๐Ÿ” Recurring Decimals & Fractions

Cambridge Lower Secondary ยท Grade 7 ยท Fractions & Decimals

Fraction to Decimal
1 รท 3 = 0.3 = 0.333โ€ฆ
The 3 repeats forever!
Algebraic Conversion
x = 0.12 โ†’ 99x = 12 โ†’ x = 12/99 = 4/33
Terminating vs Recurring
1/4 = 0.25 โœ“ Terminates
1/6 = 0.16 โœ— Recurs (factor of 3!)

Watch 1 รท 7 unfold! ๐Ÿฌ

0.

The digits 142857 repeat in a cycle of 6 โ€” forever!

What you'll learn:

  • The difference between terminating and recurring decimals
  • Recurring notation using dots above digits (e.g. 0.3ฬ„, 0.1ฬ„2ฬ„)
  • Converting fractions to decimals by long division
  • Which fractions always recur (denominator has prime factors โ‰  2 or 5)
  • Converting recurring decimals to fractions using algebra
  • Ordering fractions and decimals together on a number line
Created by Mr Josh

๐Ÿ“– Learn: Recurring Decimals

Part 1: Terminating vs Recurring Decimals

When you divide to convert a fraction to a decimal, one of two things happens:

Type What happens Example
โœ“ Terminating The division ends (remainder = 0) 3/4 = 0.75
โˆž Recurring A digit or block repeats forever 1/3 = 0.333โ€ฆ
๐Ÿ’ก A decimal can only terminate if the fraction's denominator (in lowest terms) has no prime factors other than 2 and 5. Any factor of 3, 7, 11, etc. causes the decimal to recur.

Part 2: Recurring Decimal Notation

Instead of writing 0.333โ€ฆ we use a dot above the repeating digit(s):

One digit repeating:   0.333โ€ฆ = 0.3   (dot above the 3)
Two digits repeating:   0.121212โ€ฆ = 0.12   (dots above first AND last digit of the block)
Three digits repeating:   0.142857142857โ€ฆ = 0.142857   (dots on first and last of the repeating block)
๐Ÿ“Œ Rule: Put a dot above the first and last digit of the repeating block. For a single digit, just one dot.

Part 3: Converting Fractions to Decimals by Division

Divide the numerator by the denominator. Track remainders โ€” when a remainder repeats, the decimal repeats.

Example: 2/11
20 รท 11 = 1 remainder 9  โ†’  digit: 1
90 รท 11 = 8 remainder 2  โ†’  digit: 8
20 รท 11 = 1 remainder 9  โ†’  remainder 2 appeared before! Pattern repeats.
2/11 = 0.18 = 0.181818โ€ฆ
๐Ÿ’ก The maximum number of digits in the repeating block is (denominator โˆ’ 1). So 1/7 can have at most 6 repeating digits.

Part 4: Which Fractions Give Recurring Decimals?

First simplify the fraction completely. Then check the denominator's prime factors:

If denominator = 2a ร— 5b only  โ†’  Terminates โœ“
If denominator has any other prime factor (3, 7, 11, 13, โ€ฆ)  โ†’  Recurs โœ—
Examples:   1/8 (8=2ยณ) โœ“  ยท  3/25 (25=5ยฒ) โœ“  ยท  1/6 (6=2ร—3) โœ—  ยท  5/14 (14=2ร—7) โœ—
โš ๏ธ Always simplify first!   3/6 = 1/2 โ†’ terminates even though 6 has a factor of 3.

Part 5: Converting Recurring Decimals to Fractions โ€” Algebraic Method

The key idea: multiply to shift the decimal so the repeating parts line up, then subtract.

One repeating digit: Let x = 0.3 = 0.333โ€ฆ
10x = 3.333โ€ฆ
10x โˆ’ x = 3.333โ€ฆ โˆ’ 0.333โ€ฆ
9x = 3  โ†’  x = 1/3
Two repeating digits: Let x = 0.12 = 0.121212โ€ฆ
100x = 12.121212โ€ฆ
100x โˆ’ x = 12
99x = 12  โ†’  x = 12/99 = 4/33
Mixed case: Let x = 0.16 = 0.1666โ€ฆ
10x = 1.666โ€ฆ
100x = 16.666โ€ฆ
100x โˆ’ 10x = 15
90x = 15  โ†’  x = 15/90 = 1/6
๐Ÿ“Œ Multiply by 10n where n is the length of the repeating block. If there are non-repeating digits first, use an extra multiplication to shift past them.
Created by Mr Josh

๐Ÿ’ก Worked Examples

Example 1: Fraction to Decimal โ€” Terminating

Convert 7/8 to a decimal.

Step 1: 8 = 2ยณ โ€” only factor of 2, so it terminates โœ“
Step 2: Divide: 7.000 รท 8
70 รท 8 = 8 r 6  ยท  60 รท 8 = 7 r 4  ยท  40 รท 8 = 5 r 0 โ†’ done
7/8 = 0.875

Example 2: Fraction to Decimal โ€” Recurring

Convert 5/6 to a decimal. Use dot notation.

Step 1: 6 = 2 ร— 3 โ€” has factor of 3, so it recurs โœ—
Step 2: 5 รท 6 โ†’ 50 รท 6 = 8 r 2  ยท  20 รท 6 = 3 r 2  ยท  remainder 2 keeps repeating
5/6 = 0.83 = 0.8333โ€ฆ
๐Ÿ“Œ The 8 does not repeat (it's non-recurring), only the 3 has a dot.

Example 3: Recurring Decimal to Fraction (Single Digit)

Convert 0.6 to a fraction.

Step 1: Let x = 0.666โ€ฆ
Step 2: Multiply both sides by 10:   10x = 6.666โ€ฆ
Step 3: Subtract:   10x โˆ’ x = 6.666โ€ฆ โˆ’ 0.666โ€ฆ  โ†’  9x = 6
Step 4: Solve:   x = 6/9 = 2/3
0.6 = 2/3

Example 4: Recurring Decimal to Fraction (Two Repeating Digits)

Convert 0.27 to a fraction.

Step 1: Let x = 0.272727โ€ฆ
Step 2: Repeating block has 2 digits, so multiply by 100:   100x = 27.272727โ€ฆ
Step 3: Subtract:   100x โˆ’ x = 27  โ†’  99x = 27
Step 4: x = 27/99 = 3/11   (HCF = 9)
0.27 = 3/11
Created by Mr Josh

๐Ÿ” Interactive Visualizers

๐Ÿฉ Long Division Animator

Enter a fraction and watch the long division step by step. The pattern lights up green when it starts to repeat!

รท

๐Ÿญ Terminating vs Recurring Sorter

Drag (or click) the fraction cards to the correct bin! A fraction terminates if its denominator in lowest terms has only factors of 2 and 5.

โœ“ Terminates

โˆž Recurs

๐Ÿงฎ Recurring to Fraction โ€” Step-by-Step Algebra

Choose a recurring decimal and click Next Step to watch the algebra unfold one line at a time.

๐Ÿ“ Fraction & Decimal Number Line

Click a value below to place it on the number line. See how fractions and decimals interleave!

Created by Mr Josh

โœ๏ธ Exercise 1: Convert Fractions to Decimals

Divide numerator by denominator. Give your answer as a decimal (use dot notation for recurring).

Created by Mr Josh

โœ๏ธ Exercise 2: Terminating or Recurring?

For each fraction, decide whether the decimal will terminate or recur. Type T or R.

Created by Mr Josh

โœ๏ธ Exercise 3: Write the Recurring Notation

Given the decimal expansion, state how many digits are in the repeating block, and write the answer using dot notation (type the digits of the block, e.g. "3" or "18" or "142857").

Created by Mr Josh

โœ๏ธ Exercise 4: Recurring Decimals to Fractions

Use the algebraic method. Give your answer as a fraction in its simplest form (e.g. 1/3 or 4/33).

Created by Mr Josh

๐Ÿ“ Exercise 5: Ordering Fractions & Decimals

Click the values in the pool in order from smallest to largest. Click a placed value to remove it.

Created by Mr Josh

๐Ÿ“ Practice Questions

  1. Convert 3/8 to a decimal. Does it terminate or recur?
  2. Convert 5/9 to a decimal. Use dot notation.
  3. Convert 7/12 to a decimal. Use dot notation.
  4. Convert 1/6 to a decimal. Use dot notation.
  5. Convert 11/20 to a decimal.
  6. State whether 7/14 gives a terminating or recurring decimal. Explain your reasoning.
  7. State whether 13/15 gives a terminating or recurring decimal. Explain your reasoning.
  8. Write 0.777โ€ฆ using dot notation.
  9. Write 0.363636โ€ฆ using dot notation.
  10. Write 0.2ฬ„4ฬ„ as a decimal expansion (first 6 decimal places).
  11. Convert 0.5ฬ„ to a fraction in its simplest form.
  12. Convert 0.3ฬ„6ฬ„ to a fraction in its simplest form.
  13. Convert 0.4ฬ„ to a fraction in its simplest form.
  14. Convert 0.0ฬ„9ฬ„ to a fraction. Show your working.
  15. Convert 0.13ฬ„ to a fraction. (Hint: the non-repeating part requires two multiplications.)
  16. Convert 0.41ฬ„6ฬ„ to a fraction in its simplest form.
  17. Arrange these in ascending order: 1/3, 0.35, 3/8, 0.3ฬ„
  18. Which is greater: 5/11 or 0.45? Show your reasoning.
  19. A student says "0.9ฬ„ = 1". Use the algebraic method to prove or disprove this.
  20. Explain why 1/14 gives a recurring decimal but 1/16 does not.
  1. 3/8 = 0.375 โ€” terminates (8 = 2ยณ)
  2. 5/9 = 0.5ฬ„ = 0.555โ€ฆ
  3. 7/12 = 0.583ฬ„ = 0.5833โ€ฆ
  4. 1/6 = 0.1ฬ„6ฬ„ = 0.1666โ€ฆ (note: only the 6 recurs)
  5. 11/20 = 0.55 โ€” terminates (20 = 2ยฒ ร— 5)
  6. 7/14 = 1/2 โ€” simplified denominator = 2 โ†’ terminates = 0.5
  7. 13/15 โ€” denominator 15 = 3 ร— 5 โ†’ recurs = 0.8ฬ„6ฬ„
  8. 0.7ฬ„
  9. 0.3ฬ„6ฬ„
  10. 0.242424โ€ฆ
  11. Let x = 0.555โ€ฆ; 10x = 5.555โ€ฆ; 9x = 5; x = 5/9
  12. Let x = 0.363636โ€ฆ; 100x = 36.3636โ€ฆ; 99x = 36; x = 36/99 = 4/11
  13. Let x = 0.444โ€ฆ; 10x = 4.444โ€ฆ; 9x = 4; x = 4/9
  14. Let x = 0.0909โ€ฆ; 100x = 9.0909โ€ฆ; 99x = 9; x = 9/99 = 1/11
  15. Let x = 0.1333โ€ฆ; 10x = 1.333โ€ฆ; 100x = 13.333โ€ฆ; 90x = 12; x = 12/90 = 2/15
  16. Let x = 0.41666โ€ฆ; 100x = 41.666โ€ฆ; 1000x = 416.666โ€ฆ; 900x = 375; x = 375/900 = 5/12
  17. 1/3 โ‰ˆ 0.333, 0.35, 0.3ฬ„ = 0.333, 3/8 = 0.375 โ†’ order: 1/3 = 0.3ฬ„ < 0.35 < 3/8
  18. 5/11 = 0.4ฬ„5ฬ„ โ‰ˆ 0.4545 > 0.45
  19. Let x = 0.999โ€ฆ; 10x = 9.999โ€ฆ; 9x = 9; x = 1. So 0.9ฬ„ = 1. โœ“
  20. 1/14 = 1/(2ร—7) โ€” factor of 7 causes recurrence. 1/16 = 1/2โด โ€” only 2s โ†’ terminates.
Created by Mr Josh

๐Ÿ† Challenge Questions

  1. Without long division, determine how many digits are in the repeating block of 1/13. (Hint: think about possible remainders when dividing by 13.)
  2. Show algebraically that 0.2ฬ„4ฬ„ = 8/33.
  3. A recurring decimal has the form 0.aฬ…bฬ…cฬ… where abc is a 3-digit repeating block. Write a general formula for converting it to a fraction.
  4. Prove that any fraction p/q where q = 2a5b terminates by multiplying numerator and denominator to make the denominator a power of 10.
  5. Find a fraction equivalent to 0.142857142857โ€ฆ without using long division. (Hint: the repeating block has 6 digits.)
  6. Convert 1.2ฬ„7ฬ„ (a mixed recurring decimal) to a fraction. (Hint: subtract the whole part first.)
  7. The decimal 0.3ฬ„ = 1/3 and 0.6ฬ„ = 2/3. Use these to find 0.3ฬ„ + 0.6ฬ„ and explain why the result proves that 0.9ฬ„ = 1.
  8. A student converts 3/7 and gets 0.428571428571โ€ฆ She claims the digits cycle with period 6. Verify this and find the 100th decimal digit of 3/7.
  1. When dividing by 13, remainders can be 1โ€“12, so the block has at most 12 digits. In fact 1/13 has period 6: 0.0ฬ„7ฬ„6ฬ„9ฬ„2ฬ„3ฬ„
  2. x = 0.242424โ€ฆ; 100x = 24.2424โ€ฆ; 99x = 24; x = 24/99 = 8/33 โœ“
  3. 0.aฬ…bฬ…cฬ… = abc/999 (for a 3-digit repeating block starting immediately after the decimal point)
  4. If q = 2a5b, let m = max(a,b). Multiply by 10m/10m: denominator becomes 10m, a power of 10, so decimal terminates.
  5. 0.142857142857โ€ฆ = 142857/999999 = 1/7
  6. 1.272727โ€ฆ = 1 + 0.2ฬ„7ฬ„. For 0.2ฬ„7ฬ„: 99x = 27, x = 27/99 = 3/11. Total = 1 + 3/11 = 14/11.
  7. 0.3ฬ„ + 0.6ฬ„ = 1/3 + 2/3 = 1. But digit-by-digit: 0.333โ€ฆ + 0.666โ€ฆ = 0.999โ€ฆ. Since the sum equals 1, we have 0.9ฬ„ = 1. โœ“
  8. 3/7 = 0.428571ฬ„ โ€” period 6 confirmed. 100 = 6 ร— 16 + 4, so the 100th digit is the 4th in the cycle: 428571 โ†’ 4th digit = 5.
Created by Mr Josh