Arithematics
Recurring decimals
In this note
- 1.4Read and write recurring decimal notation: \(0.\dot1\dot7=0.1717\ldots\), \(0.1\dot7=0.1777\ldots\).
- 1.4Change a fraction into a decimal and say whether it terminates or recurs.
- 1.4Change any recurring decimal into a fraction, showing all the working.
Formula bank
Notation and why decimals recur
| Written | Means | Repeating block |
|---|---|---|
| \(0.\dot4\) | 0.444 … | 4 |
| \(0.\dot4\dot7\) | 0.474 7 … | 47 |
| \(0.1\dot7\) | 0.177 7 … | 7 (the 1 does not repeat) |
| \(0.\dot12\dot3\) | 0.123 123 … | 123 |
When you divide, the same remainder keeps coming back, so the same digits come out forever: \(1\div3=0.\dot3\). A fraction (in simplest form) terminates only when its denominator's prime factors are just 2s and 5s (the factors of 10).
Try it yourself
Show answer
Answer 0.35
Show answer
Answer recurs: \(0.8\dot3\) (the denominator has a factor 3)
Recurring decimal → fraction
Multiply the decimal by 10, 100 or 1000 so that the repeating tail lines up, then subtract: the endless tails cancel exactly.
General methodHow to change any recurring decimal into a fraction
- Let \(x\) = the decimal; write a few cycles out.
- Count the digits in the repeating block (k). Multiply by \(10^{k}\) (10, 100, 1000).
- If some digits do NOT repeat, use two multiples whose tails match exactly (e.g. \(10x\) and \(100x\)).
- Write the two equations one under the other and subtract: the tails cancel.
- Solve for \(x\) and simplify.
Example Write \(0.\dot4\dot7\) as a fraction.
Two repeating digits: × 100, then subtract.
- \(x=0.4747\ldots\)
- \(100x=47.4747\ldots\)
- \(99x=47\)
Answer \(\tfrac{47}{99}\)
Example Write \(0.1\dot7\) as a fraction.
The 1 does not repeat: use 10x and 100x, whose tails are both 0.777….
- \(10x=1.777\ldots\)
- \(100x=17.777\ldots\)
- \(90x=16\)
- \(x=\tfrac{16}{90}\)
Answer \(\tfrac8{45}\)
Try it yourself
Show answer
Answer \(\tfrac23\)
Show answer
Answer \(\tfrac1{11}\)
Show answer
Answer \(\tfrac2{15}\)
Ways they test it
Type 1 Show that
"Show that \(0.\dot3\dot6=\dfrac4{11}\)." (2 marks)
Spot it "show that" with a recurring decimal.
- Two equations (\(x\) and \(100x\)) written out.
- Subtract, solve, simplify to the given fraction.
For example \(0.\dot3\dot6\)
- \(100x=36.3636\ldots\)
- \(99x=36\)
- \(x=\tfrac{36}{99}\)
Answer \(\tfrac4{11}\)
Type 2 With a non-repeating digit
"Write \(0.3\dot8\) as a fraction."
Spot it a digit before the dots.
- 10x and 100x.
- Subtract: 90x = …
For example \(0.3\dot8\)
- \(10x=3.888\ldots\)
- \(100x=38.888\ldots\)
- \(90x=35\)
Answer \(\tfrac7{18}\)
Type 3 Add recurring decimals
"Work out \(0.\dot2\dot3+0.\dot4\) as a fraction."
Spot it two recurring decimals with + or −.
- Change each to a fraction.
- Add with a common denominator.
For example \(0.\dot2\dot3+0.\dot4\)
- \(\tfrac{23}{99}+\tfrac{44}{99}\)
Answer \(\tfrac{67}{99}\)
Type 4 Terminates or recurs
"Which of these fractions gives a recurring decimal?"
Spot it a list of fractions.
- Simplify each.
- Denominator with a factor other than 2 or 5 → recurs.
Mistakes to avoid
- ✗ \(0.\dot1\dot7\) read as 0.1777…✓ 0.1717…Dots on both digits: both repeat.
- ✗ \(0.1\dot7\): \(10x-x=9x=1.6\)✓ \(100x-10x=90x=16\)Choose multiples whose tails match.
- ✗ Only the 9s shortcut written✓ show both equations and the subtraction"Show that" needs the working.
- ✗ \(\tfrac{36}{99}\) left✓ \(\tfrac4{11}\)Simplify.
Summary
- Dots mark the repeating digits (first and last of a longer block).
- Terminating decimals come from denominators made of 2s and 5s.
- Recurring → fraction: \(x\), \(10^{k}x\), subtract, solve, simplify.
- A non-repeating digit: use two multiples with matching tails.
If you see …
- "show that" with dotstwo equations and a subtraction.
- a digit before the dots10x and 100x (or 10x and 1000x).
Practice
Practice: easy, medium, hard
Easy
Show working
- \(10x=5.55\ldots\)
- \(9x=5\)
Answer \(\tfrac59\)
Show working
- \(99x=72\)
- \(\tfrac{72}{99}=\tfrac8{11}\)
Answer \(\tfrac8{11}\)
Show working
- \(3\div8\)
Answer 0.375
Medium
Guide me
What is it asking? a fraction
What do we know? two repeating digits
Plan × 100, subtract
Show working
- \(99x=36\)
- \(\tfrac{36}{99}\)
Answer \(\tfrac4{11}\)
Guide me
What is it asking? a fraction
What do we know? the 3 doesn't repeat
Plan 10x and 100x
Show working
- \(90x=35\)
- \(\tfrac{35}{90}\)
Answer \(\tfrac7{18}\)
Guide me
What is it asking? a fraction
What do we know? a 3-digit block
Plan × 1000, subtract
Show working
- \(999x=123\)
Answer \(\tfrac{41}{333}\)
Hard
Guide me
What is it asking? a fraction
What do we know? 0.21515…: the 2 doesn't repeat
Plan 10x and 1000x
Show working
- \(10x=2.1515\ldots\)
- \(1000x=215.1515\ldots\)
- \(990x=213\)
- \(\tfrac{213}{990}=\tfrac{71}{330}\)
Answer \(\tfrac{71}{330}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| two equations with matching tails | M1 | ||
| \(990x=213\) | M1 | ||
| \(\tfrac{71}{330}\) | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? the sum as a fraction
What do we know? two recurring decimals
Plan each to a fraction, then add
Show working
- \(\tfrac{23}{99}+\tfrac49=\tfrac{23}{99}+\tfrac{44}{99}\)
Answer \(\tfrac{67}{99}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\tfrac{23}{99}\) and \(\tfrac49\) | M1 | ||
| \(\tfrac{67}{99}\) | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? the value
What do we know? x = 0.999…
Plan × 10, subtract
Show working
- \(10x=9.99\ldots\)
- \(9x=9\)
- \(x=1\)
Answer 1
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(9x=9\) | M1 | ||
| 1 | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I can read and write recurring decimal notation.
- I can tell whether a fraction terminates or recurs.
- I can change a recurring decimal into a fraction, showing the subtraction.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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