Decimals and Recurring Decimals
Decimals and Recurring Decimals
Decimal questions in NTPC test three things: converting a recurring decimal into p/q, deciding whether a fraction terminates, and handling decimal places in multiplication and division without a calculator. Each has a fixed method that works in under 30 seconds.
1. Decimal Place Value
| Place after the point | Name | Value |
|---|---|---|
| 1st | Tenths | 1/10 = 0.1 |
| 2nd | Hundredths | 1/100 = 0.01 |
| 3rd | Thousandths | 1/1000 = 0.001 |
| 4th | Ten-thousandths | 0.0001 |
In 46.358 the place value of 5 is 5/100 = 0.05 and its face value is 5. Adding zeros at the right end does not change a decimal: 0.7 = 0.70 = 0.700.
2. Terminating or Recurring: the Denominator Test
Write the fraction in LOWEST terms. If the denominator has only 2s and 5s as prime factors, the decimal terminates; otherwise it recurs.
Number of decimal places = the larger of the powers of 2 and 5 in the denominator
| Fraction (lowest form) | Denominator | Type | Decimal |
|---|---|---|---|
| 13/125 | 5³ | Terminates, 3 places | 0.104 |
| 7/40 | 2³ × 5 | Terminates, 3 places | 0.175 |
| 7/30 | 2 × 3 × 5 | Recurs (factor 3) | 0.2333... |
| 11/12 | 2² × 3 | Recurs | 0.91666... |
| 21/60 = 7/20 | 2² × 5 | Terminates (simplify first!) | 0.35 |
✗ 21/60 has a 3 in 60, so it recurs | ✓ Reduce first: 21/60 = 7/20, which terminates as 0.35
3. Recurring Decimal to p/q
A bar over digits means they repeat: 0.4̅5̅ = 0.454545..., 0.46̅ = 0.4666..., 0.72̅3̅ = 0.7232323...
| Type | Rule | Example |
|---|---|---|
| Pure recurring 0.a̅ | a/9 | 0.7̅ = 7/9 |
| Pure recurring 0.a̅b̅ | ab/99 | 0.4̅5̅ = 45/99 = 5/11 |
| Pure recurring 0.a̅b̅c̅ | abc/999 | 0.1̅3̅5̅ = 135/999 = 5/37 |
| Mixed recurring 0.ab̅ | (ab − a)/90 | 0.46̅ = (46 − 4)/90 = 7/15 |
| Mixed recurring 0.ab̅c̅ | (abc − a)/990 | 0.72̅3̅ = (723 − 7)/990 = 358/495 |
| With a whole part | Convert the decimal part, then add | 2.3̅ = 2 + 3/9 = 7/3 |
General rule: (all digits written once − non-recurring digits) / (as many 9s as recurring digits, followed by as many 0s as non-recurring digits)
Useful identity: 0.9̅ = 1 exactly, because 9/9 = 1.
4. The 1/7 Family (cyclic number 142857)
| 1/7 | 2/7 | 3/7 | 4/7 | 5/7 | 6/7 |
|---|---|---|---|---|---|
| 0.142857... | 0.285714... | 0.428571... | 0.571428... | 0.714285... | 0.857142... |
All six use the same six digits in the same cyclic order; just start from the right digit (for 5/7 start with 7, because 5/7 ≈ 0.71). The digit sum of one cycle is 27.
5. Multiplying and Dividing Decimals
Multiplication: multiply as whole numbers, then give the product as many decimal places as the factors have in total
Division: make the divisor a whole number by shifting the point in both numbers by the same places
0.2 × 0.02 × 0.002 = 0.000008 (1 + 2 + 3 = 6 places). 0.0036 ÷ 0.12 = 0.36 ÷ 12 = 0.03.
हिंदी नोट: आवर्ती दशमलव को भिन्न में बदलने के लिए सभी अंकों से अनावर्ती अंक घटाइए, फिर हर में आवर्ती अंकों जितने 9 और अनावर्ती अंकों जितने 0 लिखिए।
Exam Pointer: Verified NTPC patterns: recurring-to-p/q (9 January 2021, plus a long mixed form in the 2020-21 cycle), terminating-decimal identification (7 January 2021), decimal identities (7 March 2021) and decimal-place counting in a product-quotient (Graduate CBT-1, 18 March 2026). The traps: not reducing the fraction before the denominator test, and miscounting decimal places in products.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Recurring decimal to p/q and sums of recurring decimals
[PYQ: NTPC CBT-1 9-Jan-2021 Shift-1]
EXAM LEVEL
Q. Express 0.4̅5̅ (that is, 0.454545...) as a fraction in lowest terms.
Two digits recur and none are non-recurring, so the value is 45/99. Dividing both by 9 gives 5/11.
Answer: 5/11
EXAMATLAS LEVEL
Q. Find the value of 0.46̅ + 0.72̅3̅ as a fraction (0.4666... + 0.72323...).
0.46̅ = (46 − 4)/90 = 42/90 = 7/15. 0.72̅3̅ = (723 − 7)/990 = 716/990 = 358/495. LCM of 15 and 495 is 495: 7/15 = 231/495. Sum = (231 + 358)/495 = 589/495 = 1 94/495. Treating 0.46̅ as 46/99 is the trap; only the 6 repeats.
Answer: 589/495
Pattern 2: Will it terminate, and after how many places
[PYQ: NTPC CBT-1 7-Jan-2021 Shift-2]
EXAM LEVEL
Q. Which of the following has a terminating decimal expansion? (a) 7/30 (b) 13/125 (c) 11/12 (d) 17/45
Check denominators in lowest form: 30 has a 3, 12 has a 3, 45 has 3². Only 125 = 5³ is made of 2s and 5s. 13/125 = 0.104.
Answer: (b) 13/125
EXAMATLAS LEVEL
Q. After how many decimal places will 77/(2³ × 5² × 7) terminate?
Cancel the 7 first: 77/(2³ × 5² × 7) = 11/(2³ × 5²) = 11/200. The denominator has only 2s and 5s, so it terminates, and the larger power is 3, so 3 places: 11/200 = 0.055. Students who see the 7 in the denominator and mark "non-terminating" lose the mark.
Answer: 3 places (0.055)
Pattern 3: Decimal simplification with algebraic identities
[PYQ: NTPC CBT-1 7-Mar-2021 Shift-1]
EXAM LEVEL
Q. Simplify (0.25 × 0.25 − 0.15 × 0.15) ÷ (0.25 − 0.15).
a² − b² = (a − b)(a + b), so the expression is a + b = 0.25 + 0.15 = 0.4.
Answer: 0.4
EXAMATLAS LEVEL
Q. Find the value of (0.8³ + 0.2³)/(0.8² − 0.8 × 0.2 + 0.2²) + (3.25² − 1.75²)/0.15.
The first part is (a³ + b³)/(a² − ab + b²) = a + b = 0.8 + 0.2 = 1. The second part is (3.25 − 1.75)(3.25 + 1.75)/0.15 = 1.5 × 5/0.15 = 7.5/0.15 = 50. Total = 1 + 50 = 51.
Answer: 51
Pattern 4: Ordering decimals, including recurring ones
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Arrange 0.07, 0.7, 0.077 and 0.707 in ascending order.
Write all to 3 places: 0.070, 0.700, 0.077, 0.707. Ascending: 0.070, 0.077, 0.700, 0.707.
Answer: 0.07 < 0.077 < 0.7 < 0.707
EXAMATLAS LEVEL
Q. Which is the largest: 0.3̅, 0.33, 0.3̅2̅, 0.33̅2̅?
Expand each to 5 places: 0.3̅ = 0.33333, 0.33 = 0.33000, 0.3̅2̅ = 0.32323, 0.33̅2̅ = 0.33232. Compare digit by digit: 0.33333 is the largest. The bar position decides everything; 0.33̅2̅ starts 0.332..., not 0.333.
Answer: 0.3̅
Pattern 5: Digits of a recurring decimal (the 1/7 family)
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Write 5/7 as a decimal.
5/7 is a little more than 0.7, so start the cycle 142857 at the 7: 0.714285 repeating.
Answer: 0.7̅1̅4̅2̅8̅5̅
EXAMATLAS LEVEL
Q. What is the 100th digit after the decimal point in 3/7?
3/7 = 0.428571 repeating, a cycle of 6. 100 = 6 × 16 + 4, so the 100th digit is the 4th digit of the cycle 4, 2, 8, 5, 7, 1, which is 5.
Answer: 5
Pattern 6: Decimal places in multiplication and division
[PYQ: NTPC Graduate CBT-1 18-Mar-2026 Shift-2]
EXAM LEVEL
Q. Find the value of 0.04 × 0.5 ÷ 0.002.
0.04 × 0.5 = 0.020 (3 places). Then 0.020 ÷ 0.002 = 20 ÷ 2 = 10 after shifting both by 3 places.
Answer: 10
EXAMATLAS LEVEL
Q. Find the value of (0.0036 × 1.5)/(0.012 × 0.45).
Count decimal places: numerator 4 + 1 = 5, denominator 3 + 2 = 5. Equal places cancel, so compute (36 × 15)/(12 × 45) = 540/540 = 1. Counting places once saves all the decimal shifting.
Answer: 1
60-Second Revision
- Terminates only if the LOWEST-form denominator has just 2s and 5s; places = larger power.
- Pure recurring: digits over 9s; mixed: (all − non-recurring) over 9s then 0s.
- 0.46̅ = 7/15, not 46/99; 0.9̅ = 1.
- 1/7 family uses the cycle 142857; nth digit by cycle-6 remainder.
- Product decimal places = sum of places; divide by making the divisor whole.
- Use a² − b² and a³ + b³ identities to kill decimal calculation.
Next Step: Decimals and fractions done. Practise the simplification and fraction questions in the ExamAtlas RRB NTPC 2026 mock tests; every chained-fraction slip you make there maps to a pattern on these two pages.