Recurring decimals to fractions: Class 9 practice

What you should know first

• Place value
• Simplifying fractions

The idea behind recurring decimals to fractions

A recurring decimal is rational. If x = 0.ab̅ with a block of n repeating digits, multiply by 10ⁿ and subtract x to remove the repeating part.

Step-by-step method

1. Let x = the decimal.
2. Multiply by 10ⁿ where n = length of the repeating block.
3. Subtract to cancel the repeating tail.
4. Solve for x and simplify.

Worked example 1 (Basic)

Write 0.333… (the block 3 repeats) as p/q.

1. x = 0.333…
2. 10x − x = 3
3. x = 3/9 = 1/3
4. Answer: 1/3

Worked example 2 (Standard)

Write 0.949495… (the block 94 repeats) as p/q.

1. x = 0.949495…
2. 100x − x = 94
3. x = 94/99 = 94/99
4. Answer: 94/99

Worked example 3 (Challenge)

Write 0.7777… (only 7 repeats) as a fraction.

1. 10x = 7.77…, 100x = 77.7…
2. 90x = 77 − 7 = 70
3. x = 7/9
4. Answer: 7/9

Common mistakes and misconceptions

• Multiplying by the wrong power of 10
• Not simplifying the fraction

Practice questions

1. (Basic) Write 0.555… (the block 5 repeats) as p/q.
2. (Basic) Write 0.222… (the block 2 repeats) as p/q.
3. (Standard) Write 0.969697… (the block 96 repeats) as p/q.
4. (Standard) Write 0.313131… (the block 31 repeats) as p/q.
5. (Challenge) Write 0.9222… (only 2 repeats) as a fraction.
6. (Challenge) Write 0.9888… (only 8 repeats) as a fraction.

Answer key with full working

• Q1: x = 0.555… → 10x − x = 5 → x = 5/9 = 5/9 → Answer: 5/9
• Q2: x = 0.222… → 10x − x = 2 → x = 2/9 = 2/9 → Answer: 2/9
• Q3: x = 0.969697… → 100x − x = 96 → x = 96/99 = 32/33 → Answer: 32/33
• Q4: x = 0.313131… → 100x − x = 31 → x = 31/99 = 31/99 → Answer: 31/99
• Q5: 10x = 9.22…, 100x = 92.2… → 90x = 92 − 9 = 83 → x = 83/90 → Answer: 83/90
• Q6: 10x = 9.88…, 100x = 98.8… → 90x = 98 − 9 = 89 → x = 89/90 → Answer: 89/90

Syllabus note

This skill is listed under "Number Systems" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

• Number Systems chapter notes (Class 9)
• Rationalising the denominator
• Simplifying surds
• Rational exponents
• Remainder theorem and factor theorem
• Area of a triangle by Heron's formula
• Factorising by splitting the middle term
• Cube identities
• All Class 9 practice

Frequently asked questions

Which chapter is recurring decimals to fractions in?

"Number Systems", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.