{1}{3}$ dollar ≈ $0.3333, which rounds to $0.33 for cash transactions but is exactly $\frac{1}{3}$ for accounting.
Application 3: Time and Angles
Scenario: $\frac{1}{3}$ of an hour = 20 minutes exactly. But $\frac{1}{3}$ expressed as a decimal of an hour is $0.\dot{3}$ hours. This shows why fractions are often preferred over decimals for precise time calculations.
Problem: Convert $\frac{1}{3}$ hour to minutes and to decimal hours.
Solution:
$\frac{1}{3}$ hour = 20 minutes (exact). Decimal: $0.\dot{3}$ hours.
Scenario: $\frac{1}{3}$ of an hour = 20 minutes exactly. But $\frac{1}{3}$ expressed as a decimal of an hour is $0.\dot{3}$ hours. This shows why fractions are often preferred over decimals for precise time calculations.
Problem: Convert $\frac{1}{3}$ hour to minutes and to decimal hours.
Solution:
$\frac{1}{3}$ hour = 20 minutes (exact). Decimal: $0.\dot{3}$ hours.
Application 4: Repeating Patterns in Nature
Scenario: Some natural phenomena produce repeating patterns that can be modelled by fractions with recurring decimal representations. For example, $\frac{1}{7} = 0.\dot{1}4285\dot{7}$ appears in cyclic numbers.
Problem: The repeating block "142857" is known as a cyclic number — multiplying it by 1 to 6 produces rotations of the same digits.
Scenario: Some natural phenomena produce repeating patterns that can be modelled by fractions with recurring decimal representations. For example, $\frac{1}{7} = 0.\dot{1}4285\dot{7}$ appears in cyclic numbers.
Problem: The repeating block "142857" is known as a cyclic number — multiplying it by 1 to 6 produces rotations of the same digits.
Cross-Curricular Connections
- Computer Science: Floating-point arithmetic and representing repeating decimals in binary.
- Physics: Measurements that yield rational numbers with repeating decimal expansions.
- Music Theory: Ratios of frequencies (e.g., perfect fifth = 3:2 = 1.5 terminating; some intervals produce recurring decimals).
- Economics: Interest rates and currency conversions that involve repeating decimals.
Cumulative Practice Exercises
Try these problems on your own. Show all working steps. Use the verification strategies to check your answers.
- Convert $\frac{4}{9}$ to a recurring decimal.
- Convert $\frac{2}{11}$ to a recurring decimal.
- Convert $\frac{7}{12}$ to a recurring decimal.
- Convert $0.\dot{8}$ to a fraction.
- Convert $0.\dot{4}\dot{5}$ to a fraction.
- Convert $0.\dot{1}23\dot{4}$ to a fraction.
- Convert $0.3\dot{6}$ to a fraction.
- Convert $0.12\dot{3}$ to a fraction.
- Convert $0.01\dot{8}$ to a fraction.
- Convert $0.2\dot{4}5\dot{7}$ to a fraction.
- Which is larger: $0.\dot{3}$ or $\frac{1}{3}$? Explain.
- Find the fraction equivalent to $0.4\dot{2}\dot{8}$ (repeating block "28").
- Error analysis: A student said $0.1\dot{6} = \frac{16}{99}$. Is this correct? If not, what is the correct fraction?
- The recurring decimal $0.\dot{1}\dot{4}$ represents what fraction? Simplify your answer.
- Prove that $0.\dot{9} = 1$ using the algebraic method.
Answers to Cumulative Exercises
- Problem: $\frac{4}{9}$ to decimal.
Answer: $0.\dot{4}$ - Problem: $\frac{2}{11}$ to decimal.
Answer: $0.\dot{1}\dot{8}$ (0.181818...) - Problem: $\frac{7}{12}$ to decimal.
Answer: $0.58\dot{3}$ (0.583333...) - Problem: $0.\dot{8}$ to fraction.
Answer: $\frac{8}{9}$ - Problem: $0.\dot{4}\dot{5}$ to fraction.
Answer: $\frac{45}{99} = \frac{5}{11}$ - Problem: $0.\dot{1}23\dot{4}$ to fraction.
Answer: $\frac{1234}{9999}$ (check simplification: no common factors with 9999? 9999=9×1111=9×101×11; 1234 even? 1234÷2=617, 617 prime? 617÷617=1; no common factors → $\frac{1234}{9999}$) - Problem: $0.3\dot{6}$ to fraction.
Answer: Let $x=0.3666...$, $10x=3.666...=3+\frac{2}{3}=\frac{11}{3}$, $x=\frac{11}{30}$ - Problem: $0.12\dot{3}$ to fraction.
Answer: $\frac{123-12}{900}=\frac{111}{900}=\frac{37}{300}$ - Problem: $0.01\dot{8}$ to fraction.
Answer: $\frac{18-1}{900}=\frac{17}{900}$ - Problem: $0.2\dot{4}5\dot{7}$ to fraction.
Answer: $\frac{2457-2}{9990}=\frac{2455}{9990}=\frac{491}{1998}$ - Problem: Compare $0.\dot{3}$ and $\frac{1}{3}$.
Answer: They are equal. $0.\dot{3} = \frac{1}{3}$ exactly. - Problem: $0.4\dot{2}\dot{8}$ to fraction (repeating "28", non-repeating "4").
Answer: $\frac{428-4}{990}=\frac{424}{990}=\frac{212}{495}$ - Problem: Error analysis: $0.1\dot{6} = \frac{16}{99}$?
Answer: Incorrect. $0.1\dot{6} = \frac{16-1}{90} = \frac{15}{90} = \frac{1}{6}$ - Problem: $0.\dot{1}\dot{4}$ to fraction.
Answer: $\frac{14}{99}$ - Problem: Prove $0.\dot{9}=1$.
Answer: Let $x=0.\dot{9}=0.999...$; $10x=9.999...$; $10x-x=9.999...-0.999...=9$; $9x=9$; $x=1$
Conclusion & Summary
Recurring decimals are an important concept in understanding rational numbers. Every fraction either terminates or repeats, and every repeating decimal can be expressed as a fraction. The algebraic method provides a reliable way to convert between the two representations.
Key Takeaways:
1. Recurring decimals occur when a fraction's denominator (in simplest form) has prime factors other than 2 and 5.
2. Pure recurring decimals (e.g., $0.\dot{a}\dot{b}$) convert to $\frac{ab}{99}$ (where ab is the repeating block).
3. Mixed recurring decimals (e.g., $0.a\dot{b}$) convert using $\frac{ab - a}{90}$ or the general formula.
4. Algebraic method: Let $x$ = decimal, multiply by powers of 10, subtract, solve for $x$.
5. Verification: Always check by converting the fraction back to decimal.
6. $0.\dot{9} = 1$ exactly — an important mathematical fact.
Keep practicing conversions until they become automatic. Understanding recurring decimals deepens your grasp of rational numbers!
Video Resource
Watch this video for more examples of converting recurring decimals to fractions.
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