Paper I · 9 Logical Reasoning & Analytical Ability · 9.1 Number system and simplification
9.1.6 Decimal fractions and recurring decimals: conversion and comparison
Every fraction is a terminating or a recurring decimal; know when each happens, convert both ways by rule, and handle decimal points by counting places rather than guessing.
- APPSC Paper I · Paper I
- Standard level
- 6 min read
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Core facts
- Decimal fraction: a fraction whose denominator is a power of 10 (0.07 = 7/100).
- Decimal to fraction: write the digits over 1 followed by as many zeros as there are decimal places, then reduce: 0.875 = 875/1,000 = 7/8.
- Terminating test: a fraction in lowest terms terminates only if its denominator has no prime factor other than 2 and 5. The number of decimal places = the larger of the powers of 2 and 5.
- Notation: in this note, digits in brackets recur: 0.(36) = 0.363636…; 0.2(45) = 0.2454545…
- Pure recurring (all digits after the point recur): numerator = the recurring block; denominator = as many 9s as digits in the block. 0.(7) = 7/9; 0.(36) = 36/99.
- Mixed recurring: numerator = (all digits after the point) − (non-recurring digits); denominator = as many 9s as recurring digits, followed by as many 0s as non-recurring digits. 0.2(45) = (245 − 2)/990.
- Multiplying decimals: multiply as whole numbers; decimal places in the answer = sum of decimal places.
- Dividing decimals: shift the point in both numbers until the divisor is a whole number.
- Adding zeros at the right end of a decimal does not change it: 0.5 = 0.50 = 0.500.
Example 1. Write 0.875 as a fraction in lowest terms. Solution. 875/1,000; divide by 125. Answer: 7/8
Example 2. Write 0.(36) as a fraction. Solution. 36/99 = 4/11. Answer: 4/11
Example 3. Find 0.18 × 0.05. Solution. 18 × 5 = 90; places 2 + 2 = 4, so 0.0090. Answer: 0.009
Explained
Type 1: mixed recurring to a fraction.
Example 4. Write 0.2(45) as a fraction in lowest terms. (a) 27/110 (b) 243/999 (c) 245/990 (d) 27/99 Solution. (245 − 2)/990 = 243/990; divide by 9: 27/110. Answer: (a) 27/110
Example 5. Write 3.1(6) as a fraction. Solution. Whole part 3 stays. 0.1(6) = (16 − 1)/90 = 15/90 = 1/6. So 3 + 1/6. Answer: 19/6
Type 2: will it terminate, and after how many places? Reduce the fraction first.
Example 6. Which of these does NOT give a terminating decimal? (a) 17/80 (b) 21/56 (c) 9/75 (d) 13/60 Solution. 17/80: 80 = 2⁴ × 5, terminates. 21/56 = 3/8, terminates. 9/75 = 3/25, terminates. 13/60: 60 = 2² × 3 × 5 has a 3. Answer: (d) 13/60
Example 7. After how many decimal places does 23/(2³ × 5⁴) terminate? Solution. The larger power is 4, so 4 places (it equals 0.0046). Answer: 4
Type 3: dividing decimals.
Example 8. Find 0.0084 ÷ 0.12. Solution. Multiply both by 100: 0.84 ÷ 12 = 0.07. Answer: 0.07
Type 4: arranging fractions. Convert each to a decimal to 3 places, or compare pairs by cross-multiplying.
Example 9. Arrange 4/7, 5/9, 7/12 and 9/16 in ascending order. Solution. 4/7 ≈ 0.571, 5/9 ≈ 0.556, 7/12 ≈ 0.583, 9/16 = 0.5625. Answer: 5/9 < 9/16 < 4/7 < 7/12
Type 5: sums of recurring decimals. Convert to fractions, add, convert back if needed.
Example 10. Find 0.(4) + 0.(5) + 0.(6). Solution. (4 + 5 + 6)/9 = 15/9 = 5/3 = 1.(6). Answer: 5/3
Example 11. Simplify (0.6 × 0.6 − 0.4 × 0.4) ÷ (0.6 − 0.4). Solution. (a² − b²)/(a − b) = a + b = 1. Answer: 1
Speed tips.
- Learn the common decimals: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875, 1/16 = 0.0625, 1/7 = 0.(142857), 1/9 = 0.(1), 1/11 = 0.(09), 1/12 = 0.08(3).
- Counting decimal places is safer than "moving the point" by eye; write the count down.
Common traps
- ✗ "0.2(45) = 245/999." ✓ Only "45" recurs: (245 − 2)/990 = 27/110.
- ✗ "21/56 does not terminate because 56 has a 7." ✓ Reduce first: 21/56 = 3/8, which terminates.
- ✗ "0.18 × 0.05 = 0.09." ✓ Places add: 2 + 2 = 4, so 0.0090 = 0.009.
- ✗ "0.(9) is slightly less than 1." ✓ 0.(9) = 1 exactly.
- ✗ "0.0084 ÷ 0.12 = 0.7." ✓ Shift both points two places: 0.84 ÷ 12 = 0.07.
Quick revision
- Terminates iff the reduced denominator has only 2s and 5s.
- Decimal places = larger of the powers of 2 and 5 in the denominator.
- Pure recurring: block over as many 9s.
- Mixed recurring: (all digits − non-recurring digits) over 9s then 0s.
- Product of decimals: add the decimal places.
- Division: make the divisor a whole number first.
- 0.(ab) + 0.(ba) = (a + b)/9.
- 1/7 = 0.(142857); nth digit from n mod 6.
- 0.(9) = 1.
- 1/8 = 0.125, 1/16 = 0.0625, 1/12 = 0.08(3).
Go deeper: Deeper (beyond what APPSC Paper I usually asks)
Deeper Advanced
Example 12. Which digit is in the 100th place after the decimal point in 1/7? Solution. 1/7 = 0.(142857), period 6. 100 = 6 × 16 + 4, so the 100th digit is the 4th digit of the block, 8. Answer: 8
Example 13. Find the sum of the first 20 digits after the decimal point in 3/11. Solution. 3/11 = 0.(27). Twenty digits = ten blocks of "27", each adding 9. Answer: 90
Example 14. If 1/3.718 = 0.26896 (to 5 places), find 1/0.0003718. Solution. 0.0003718 = 3.718 ÷ 10⁴, so its reciprocal = 0.26896 × 10⁴. Answer: 2,689.6
Example 15. Find 0.(ab) + 0.(ba) in terms of the digits a and b, and use it for 0.(38) + 0.(83). Solution. (10a + b)/99 + (10b + a)/99 = 11(a + b)/99 = (a + b)/9. For a = 3, b = 8: 11/9. Answer: (a + b)/9; 11/9
- 0.(9) = 1 exactly, not "just below 1": 9/9 = 1. Likewise 0.4(9) = 0.5.
- Period length (CSAT-level fact): for a prime p other than 2 and 5, the period of 1/p divides p − 1. 1/7 has period 6; 1/13 = 0.(076923) also has period 6; 1/11 has period 2.
- Multiplying recurring decimals: convert first. 0.(7) × 0.(3) = 7/9 × 3/9 = 7/27.
Practise this topic. Questions on Numerical ability, with answers explained.
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