Simplification is the largest self-contained topic in the Punjab Police SI's Quantitative Aptitude section: 85 questions since 2016, and not one of them needs a concept. They need order — knowing which operation to do first — and they need you to spot the two or three places where an ugly-looking expression collapses to almost nothing. Thirty of the 85 are straight BODMAS. Fifteen are fractions and decimals. Six hide a factorisation identity, four are about shifting decimal points, and thirty are indices and surds. You will learn (1) the BODMAS order including the step most candidates skip, (2) the left-to-right rule, (3) the four identities worth memorising, (4) the decimal and fraction habits that turn a two-minute question into twenty seconds, and (5) the seven laws of indices.
BODMAS, properly
Thirty questions rest on this one order.
In an exam expression, “of” binds tighter than ÷ and ×. So 1/4 ÷ 1/4 of 1/4 means 1/4 ÷ (1/16) = 4, while 1/4 ÷ 1/4 × 1/4 means (1/4 ÷ 1/4) × 1/4 = 1/4. The SI builds whole questions out of that single difference, putting the two side by side so you have to notice.
Find the value of 15 − [(7 × 8 ÷ 2) ÷ {44 − (32 − 20 ÷ 10)}] (SI 2 Jul 2023)
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Work from the innermost bracket outwards. 20 ÷ 10 = 2, so 32 − 2 = 30, so 44 − 30 = 14. Next, 7 × 8 ÷ 2 = 56 ÷ 2 = 28. Then 28 ÷ 14 = 2, and finally 15 − 2 = 13. Innermost first, every time — never try to hold two brackets in your head at once.
The value of the expression 75 − 2(24 + 12 ÷ 3 × 6 − 4 × 5) + 9 is : (SI 18 Aug 2021)
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Inside the bracket: 12 ÷ 3 × 6 = 4 × 6 = 24, and 4 × 5 = 20. So the bracket is 24 + 24 − 20 = 28. Then 75 − 2(28) + 9 = 75 − 56 + 9 = 28. Note the 12 ÷ 3 × 6 — left to right again, not 12 ÷ 18.
- “of” comes before ÷. It is the O in BODMAS, not another name for ×.
- ÷ and × are equal. 12 ÷ 3 × 6 is 24, not 12 ÷ 18.
- A minus sign in front of a bracket flips every sign inside it. 75 − 2(24 + 4) is 75 − 48 − 8, not 75 − 48 + 8.
- Mixed numbers must become improper fractions first. 3¼ is 13/4, and you cannot divide “three and a quarter” in parts.
“P means +, Q means −, R means ×, S means ÷.” Rewrite the line with real symbols, then apply BODMAS normally — the letters change nothing about the order. 70 Q 5 R 81 S 9 P 20 becomes 70 − 5 × 81 ÷ 9 + 20 = 70 − 45 + 20 = 45.
When an ugly fraction is really an identity
Six questions, and each is over in ten seconds once you see it.
When you see two awkward decimals appearing over and over in a fraction — 0.943 and 0.057, 0.52 and 0.48, 2.3 and 0.3 — stop calculating. The paper has hidden a factorisation. Name the two numbers a and b, look at the shape of the top and the bottom, and the whole fraction will cancel to something tiny, usually 1 or a + b.
(7.85×7.85 − 2.15×2.15)/(7.85 − 2.15) + ((2.3)³ − 0.027)/((2.3)² + 0.69 + 0.09) is equal to ______. (SI 19 Aug 2021)
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First part: (7.85² − 2.15²)/(7.85 − 2.15) = (a² − b²)/(a − b) = a + b = 10. Second part: 0.027 = 0.3³ and 0.69 = 2.3 × 0.3, 0.09 = 0.3², so the bottom is a² + ab + b² with a = 2.3, b = 0.3 — and the top is a³ − b³. That part is a − b = 2. Total = 10 + 2 = 12.
Decimals and fractions
Fifteen questions, and most of them are about keeping track of the point.
Ramesh spends 3/5 of his monthly salary on food and 1/3 of his salary on clothes and travel, and keeps the rest as savings. If his monthly salary is 45,000, how much will he save in 3 months? (SI 20 Jun 2023)
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Fraction spent = 3/5 + 1/3 = 9/15 + 5/15 = 14/15, so the saving is 1/15 of the salary. Monthly saving = 45,000/15 = ₹3,000. The question asks for three months: 3 × 3,000 = ₹9,000. The distractor ₹3,000 is the one-month figure — the commonest way to lose this mark.
A tailor has 87.5 metres of cloth and has to make 8 pieces out of every one metre of cloth. How many pieces of cloth can be made in all? (SI 14 Jul 2023)
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87.5 metres, 8 pieces from every metre. 87.5 × 8 = 700 pieces. A one-line question, but note that the answer is a count, so it must be a whole number — and 87.5 × 8 obligingly is.
Also asked onceTap to open
Simplify the following. 325 × 325 + 275 × 275 − 12 + (4140 ÷ 345) (SI 20 Jun 2023)
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325² + 275² − 12 + (4140 ÷ 345). 325² = 105,625 and 275² = 75,625, which add to 181,250. 4140 ÷ 345 = 12, so the −12 and +12 cancel exactly. The answer is 181,250.
Find the value of '?' in the equation given below. (1/2) of 2558 + 42% of 1350 = (?)² − 7605 (SI 19 Aug 2021)
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Work the left side: half of 2558 = 1279, and 42% of 1350 = 567, so the left side is 1,846. That makes (?)² = 1846 + 7605 = 9,451 and ? ≈ 97.2 — which is not among the four choices. The printed figures do not close. The official key (the green tick on the response sheet) is 95, so treat this as a paper defect and move on; in the exam, pick the key.
The value of (287.0134 × 0.0491 × 1.902)/(41.004 × 99.89) is closest to ______. (SI 19 Aug 2021)
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Round everything first: (287 × 0.05 × 1.9)/(41 × 100) ≈ 27.3/4100 ≈ 0.007. ‘Closest to’ questions are an invitation to approximate, not to multiply exactly.
If A = (3/4) ÷ (5/6), B = 3 ÷ [(4 ÷ 5) ÷ 6], C = [3 ÷ (4 ÷ 5)] ÷ 6, D = (3 ÷ 4) ÷ (5 ÷ 6), then : (SI 29 Jun 2023)
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A = (3/4) ÷ (5/6) = 9/10. D = (3 ÷ 4) ÷ (5 ÷ 6) = (3/4) ÷ (5/6) = 9/10 — the same thing written differently. B and C come out as 45/2 and 45/8. So A and D are equal. Brackets, not the letters, decide the value.
Find the value of '?' in the equation given below. (0.3³ + 0.33³) ÷ (0.3³ + 0.03³) = ? (SI 24 Aug 2021)
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Top: 0.3³ + 0.33³. Bottom: 0.3³ + 0.03³. Factor 0.03³ out of the bottom and 0.03³ out of the top as well: the ratio reduces to (10³ + 11³)/(10³ + 1³) = 2331/1001 = 2.33. Pulling out the common power of ten is the move.
The rest of the paper, question by question
Thirty-one more, grouped by the shape they take.
Four SI questions share the exact shape [A ÷ {B − C(D − E)}] ÷ F, always with mixed numbers. Work strictly from the inside out: 1. Turn every mixed number into an improper fraction first — 3¼ becomes 13/4. 2. Do the innermost round bracket (D − E). 3. Multiply by C, subtract from B inside the curly bracket. 4. Divide A by that, then divide by F last. If the final answer is not a tidy number, you have almost certainly done step 4 before step 3.
More nested-bracket and BODMAS questionsTap to open
The value of [3 1/4 ÷ {1 1/4 − 1/2(2 1/2 − 1/4 − 1/6)}] ÷ (1/2 of 4 1/3) is : (SI 14 Jul 2023)
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Round bracket: 2½ − ¼ − ⅙ = 5/2 − 1/4 − 1/6 = (30 − 3 − 2)/12 = 25/12. Half of that is 25/24, and 1¼ − 25/24 = 30/24 − 25/24 = 5/24. Then 13/4 ÷ 5/24 = 78/5, and ½ of 4⅓ = 13/6. Finally 78/5 ÷ 13/6 = 36/5 = 7 1/5.
Simplify : [2 3/4 ÷ {1 1/6 − 1/3 (1 1/3 − |1/6 − 1/12|)}] ÷ 1/3 of 5 1/2 (SI 20 Jun 2023)
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Innermost: |1/6 − 1/12| = 1/12. Then 1⅓ − 1/12 = 4/3 − 1/12 = 15/12 = 5/4. A third of that is 5/12, and 1⅙ − 5/12 = 7/6 − 5/12 = 9/12 = 3/4. So 11/4 ÷ 3/4 = 11/3, and ⅓ of 5½ = 11/6. Finally 11/3 ÷ 11/6 = 2.
Let a = [1/2 ÷ 1/3 + 3/4 × 10/6] × (2/3 ÷ 2/9 + 1/2) and b = [3(1/6 − 5/12 × 6/5) − (−3)] × 2 ÷ (5/2). The value of a/b is : (SI 9 Jul 2023)
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a = [½ ÷ ⅓ + ¾ × 10/6] × (⅔ ÷ 2/9 + ½) = [3/2 + 5/4] × [3 + 1/2] = (11/4)(7/2) = 77/8. b = [3(⅙ − 5/12 × 6/5) − (−3)] × 2 ÷ 5/2 = [3(1/6 − 1/2) + 3] × 4/5 = [−1 + 3] × 4/5 = 8/5. So a/b = (77/8) ÷ (8/5) = 385/64.
Find the value of the following expression. 316 − 3(20 + 16 ÷ 4 × 3 − 2 × 2) + 101 (SI 9 Jul 2023)
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Inside the bracket, division and multiplication come before the additions: 16 ÷ 4 × 3 = 12 and 2 × 2 = 4, so the bracket is 20 + 12 − 4 = 28. Then 316 − 3(28) + 101 = 316 − 84 + 101 = 333.
Simplify the following. 12.0737 − [5 + 0.05 of (4.1 − 1.3 × 2.02)] (SI 15 Jul 2023)
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Inner bracket: 4.1 − 1.3 × 2.02 = 4.1 − 2.626 = 1.474. “0.05 of 1.474” = 0.0737. So the square bracket is 5 + 0.0737 = 5.0737, and 12.0737 − 5.0737 = 7.0.
Find the value of 1 ÷ 3/7 of (6 + 8 × 1) + [1/5 ÷ 7/25 − 1]. (SI 15 Jul 2023)
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“of” first: 3/7 of (6 + 8) = 3/7 × 14 = 6, so 1 ÷ 6 = 1/6. The square bracket: 1/5 ÷ 7/25 − 1 = 5/7 − 1 = −2/7. Total = 1/6 − 2/7 = (7 − 12)/42 = −5/42.
Simplify the following using BODMAS rule. 26 + 11 − 5 × 1/2 of 12 − {36 ÷ (14 + 4)} (SI 16 Oct 2022)
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“½ of 12” = 6, so the middle term is 5 × 6 = 30. The curly bracket is 36 ÷ 18 = 2. Then 26 + 11 − 30 − 2 = 5.
What is the value of the following expression? (4/3 + 5/9) of 4/17 ÷ 2 1/3 × 1 3/4 − 1/4 (SI 16 Oct 2022)
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4/3 + 5/9 = 17/9, and “of 4/17” gives 17/9 × 4/17 = 4/9. Then 4/9 ÷ 7/3 × 7/4 = 4/9 × 3/7 × 7/4 = 1/3, and 1/3 − 1/4 = 1/12.
What do we get on simplifying (9/−8) × ((7/3) − 2 + (5/6))? (SI 19 Aug 2021)
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Bracket first: 7/3 − 2 + 5/6 = (14 − 12 + 5)/6 = 7/6. Then (−9/8)(7/6) = −63/48 = −21/16.
Solve the equation given below. [10 + 8 − 2 × (2 + 78) − 45÷9 + (36÷9 + 5×3 − 20)] × (−3/2) (SI 19 Aug 2021)
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Work through the square bracket: 10 + 8 = 18; 2 × 80 = 160; 45 ÷ 9 = 5; and the inner bracket 4 + 15 − 20 = −1. So the square bracket is 18 − 160 − 5 − 1 = −148, and −148 × (−3/2) = 222.
Simplify the given expression. ((−4 − 6)⁴ − 6³ + 10/2)/(18 − 7 ÷ 7 − 4) (SI 20 Jun 2023)
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Top: (−10)⁴ = 10,000; minus 6³ = 216; plus 10/2 = 5. That gives 9,789. Bottom: 18 − 1 − 4 = 13. And 9,789 ÷ 13 = 753. An even power of a negative number is positive — that is the only trap here.
Find the value of 96 ÷ 48 − {(4² − 6 × 2)² ÷ (4 ÷ 2) + 7} + (22/9 ÷ 11/8 × 2 1/4). (SI 20 Jun 2023)
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96 ÷ 48 = 2. Inside the curly bracket: 4² − 6 × 2 = 16 − 12 = 4, so 4² = 16, then 16 ÷ 2 = 8 and 8 + 7 = 15. The last bracket: 22/9 ÷ 11/8 × 9/4 = (22/9)(8/11)(9/4) = 4. Total = 2 − 15 + 4 = −9.
Solve : [75 − 12 ÷ 3 of 2] + (18 − 6) ÷ 4 (SI 24 Aug 2021)
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“3 of 2” = 6, so 12 ÷ 6 = 2 and the square bracket is 75 − 2 = 73. Then (18 − 6) ÷ 4 = 3, and 73 + 3 = 76.
8 − 1 1/6 of 1 5/7 ÷ 2 2/5 of 5/6 is equal to : (SI 29 Jun 2023)
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Both “of”s bind first: 1⅙ of 1 5/7 = (7/6)(12/7) = 2, and 2⅖ of ⅚ = (12/5)(5/6) = 2. So the expression is 8 − 2 ÷ 2 = 8 − 1 = 7.
(1/3 ÷ 6 + 12)/(1/3 × 6 + 12) = ? (SI 29 Jun 2023)
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Top: 1/3 ÷ 6 = 1/18, so 1/18 + 12 = 217/18. Bottom: 1/3 × 6 = 2, so 2 + 12 = 14. Then (217/18) ÷ 14 = 217/252 = 31/36.
What is the value of 5 1/2 ÷ (3 1/2 + 7 1/2)? (SI 29 Jun 2023)
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3½ + 7½ = 11, and 5½ ÷ 11 = (11/2) ÷ 11 = 1/2.
Fractions and decimalsTap to open
What is the value of 5/4 − 0.45 + 3.75 − 1/2 − 0.05 ? (SI 16 Oct 2022)
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Convert the fractions: 5/4 = 1.25 and 1/2 = 0.5. Then 1.25 − 0.45 + 3.75 − 0.5 − 0.05 = 4. Group the numbers that make whole values: (1.25 + 3.75) = 5 and (0.45 + 0.5 + 0.05) = 1.
Find the value of 'y' in the equation given below. −774.67 + 1742.52 − 873.43 − y = − 7472.78 (SI 19 Aug 2021)
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Collect the known terms first: −774.67 + 1742.52 − 873.43 = 94.42. So 94.42 − y = −7472.78, giving y = 94.42 + 7472.78 = 7567.20.
The length of a field is divided into three parts. The first part is 1/3 of the whole length, the second part is 3/5 of the first part, and the third part is 3.50 metres. Find the length of the field. (SI 20 Jun 2023)
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Let the length be L. The first part is L/3 and the second is 3/5 of that, which is L/5. Together the first two are L/3 + L/5 = 8L/15. The third part is the rest: L − 8L/15 = 7L/15 = 3.50, so L = 3.50 × 15/7 = 7.50 metres.
If (2 + 1/(1 + 4/5)) / (2 + 1/(1 + 1/(1 + 1/4))) = x, then what is the value of x? (SI 24 Aug 2021)
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Top: 1 + 4/5 = 9/5, so 1 ÷ (9/5) = 5/9 and the numerator is 2 + 5/9 = 23/9. Bottom, from the inside: 1 + ¼ = 5/4; 1 ÷ (5/4) = 4/5; 1 + 4/5 = 9/5; 1 ÷ (9/5) = 5/9; so the denominator is also 2 + 5/9 = 23/9. The two are equal, so x = 1. Continued fractions are always built from the bottom up.
Raghu consumes 0.75 litre of milk every day, and Rani consumes 1/4 litre of milk every day. If the price of milk is ₹42.25 per litre, then find the total amount (in ₹) spent by the two of them on milk in 15 weeks. (SI 24 Aug 2021)
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Together they drink 0.75 + 0.25 = 1 litre a day — the fractions were chosen to add to one. 15 weeks is 105 days, so 105 litres at ₹42.25 costs 105 × 42.25 = ₹4,436.25.
Simplify the following. 5 − [3/4 + {3½ − (0.7 + (1/5 − 1/3))}] = ? (SI 29 Jul 2023)
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Innermost: 1/5 − 1/3 = −2/15. Then 0.7 + (−2/15) = 7/10 − 2/15 = (21 − 4)/30 = 17/30. Next 3½ − 17/30 = 105/30 − 17/30 = 88/30 = 44/15, and 3/4 + 44/15 = (45 + 176)/60 = 221/60. Finally 5 − 221/60 = (300 − 221)/60 = 79/60.
If x = 0.3/10, then the value of (5x + 25x² + 125x³)/0.1³ is ________. (SI 29 Jul 2023)
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x = 0.3/10 = 0.03. Then 5x = 0.15, 25x² = 25(0.0009) = 0.0225 and 125x³ = 125(0.000027) = 0.003375. Their sum is 0.175875, and dividing by 0.1³ = 0.001 shifts the point three places: 175.875.
Simplify the following. (5 × 0.5 × 0.05 × 0.005 × 50)/(6 × 0.6 × 0.06 × 0.006 × 60) (SI 31 Jul 2023)
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Every factor on the bottom is 6/5 of the one above it, and there are five of them. So the whole fraction is (5/6)⁵ = 3125/7776 = 3125/7776. Spotting the matching pattern saves multiplying ten decimals.
3/4 of the difference between 2 1/6 and 1 1/9 is subtracted from 3/4 of 4 1/3. The resultant value is : (SI 31 Jul 2023)
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The difference: 2⅙ − 1 1/9 = 13/6 − 10/9 = (39 − 20)/18 = 19/18, and ¾ of that is 19/24. Three-quarters of 4⅓ = ¾ × 13/3 = 13/4. Subtracting, 13/4 − 19/24 = (78 − 19)/24 = 59/24.
Roots and powersTap to open
Simplify the given expression. (3² − 3) + (2² − 7) + (7² − 5²) + (9² − 1) (SI 14 Jul 2023)
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Four small brackets: (9 − 3) = 6, (4 − 7) = −3, (49 − 25) = 24 and (81 − 1) = 80. Adding, 6 − 3 + 24 + 80 = 107. The second bracket is negative, which is where the 131 option comes from.
What is the approximate value that will come in place of (?) in the equation given below? 80% of (280 + √?) = 240 (SI 19 Aug 2021)
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Undo the percentage first: if 80% of something is 240, that something is 240 ÷ 0.8 = 300. So 280 + √? = 300, giving √? = 20 and ? = 400.
Indices and surds
Thirty more questions, and only seven laws behind them.
10^0 = ? (SI 2016)
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Any non-zero number raised to the power zero is 1, so 10⁰ = 1. It follows from the division rule: aⁿ ÷ aⁿ = a⁰, and anything divided by itself is 1.
If 2^a = 3^b = 6^c, take logarithms of the common value k. Then 1/a = log2 / log k, 1/b = log3 / log k and 1/c = log6 / log k. Because log 2 + log 3 = log 6, the combination 1/a + 1/b − 1/c is always 0. The SI paper has asked this twice, once directly and once dressed up as (xy + yz + zx)/xyz — which is just 1/x + 1/y + 1/z.
If ((4^8)²)/(64(7)^8) × (36(6^9)^{1/4})/((4)^{15}) × 49 × (7^6)/(6^{1/4}) = (6/x)⁴ × 7^y, then find x and y. (SI 29 Jun 2023)
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Collect each base separately. Powers of 4: (4⁸)² = 4¹⁶, and 64 = 4³, so 4¹⁶ ÷ (4³ × 4¹⁵) = 4⁻² = 1/16. Powers of 6: 36 = 6² and (6⁹)^(1/4) = 6^(9/4), divided by 6^(1/4) gives 6^(2 + 9/4 − 1/4) = 6⁴. Powers of 7: (1/7⁸) × 49 × 7⁶ = 7^(−8 + 2 + 6) = 7⁰ = 1. So the left side is 6⁴/16. The right side is (6/x)⁴ × 7^y, so x⁴ = 16 giving x = 2, and 7^y = 1 giving y = 0. The answer is x = 2 and y = 0.
If [(√(1455) − √(1239))^{2/3} × (√(1239) + √(1455))^{2/3}] / [(8^{1/6} + 1/8^{1/6})(8^{1/3} − 1 + 1/8^{1/3})(8^{1/6} − 1/8^{1/6})(8^{1/3} + 1 + 1/8^{1/3})] = m m/n (a mixed number), where m and n are natural numbers, then find the value of 2m(n + 1) − n²? (SI 15 Jul 2023)
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Take the two halves separately. Top: (√1455 − √1239)^(2/3) × (√1455 + √1239)^(2/3) = [(√1455 − √1239)(√1455 + √1239)]^(2/3) = (1455 − 1239)^(2/3) = 216^(2/3) = 36. Bottom: let t = 8^(1/6), so 8^(1/3) = t² and t × (1/t) = 1. Then (t + 1/t)(t² − 1 + 1/t²) = t³ + 1/t³ and (t − 1/t)(t² + 1 + 1/t²) = t³ − 1/t³ — both are the cube identities. Their product is t⁶ − 1/t⁶ = 8 − 1/8 = 63/8. So the fraction is 36 ÷ (63/8) = 288/63 = 32/7 = 4 4/7, which matches the stated form m m/n with m = 4 and n = 7. Finally 2m(n + 1) − n² = 2(4)(8) − 49 = 64 − 49 = 15.
- Reading a⁶ ÷ a⁻⁵ as a¹. Subtracting a negative index adds, so the answer is a¹¹.
- Expanding 3^(x−2) + 3^x instead of factoring out the smaller power. The bracket is always a tidy fraction; the expansion never is.
- Leaving a surd under the root. √252 is 6√7, and once every term is a multiple of √7 the root cancels out of the fraction entirely.
- Forgetting that a negative index flips a fraction rather than making it negative. (1/3)⁻³ is 27, not −27.
One-page revision
Read only this the night before.
- B → O → D → M → A → S, and O includes the word “of”.
- ÷ and × left to right. + and − left to right.
- Brackets from the inside out: ( ) then { } then [ ].
- Two repeated decimals in a fraction → look for a³ ± b³ before you calculate.
- (a − b)² + (a + b)² = 2(a² + b²) — appears every other year.
- Every number on the bottom a tenth of the top → the squared fraction is 100.
- Letter codes (P, Q, R, S) change the symbols, never the order.
- aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1.
- Two powers of one base added → factor out the smaller power.
- Learn 3⁵ = 243, 3⁸ = 6561, 2¹⁵ = 32768, 7⁴ = 2401, 4⁴ = 256.
- Pull the perfect square out of every surd before you divide.
- 2^a = 3^b = 6^c → 1/a + 1/b − 1/c = 0.
Brackets, of, divide, multiply, add, subtract — and when two share a rank, read left to right. That one sentence answers thirty of the eighty-five, and the seven index laws answer thirty more.