Everything the Decimal and Fraction chapter asks in SSC CGL, CHSL, CPO and other competitive exams — terminating and recurring decimals, why the 9s-and-0s rule works, comparing and ordering fractions, continued fractions, the 99-type products and every fraction word problem pattern. Every concept used across the chapter tests is explained here from zero.
Decimal and Fraction is one of the most reliable marks in SSC Quantitative Aptitude. Almost every shift carries a question from it — a recurring decimal to be written as a fraction, a continued fraction to be evaluated, four fractions to be put in order, or a fraction that changes when something is added to its numerator and denominator. None of these needs heavy calculation; each needs one rule, applied in the right order, in under thirty seconds.
These notes explain every rule from the beginning — not just what the rule says but why it works, because the "why" is what lets you handle a decimal or a continued fraction you have never seen before. Each rule comes with a worked example, the exam shortcut, and the mistake that most often costs a mark.
Notes fix the method; only practice fixes the speed. Keep these open in one tab and a test in another — start with the chapter tests for this topic, then move to full papers once the rules feel automatic.
Graded across two levels, matched to these notes. Every question carries a step-by-step solution.
Start practising Previous yearReal shift-wise CGL papers with answer keys and full solutions. The best guide to what actually gets asked.
Attempt PYQs Previous yearShift-wise CHSL papers, solved. Useful for CHSL, CPO and MTS aspirants alike.
Attempt PYQs Full lengthFull-length papers under real exam timing, with current affairs and detailed solutions after every attempt.
Take a mock testHow to use these notes: read a section, cover the example with your hand, solve it yourself, then check. A rule you have only read is forgotten by evening. A rule you have used once stays.
Every fraction of whole numbers turns into a decimal of exactly one of three kinds. Recognising the kind on sight is the first skill of the chapter, because the conversion rule and the comparison trick both depend on it.
Terminating — the digits stop: 3/8 = 0.375.
Pure recurring — a block repeats for ever, starting immediately after the point: 4/11 = 0.36 = 0.363636…
Mixed recurring — some digits come first and do not repeat, then a block repeats: 7/30 = 0.23 = 0.2333…
The bar (called a vinculum) sits over exactly the digits that repeat. 0.54 is 0.5454…, while 0.54 is 0.5444… — different numbers, and the bar position is the only clue.
A number is rational when it can be written as one whole number over another. Every terminating decimal is rational (0.375 = 375/1000) and every recurring decimal is rational (Sections 3 and 4 show how). A decimal that never ends and never settles into a repeating block — 0.101001000100001… — is irrational.
So "non-terminating" is not the same as "irrational": 0.3 never ends, yet it is exactly 1/3.
Which of these is a pure recurring decimal: 0.125, 0.27, 0.36, 0.101001000…?
0.125 stops — terminating. 0.27 has a non-repeating 2 first — mixed. 0.101001000… has no block — irrational. Only 0.36 starts repeating right after the point.
The whole-number part changes nothing. 3.1245 is mixed recurring for the same reason 0.1245 is — classify by the digits after the point only.
You never need to divide to find out what kind of decimal a fraction gives. The denominator tells you — but only after the fraction is in lowest terms.
Reduce first, then factor the denominator.
| Lowest-terms denominator has | Decimal | Example |
|---|---|---|
| only 2s and 5s | terminating | 7/40 = 0.175 (40 = 23 × 5) |
| no 2 and no 5 | pure recurring | 4/11 = 0.36 |
| some 2 or 5 and another prime | mixed recurring | 7/30 = 0.23 (30 = 2 × 3 × 5) |
Why. 10 = 2 × 5, so a denominator made of 2s and 5s can be multiplied up to a power of 10 — that is what a terminating decimal is. Any other prime in the denominator can never be cleared by a power of 10, so the digits go on for ever. The 2s and 5s produce the non-repeating digits; the other prime produces the repeating block.
For a lowest-terms denominator 2m × 5n, the decimal ends after max(m, n) places — the larger exponent, not the sum. 7/320: 320 = 26 × 5, so six places: 0.021875. The reason is that 10max(m,n) is the smallest power of 10 the denominator divides.
How many of 3/8, 5/24, 7/50, 9/45, 11/40 terminate?
8 = 23 ✔ 24 = 23 × 3 ✘ 50 = 2 × 52 ✔ 9/45 = 1/5 ✔ 40 = 23 × 5 ✔ — four. The fourth one is the whole question: 45 contains a 9, but the fraction reduces.
For how many n from 1 to 30 is n/30 a terminating decimal?
30 = 2 × 3 × 5. The 2 and the 5 may stay; the 3 must be cancelled, so n must be a multiple of 3: 3, 6, …, 30 — 10 values. Note 10/30 = 1/3 recurs even though 10 cancels both the 2 and the 5; cancelling the allowed primes achieves nothing.
Never judge the parts of a sum. 1/3 and 1/6 both recur, but 1/3 + 1/6 = 1/2 terminates. Add first, reduce, then read the denominator.
Write the repeating block over as many 9s as it has digits.
0.a = a/9 0.ab = ab/99 0.abc = abc/999
Why. Let x = 0.36. Then 100x = 36.36, and subtracting, 100x − x = 36, so 99x = 36 and x = 36/99 = 4/11. Multiplying by 10k (k = length of the block) shifts the decimal by exactly one block, so the tails cancel. Every 9s rule in this chapter is this one subtraction.
Reduce at the end, and know the factorisations: 99 = 9 × 11 and 999 = 27 × 37. So two-digit blocks often become elevenths (0.27 = 3/11, 0.63 = 7/11) and three-digit blocks become 27ths or 37ths (0.135 = 5/37, 0.296 = 8/27). A whole-number part is simply added back: 2.18 = 2 + 2/11 = 24/11.
Express 2.18 as a fraction in lowest terms.
0.18 = 18/99 = 2/11, so 2.18 = 2 + 2/11 = 24/11. The distractor 218/99 treats 218 as the block — but the 2 does not repeat.
0.27 is not 27/100, and 0.027 is not 27/99. The first is 27/99 (both digits repeat); the second has a non-repeating 0 and belongs to Section 4, where it becomes 27/990.
Numerator: all the digits after the point (non-repeating and one block), minus the non-repeating digits.
Denominator: one 9 for each repeating digit, followed by one 0 for each non-repeating digit.
0.ab = (ab − a)/90 0.abc = (abc − a)/990 0.abc = (abc − ab)/900
Why. x = 0.46. Then 10x = 4.6 and 100x = 46.6; subtracting, 90x = 46 − 4 = 42, so x = 42/90 = 7/15. The first multiplication moves past the non-repeating digits (giving the 0s), the second moves one block further (giving the 9s).
0.46 = ? (46 − 4)/90 = 42/90 = 7/15. Check: 7 ÷ 15 = 0.4666… ✔
0.327 = ? (327 − 3)/990 = 324/990 = 18/55.
1.23 = ? 1 + (23 − 2)/90 = 1 + 7/30 = 37/30. The distractor 11/9 is 1.2, a different number.
0.45 − 0.45 = ?
Same digits, different bars. 0.45 = 41/90 and 0.45 = 45/99 = 5/11. Difference = (451 − 450)/990 = 1/990. Never subtract the bars digit by digit — convert both first.
A leading zero after the point is a non-repeating digit: 0.045 = (045 − 0)/990 = 45/990 = 1/22. And a pair with swapped digits sums neatly: 0.27 + 0.72 = 25/90 + 65/90 = 1 whenever the two digits add to 9.
"Simplest form" means reduce before you read the numerator and denominator. 0.73 = 66/90 = 11/15, so "numerator + denominator" is 26, not 156.
The period is the length of the repeating block. For p/q with q co-prime to 10, the period is the smallest k for which q divides 10k − 1 — that is, the smallest number of 9s that q divides. It depends only on q, never on p.
| Denominator | Period | Because |
|---|---|---|
| 3, 9 | 1 | divide 9 |
| 11 | 2 | divides 99 |
| 27, 37 | 3 | divide 999 |
| 7, 13 | 6 | divide 999999 but no shorter run of 9s |
| 81 | 9 | 1/81 = 0.012345679 — no 8 in the block |
For a mixed recurring fraction, strip the 2s and 5s first: 3/70 → 70 = 2 × 5 × 7, so the period is that of 7, which is 6. For co-prime factors the period of the product is the LCM of the periods: 1/77 has period LCM(6, 2) = 6, not 12.
For a pure recurring decimal with block length L: the n-th digit is digit number (n mod L) of the block, where "mod" means the
remainder on division, and a remainder of 0 means the last digit of the block.
For a mixed recurring decimal with s non-repeating digits, subtract s first and use (n − s) mod L.
The 40th digit of 2/7? 2/7 = 0.285714, L = 6; 40 = 6 × 6 + 4 → the 4th digit of the block, 7.
The 50th digit of 5/22? 5/22 = 0.227, s = 1, L = 2; 49 mod 2 = 1 → the 1st digit of the block, 2. Forgetting the offset gives 7, which is wrong.
Sum of the first 20 digits of 4/7? Block 571428 has digit sum 27; 20 = 3 × 6 + 2, so 3 × 27 + (5 + 7) = 93.
Do not find the n-th digit of two fractions separately and add. Carries between digits break it. For 1/7 + 1/11, add first (18/77 = 0.233766), then locate the digit.
Convert, then compute. Recurring decimals obey ordinary fraction arithmetic exactly — because they are fractions. Same-length blocks can be added straight over 99 or 999: 0.12 + 0.23 + 0.34 = 69/99 = 23/33. Quotients of single-digit recurring decimals cancel the 9s: 0.2 ÷ 0.6 = 2/6 = 1/3.
0.9 = 1 exactly. 10x − x = 9 gives x = 1. So 0.3 + 0.6 = 1, and multiplying by 0.9 changes nothing.
Convert first: √(0.4) = √(4/9) = 2/3 = 0.6 — not 0.2. And (0.3)2 = 1/9 = 0.1, (0.3)2 + (0.6)2 = 1/9 + 4/9 = 5/9.
The least natural number that turns a recurring decimal into a whole number is the denominator of its lowest-terms fraction. 0.24 = 8/33 → 33 (not 99). 0.23 = 7/30 → 30 (not 90). For a sum or a quotient, combine first, reduce, then read the denominator: 0.6 + 0.15 = 9/11 → 11.
(0.45 + 0.54) ÷ 0.3 = ? The blocks are complements: 45 + 54 = 99, so the numerator is exactly 1. 1 ÷ (1/3) = 3.
What must be subtracted from 0.7 to get 0.07? 77/99 − 7/99 = 70/99.
"Slightly less than 1." Every year students mark 0.9 < 1 as true. It is false; 0.9 − 0.98 = 1 − 89/90 = 1/90, not 0.
Run the 9s rule backwards. To find the repeating block of p/q, rewrite p/q with denominator 9, 99 or 999 — whichever q divides — and read the block off the numerator.
If 8/27 = 0.abc, find a + b + c. 8 × 37 = 296, so the digits are 2, 9, 6 and the sum is 17. Dividing by hand and stopping at 0.29 hides the 6.
If 0.ab = 17/90, find a + b. 9a + b = 17 with digits → a = 1, b = 8 → 9. Check: 0.18 = 0.1888… = 17/90 ✔
a − b = 1 and 0.ab + 0.ba = 5/9. Find 0.ab. The sum is 11(a + b)/99 = (a + b)/9, so a + b = 5; with a − b = 1, a = 3, b = 2: 32/99.
0.ab + 0.ba = (a + b)/9 always, and 0.ab × 11 = ab/9. Both are asked as "find the digits" questions; both collapse to one line.
For positive fractions, a/b > c/d exactly when ad > bc. Multiplying both sides by the positive number bd keeps the direction, and it removes the denominators. 9/13 vs 11/16: 9 × 16 = 144 against 13 × 11 = 143, so 9/13 is larger — by a margin a decimal estimate (0.692 vs 0.6875) can easily get wrong.
Same numerator: the larger denominator is the smaller fraction — 7/15 < 7/12 < 7/11 < 7/9. The same quantity split into more parts.
Same denominator: the larger numerator wins. This is why bringing everything to the LCM of the denominators works — and it is the
method to use when the question then asks for a difference or a sum of the extremes.
For a proper fraction a/b (a < b): (a + k)/(b + k) > a/b for any positive k — it moves up towards 1. For an improper fraction it moves down towards 1. Proof in one line: b(a + k) − a(b + k) = k(b − a), positive when a < b. So 7/9 < 9/11, but 5/3 > 6/4 > 7/5.
Difference between the largest and smallest of 4/9, 5/12, 7/18, 1/2. LCM 36: 16, 15, 14, 18 over 36. Largest 18/36, smallest 14/36, difference 4/36 = 1/9.
Which is true: I. 4/9 < 5/11 < 6/13; II. 7/12 > 8/13 > 9/14? Cross-multiply each pair: 44 < 45 and 65 < 66 make I true; 91 < 96 means 7/12 < 8/13, so II is false. Only I.
Compare recurring decimals as fractions or as equal-length expansions, never by "how many digits repeat". 0.72 looks longer than 0.7 but is smaller (0.7272… vs 0.7777…). And 0.54 < 0.54 because the third digits are 4 and 5.
When every fraction is a little below 1, compare the gaps 1 − a/b = (b − a)/b instead. The smallest gap is the largest fraction. 23/24, 19/20, 15/16, 11/12 have gaps 1/24, 1/20, 1/16, 1/12, so 23/24 is largest. When a gap has numerator 2 (13/15 → 2/15), halve it before comparing with unit gaps: 2/15 = 1/7.5, which sits between 1/8 and 1/6.
Fractions of the form n/(n + k) with the same k increase with n, because the gap k/(n + k) shrinks. 13/17, 17/21, 21/25, 25/29 all have k = 4, so the last is largest. And 5/7, 6/8, 7/9, 8/10 all have k = 2, so the list is already ascending.
If numerators rise by a and denominators by b at each step, the fractions move steadily towards a/b. 2/3, 3/5, 4/7, 5/9 have a = 1, b = 2, so they head towards 1/2; since 2/3 > 1/2 they decrease: 2/3 > 3/5 > 4/7 > 5/9. A chain in such a list can only run one way — a statement like "7/12 > 8/13 > 9/14" that claims the opposite direction is false before you calculate.
Descending order of 11/14, 14/17, 0.8, 17/20. The three fractions are n/(n+3), so 17/20 > 14/17 > 11/14. Place 0.8: 17/20 = 0.85 and 14/17 ≈ 0.8235 are above it, 11/14 ≈ 0.786 is below. 17/20 > 14/17 > 0.8 > 11/14. The close call (0.8235 vs 0.8) is the whole question — cross-multiply 14 × 5 = 70 > 68.
Ascending order of 0.5, 0.5, 0.54, 0.54. Expand to five places: 0.50000, 0.55555, 0.54545, 0.54444. 0.5 < 0.54 < 0.54 < 0.5.
For a proper fraction x: x2 < x < √x. Squaring a number below 1 shrinks it; rooting it grows it. For x > 1 the order flips. So with x = 4/9: 16/81 < 4/9 < 2/3.
To find n/d between p/q and r/s, rewrite both bounds with denominator d — p/q = (pd/q)/d — and pick the whole numerators strictly between. The count of such fractions is (upper numerator) − (lower numerator) − 1 when both bounds are exact d-ths.
To find N/d between two fractions, rewrite the bounds with numerator N instead — and remember the inequality on d runs the opposite way, because a bigger denominator makes a smaller fraction.
A fraction with denominator 40 between 3/8 and 5/12. 3/8 = 15/40 and 5/12 = 16.67/40; the only whole number strictly between is 16, so 16/40. 15/40 fails — it equals 3/8.
How many fractions with denominator 24 lie strictly between 1/3 and 3/4? 8/24 and 18/24; count = 18 − 8 − 1 = 9. Counting the ends gives the wrong 11.
A fraction with numerator 5 between 3/8 and 2/5. 3/8 = 5/13.33 and 2/5 = 5/12.5, so d = 13: 5/13. Check that 5/12 (0.417) and 5/14 (0.357) both fall outside.
A bound in recurring form is still exact. Between 0.3 and 0.6 with denominator 36 means between 12/36 and 24/36: 11 fractions. And 0.9 as an upper bound means 1 itself — so 12/12 is a boundary, not inside.
A continued fraction is a fraction whose denominator itself contains a fraction, repeated in layers. Start at the innermost layer; at each layer add the whole number to the reciprocal (the flip) of the value below it, and move out.
3 + 1/(2 + 1/(1 + 1/4)): 1 + 1/4 = 5/4 → 2 + 4/5 = 14/5 → 3 + 5/14 = 47/14.
Respect each layer's own sign and numerator: a minus stays a minus, and a layer written 2/(…) is twice the flip. 2 + 1/(3 − 1/(1 + 1/2)) = 2 + 1/(3 − 2/3) = 2 + 3/7 = 17/7.
All ones: 1 + 1/(1 + 1/(1 + …)) gives 3/2, 5/3, 8/5, 13/8, … — each numerator is the sum of the previous two, and each new denominator is the old numerator.
All twos: 2 + 1/(2 + 1/(2 + …)) gives 5/2, 12/5, 29/12, 70/29, … — each numerator is twice the previous plus the one before.
Count the layers and read the term off; no division needed.
1 + 1/(1 + 1/0.3) = ? Replace the decimal by its fraction first: 1/0.3 = 3, then 1 + 3 = 4, flip to 1/4, and 1 + 1/4 = 5/4. Using 0.3 instead of 1/3 gives a mess.
1/(1 + 1/(1 + 1/(1 + 1/3))) as a recurring decimal. 4/3 → 7/4 → 11/7 → flip: 7/11 = 63/99 = 0.63.
A car covers 1/(1 + 1/(1 + 1/3)) of a journey in 20 minutes. Time for the rest? The fraction is 4/7; the rest is 3/7, so 20 × 3/4 = 15 minutes. 35 minutes is the whole journey, a distractor.
The last flip. When the whole thing is 1/(…), the very last step is a flip — 11/7 becomes 7/11. Most wrong answers in this section are the value one flip early.
x + CF = k, x − CF = k, x × CF = k, CF ÷ x = k: evaluate the continued fraction to a number C first, then it is a one-line equation. x − 1/(2 + 1/(3 + 1/2)) = 1 → C = 7/16 → x = 1 + 7/16 = 23/16.
Peel the layers from the outside: flip both sides, subtract the layer's whole number, flip again, and repeat until only x is left.
1/(2 + 1/(1 + 1/x)) = 4/11: flip → 2 + 1/(1 + 1/x) = 11/4; subtract 2 → 1/(1 + 1/x) = 3/4; flip → 1 + 1/x = 4/3; so 1/x = 1/3 and x = 3.
Flip the fraction, split off the whole part, flip the remainder, repeat until the remainder is a unit fraction. The whole parts are the layer numbers.
7/24 → 24/7 = 3 + 3/7 → 7/3 = 2 + 1/3, so 7/24 = 1/(3 + 1/(2 + 1/3)). For an improper fraction take the whole part first: 19/7 = 2 + 5/7 → 7/5 = 1 + 2/5 → 5/2 = 2 + 1/2.
At a layer written 2/(…), the layer's value is 2 ÷ remainder: for 1/(x + 2/(y + 1/z)) = 7/18, 18/7 = 2 + 4/7 gives x = 2 and y + 1/z = 2 ÷ 4/7 = 7/2, so y = 3, z = 2.
1/(a + 1/(b + 1/c)) = 7/16, natural a, b, c. Find a + 2b + c. 16/7 = 2 + 2/7; 7/2 = 3 + 1/2 → a = 2, b = 3, c = 2 → 10.
1/(2 + 1/(1 + 1/x)) = 3/8. Find x2 + 1/x2. Peel: 8/3 − 2 = 2/3 → 3/2 − 1 = 1/2 → x = 2. Then 4 + 1/4 = 17/4.
Algebraic: 1/(1 + 1/(1 + 1/x)) = ? (x + 1)/x → 1 + x/(x + 1) = (2x + 1)/(x + 1) → flip: (x + 1)/(2x + 1). Test with x = 1: both sides give 2/3.
1 + y/(1 − y) = 1/(1 − y), so its reciprocal collapses to 1 − y. Any continued fraction built on y/(1 − y) or a/(1 + a) shrinks to one line by this identity: (1 − 1/(1 + x/(1 − x)))/x = 1.
A mixed number just below a round number is (round number − small fraction): 99 k−1⁄k = 100 − 1/k. So
99 k−1⁄k × m = 100m − m/k, and the same with 1000 and 10000.
99 5/6 × 36 = 3600 − 6 = 3594. 9999 3/4 × 8 = 80000 − 2 = 79998. 999 7/8 × 16 + 2 = 15998 + 2 = 16000 — the added fraction is chosen to restore the round number.
The product is a whole number exactly when k divides m: 99 3/4 × 12 is whole (4 | 12), 99 3/4 × 10 is not.
Add the mixed numbers as (whole parts) + (fractional parts over the LCM). If the total is W + r/L, the least to subtract for a whole number is r/L, and the least to add is 1 − r/L. They sum to 1 — a quick check. If the target is "the next multiple of 5", aim for that multiple, not the next integer.
Least fraction to add to 1 1/4 + 2 2/3 + 3 1/2 for a whole number? Fractions: 3/12 + 8/12 + 6/12 = 17/12 = 1 5/12; total 7 5/12; add 1 − 5/12 = 7/12.
If 99 k−1⁄k × 48 = 4794, find k. 4800 − 48/k = 4794 → 48/k = 6 → k = 8.
Least fraction to add to 0.6 + 0.7 + 0.8? 21/9 = 2 1/3, so add 2/3.
Subtract or add? 5/12 and 7/12 are both offered every time. Subtracting the fractional part goes down to the integer; adding its complement goes up. Read which one the question wants.
Call the fraction n/d, turn each sentence into an equation by cross-multiplying, and solve the pair. "Becomes 1" means n = d after the change. If the parts are in a ratio, write them as 3k and 4k; if the denominator is "1 more than twice the numerator", write d = 2n + 1.
A fraction becomes 2/3 when 1 is added to the numerator and 1/2 when 1 is added to the denominator. 3n + 3 = 2d and 2n = d + 1 → d = 2n − 1 → 3n + 3 = 4n − 2 → n = 5, d = 9: 5/9.
The denominator is 1 more than twice the numerator; add 5 to the numerator and subtract 1 from the denominator and it becomes 3/4. (n + 5)/(2n) = 3/4 → 4n + 20 = 6n → n = 10, d = 21; sum 31.
Parts in ratio 3 : 4; add 4 to each and it becomes 7/9. (3k + 4)/(4k + 4) = 7/9 → 27k + 36 = 28k + 28 → k = 8; the fraction is 24/32 = 3/4.
Sum S and difference D: larger = (S + D)/2, smaller = (S − D)/2.
Sum S and product P: (x − y)2 = S2 − 4P gives the difference without solving a quadratic.
Product and sum of squares: (x + y)2 = (x2 + y2) + 2P.
Ratio p : q and product P: write pk, qk; then pq k2 = P.
Sum and sum of reciprocals: 1/x + 1/y = (x + y)/xy, so product = sum ÷ (sum of reciprocals).
Translate each clause: "c divided by a gives 5/3" → c = 5a/3; "which is 7/6 more than b" → b = 5/3 − 7/6; then use the sum. For a chain of ratios a : b = 2 : 3, b : c = 4 : 5, make the b-parts equal: 8 : 12 : 15, and split the total into 35 parts. Pairwise sums a + b, b + c, a + c: add all three, halve to get the total, subtract each pair.
Divided by f instead of multiplying: error = x(1/f − f). Multiplied by 5/6 instead of dividing: shortfall = x(6/5 − 5/6) = 11x/30. Used the reciprocal q/p instead of p/q: error = x(q2 − p2)/pq. Added f instead of multiplying: error = x(1 − f) + f — this one is not proportional to x, so write the full equation.
Sum 11/12, difference 1/4. Larger fraction? (11/12 + 3/12)/2 = 7/12.
Sum 5/6, product 1/6. Larger minus smaller? (x − y)2 = 25/36 − 24/36 = 1/36 → 1/6.
A student divided a number by 5/6 instead of multiplying and got 11/24 more. The correct answer? 11x/30 = 11/24 → x = 5/4; correct = 5/4 × 5/6 = 25/24.
Sum of three fractions 5/4; largest ÷ smallest = 3/2, which is 13/12 more than the middle. The largest? middle = 5/12; s + l = 5/6 with l = 3s/2 → s = 1/3, l = 1/2.
"2/3 of 3/5 of N is 48" means (2/3)(3/5)N = 48 → N = 120. "3/4 of N exceeds 2/3 of N by 6" means (3/4 − 2/3)N = N/12 = 6 → N = 72. Multiply the fractions first, then invert.
Answer what was asked. After solving for the fraction, the question may want the sum of the parts, their product, the reciprocal, the decimal form, or the period of its decimal. Half the marks lost in this section are lost after the fraction is already correct.
Multiply the whole numbers, then place the point after as many places as the factors have in total: 0.04 × 0.4 × 0.004 → 64 with 2 + 1 + 3 = 6 places → 0.000064. Then drop trailing zeros: 0.25 × 0.5 × 0.16 → 2000 with 5 places → 0.02000 = 0.02, which has 2 decimal places, not 5.
| Decimal | Fraction | Decimal | Fraction |
|---|---|---|---|
| 0.125 | 1/8 | 0.0625 | 1/16 |
| 0.375 | 3/8 | 0.3 | 1/3 |
| 0.625 | 5/8 | 0.6 | 2/3 |
| 0.875 | 7/8 | 0.1 | 1/9 |
| 0.75 | 3/4 | 0.09 | 1/11 |
| 0.2 | 1/5 | 0.142857 | 1/7 |
0.375 ÷ 0.0625 = 3/8 ÷ 1/16 = 6; 0.6 × 0.375 ÷ 0.25 = 2/3 × 3/8 × 4 = 1. The fraction forms make the arithmetic exact and short.
Shifting the decimal point. In a division move both points by the same number of places: 0.375 ÷ 0.0625 = 3750 ÷ 625. Moving them by different amounts changes the value.
Test every claim on the fraction form, never on the truncated decimal. The statements SSC likes:
| Claim | Verdict | Because |
|---|---|---|
| 0.3 × 3 = 1 | true | 1/3 × 3 |
| 0.9 < 1 | false | 0.9 = 9/9 = 1 |
| 0.45 = 5/11 | true | 45/99 |
| 0.5 > 5/9 | false | they are equal |
| 1 ÷ 0.5 = 1.8 | true | 9/5 |
| (0.3)2 = 0.1 | true | 1/9 |
| every non-terminating decimal is irrational | false | 0.3 = 1/3 |
| 1/6 − 1/7 = 1/13 | false | it is 1/42 = 1/(6 × 7) |
Three separate verdicts, in order: is A true, is R true, does R explain A. "7/8 > 6/7" with Reason "(n − 1)/n grows with n" — both true, and R explains A because both fractions really are of that form. Check the form before accepting the explanation.
Fix the pairing you are surest of first — usually the pure recurring decimal, which reads straight off the 9s rule — and throw away every option that disagrees. Leave the mixed recurring ones until the field has narrowed, then confirm by conversion.
The night before the exam, read only this page.
| Decimal | Fraction | Because |
|---|---|---|
| 0.a / 0.ab / 0.abc | a/9 / ab/99 / abc/999 | 10kx − x cancels the tails |
| 0.ab | (ab − a)/90 | 100x − 10x |
| 0.abc | (abc − a)/990 | 1000x − 10x |
| 0.abc | (abc − ab)/900 | 1000x − 100x |
| 0.9 | 1 | 10x − x = 9 |
| p/11 | 0.9p (two digits) | 99 = 9 × 11 |
| p/27, p/37 | 0.37p, 0.27p | 999 = 27 × 37 |
| terminates? | lowest-terms denominator = 2m5n | 10 = 2 × 5 |
| decimal places | max(m, n) | smallest power of 10 it divides |
| Situation | Formula |
|---|---|
| Period of p/q (q co-prime to 10) | least k with q | 10k − 1; 11→2, 27, 37→3, 7, 13→6, 81→9 |
| Period of a product of co-prime q's | LCM of the periods |
| n-th digit (pure / mixed) | block digit (n mod L) / ((n − s) mod L); 0 = last |
| Least multiplier for a whole number | lowest-terms denominator |
| Compare a/b, c/d | ad vs bc |
| Same numerator | bigger denominator = smaller fraction |
| Near 1 | smallest gap (b − a)/b wins |
| n/(n + k), fixed k | increases with n |
| (a + k)/(b + k) | > a/b if proper, < a/b if improper |
| Proper fraction x | x2 < x < √x |
| Count of d-ths strictly between p/d and q/d | q − p − 1 |
| Continued fraction | evaluate inside-out; solve outside-in (flip, subtract, flip) |
| All-ones / all-twos ladder | 3/2, 5/3, 8/5, 13/8 / 5/2, 12/5, 29/12, 70/29 |
| 99 k−1⁄k × m | 100m − m/k; whole when k | m |
| Least to subtract / add | r/L / 1 − r/L |
| Sum S, difference D | (S ± D)/2 |
| Sum S, product P | (x − y)2 = S2 − 4P |
| Product P, squares Q | (x + y)2 = Q + 2P |
| Sum, sum of reciprocals R | product = S/R |
| Divided by f instead of × | error = x(1/f − f) |
| Added f instead of × | error = x(1 − f) + f |
| Decimal places of a product | sum of places, then drop trailing zeros |
Answers: 1) 7/15 2) 4 3) 7 4) 5/9 < 4/7 < 3/5 < 2/3 5) 47/14 6) 3 7) 3594 8) 5/9
For a pure recurring decimal, write the repeating block over as many 9s as it has digits: 0.36 = 36/99 = 4/11. For a mixed recurring decimal, subtract the non-repeating digits from all the digits and write the result over 9s for the repeating digits followed by 0s for the non-repeating digits: 0.46 = (46 − 4)/90 = 7/15.
Reduce it to lowest terms first. It terminates only when the denominator has no prime factor other than 2 and 5, and it then has as many decimal places as the larger of the two exponents. 7/40 terminates in three places; 7/30 recurs because 30 contains a 3.
Yes, exactly. If x = 0.9 then 10x − x = 9, so x = 1. It is the same fact as 0.3 = 1/3 multiplied by 3. Statement questions that claim 0.9 is less than 1 are false.
Cross-multiply for a pair: a/b is greater than c/d exactly when ad is greater than bc. For a list, compare gaps from 1 when the fractions are all near 1, remember that same-numerator fractions shrink as the denominator grows, and that fractions of the form n/(n + k) increase with n.
Work from the innermost layer outwards, adding the whole number to the reciprocal of the value below at each step. If the unknown is inside, peel from the outside in: flip both sides, subtract the layer's whole number, flip again, and repeat until only the unknown is left.
After finishing a section here, attempt the matching Decimal & Fraction practice test on TrickySSC. The tests are graded across two levels, and every question carries a step-by-step solution in English and Hindi.
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