Mathematics — Chapter 1

Numbers & Simplification — BODMAS, Fractions & Decimals

Why this chapter matters: Simplification is the gate skill of the whole mathematics paper — every later topic (trains, ages, partnership, percentages) is just BODMAS, fractions and decimals dressed in a story. Our verified banks hold 265 topic-relevant items for this chapter: set 133’s 29 simplification questions (book pp. 740–742), the decimal-fractions set’s 25 questions, plus 167 fraction items, 23 BODMAS/simplification items and 21 decimal items scattered through the PPSC and FPSC papers. The taxonomy’s three sub-topics are Section 1 (BODMAS), Sections 2–3 (fractions) and Section 4 (decimals); Section 5 turns set 133’s word problems into reusable methods, because in the papers simplification rarely arrives naked — it arrives as a story about a pencil, a crate of mangoes, or a salary bill.

1. BODMAS — the order that decides everything

When an expression mixes operations, the order is fixed. BODMAS is the ladder:

Division and multiplication are equals — do them left to right, in the order they appear. Addition and subtraction are equals — likewise left to right. This left-to-right rule is where most marks are lost (Section 8).

Worked example — paper question: Solve: 16 × 5 − 2 + (84 − 3) ÷ 3² = ? Brackets first: (84 − 3) = 81. Orders: 3² = 9. Then multiplication and division left to right: 16 × 5 = 80, 81 ÷ 9 = 9. Then addition/subtraction left to right: 80 − 2 + 9 = 87.

Worked example — paper question: 5 − 5 ÷ 5 × 5 + 5 = ? Division and multiplication before addition and subtraction: 5 ÷ 5 = 1, 1 × 5 = 5. Then 5 − 5 + 5 = 5. (Working left to right as written — 5 − 5 = 0, then 0 × 5, then + 5 — is the classic trap; it gives a wrong answer.)

Worked example — paper question: Solve: 100 ÷ 50 × 2 = ? Division and multiplication are equals, so go left to right: 100 ÷ 50 = 2, then 2 × 2 = 4. (Dividing by 50 × 2 = 100 first gives 1 — wrong, because the × 2 is not inside a bracket.)

Worked example — paper question: Solve: 6 − 4 + 4 × 8 ÷ 4 = ? Multiplication/division first: 4 × 8 = 32, 32 ÷ 4 = 8. Then left to right: 6 − 4 + 8 = 10.

Worked example — paper question: Solve: 840 ÷ 14 + 40 = ? 840 ÷ 14 = 60, then 60 + 40 = 100.

Worked example — paper question: Solve: 5 + 6 × 3 − 2 + 2 = ? 6 × 3 = 18, then 5 + 18 − 2 + 2 = 23.

The “of” step: in 4/15 of 5/7 of a number, every “of” is a multiplication — (4/15) × (5/7) × number. Set 133, Q1 uses exactly this: (4/15 × 5/7 − 4/9 × 2/5) of the number = 8. The bracket is (4/21 − 8/45) = 4/315, so the number is 8 ÷ (4/315) = 630, and half of it is 315.

2. Fractions — the four operations and the “of a number” pattern

2.1 What a fraction is

A fraction a/b means a parts out of b equal parts. a is the numerator, b the denominator. A fraction with numerator smaller than denominator is proper (3/4); with numerator larger, improper (7/4 = 1¾, a mixed number). Two fractions are equivalent when cross-multiplication agrees: 5/8 = 25/40 because 5 × 40 = 8 × 25 = 200. Paper question: The fraction 5/8 with a denominator 40 is 25/40.

2.2 Adding and subtracting — common denominator first

Paper question: Solve: 2/3 + 3/4 = ? Common denominator 12: 8/12 + 9/12 = 17/12 (= 1 5/12). Paper question: Solve: 1/4 − 1/4 = ? — 0. Paper question: Solution of 16/4 + 9 − 6 = ? 16/4 = 4, then 4 + 9 − 6 = 7.

2.3 Multiplying and dividing — and the “invert and multiply” rule

Multiply straight across: 1/3 × 3/2 = 3/6 = 1/2. Paper question: 1/3 × 3/2 ÷ 1/2 = ? First 1/3 × 3/2 = 1/2, then 1/2 ÷ 1/2 = 1 (dividing by a fraction means multiplying by its reciprocal).

2.4 Comparing fractions — three ways, fastest first

The decimal-fractions set tests comparison four times, so learn all three methods:

1. Convert to decimals (most reliable): 2/3 ≈ 0.667, 3/4 = 0.75, 4/5 = 0.8, 5/6 ≈ 0.833. So the biggest of the four is 5/6, the smallest 2/3, and their difference is 5/6 − 2/3 = 1/6 (set question Q4).
2. Cross-multiplication for two fractions: compare 13/16 with the other options by 13 × (other denominator) vs (other numerator) × 16 — the set’s Q5 answer is 13/16 = 0.8125, the smallest of its group.
3. The unit-difference shortcut: when the numerator is exactly one less than the denominator (2/3, 3/4, 4/5, 5/6), the larger the numbers, the larger the fraction — 5/6 > 4/5 > 3/4 > 2/3. It works because each fraction is 1 minus a smaller and smaller piece (1/6 < 1/5 < 1/4 < 1/3).

Between two fractions: convert both bounds to decimals and test. Set Q6: between 3/4 = 0.75 and 5/6 ≈ 0.833 lies 4/5 = 0.8. Set Q7: between 1/3 ≈ 0.333 and 7/8 = 0.875 lies 17/24 ≈ 0.708. Set Q8: between 7/13 ≈ 0.538 and 4/5 = 0.8, 1/2 = 0.5 does not lie inside — the interval’s lower bound is 7/13, not 4/5, so read both ends before testing.

2.5 “Of a number” — the set-133 workhorse

When a question says “3/14 of a certain number”, write it as (3/14) × N and let the equation do the work:

• Difference form (Q26): a student found 3/4 of a number instead of 3/14, and his answer was 150 too high. (3/4 − 3/14) × N = 150. 3/4 − 3/14 = 21/28 − 6/28 = 15/28. So N = 150 × 28/15 = 280.
• Wrong-multiplier form (Q12): a boy multiplied by 52 instead of 25 and got 324 too much. (52 − 25) × N = 324, so N = 324 ÷ 27 = 12. The method is identical — wrong value minus right value, times the number, equals the stated error.
• Remainder form (Q3): 1/8 of a pencil is black, 1/2 of the remaining is white, and 3½ cm is blue. After black, 7/8 remains; half of that (7/16) is white, leaving 7/16 = 3.5 cm, so the whole pencil is 3.5 × 16/7 = 8 cm. The trap is taking 1/2 of the whole instead of 1/2 of the remainder.
• Share-of-the-whole form (Q25): A gets 3/16, B gets 1/4, and C gets Rs. 81. C’s share is 1 − 3/16 − 1/4 = 9/16 of the total, so the total is 81 × 16/9 = Rs. 144, and B gets 1/4 × 144 = Rs. 36.
• Repeated-fraction form (Q20): a ball rises to 3/4 of the height it fell from. Dropped from 32 m, after three bounces it rises 32 × (3/4)³ = 32 × 27/64 = 13.5 m. Each bounce multiplies by the same fraction — powers, not repeated subtraction.
• Unit form (Q19): one second is 1/3600 of an hour (60 × 60 seconds). (Q22: one-third of a tank holds 80 litres, so the full tank holds 240 litres and half holds 120 litres. Q21: March earnings double the other months, so the year is 13 equal parts and March is 2/13 of the year’s earnings.)

3. Ratio and proportion — the distribution engine

A ratio a : b compares two quantities; a proportion says two ratios are equal. Set 133’s distribution questions all reduce to one unknown share:

• Q2: a man divides Rs. 8,600 among 5 sons, 4 daughters and 2 nephews; each daughter gets 4× a nephew’s share, each son 5×. Let a nephew’s share be x: 5(5x) + 4(4x) + 2x = 43x = 8,600, so x = 200 and a daughter gets 4x = Rs. 800.
• Q17: a Rs. 1,000 bonus; Rashid gets twice Sohail, Sohail gets one-fifth of Ghani. Ghani + (1/5)Ghani + (2/5)Ghani = 1.6 Ghani = 1,000, so Ghani gets Rs. 625.
• Q14: English marks are 2½ times History marks, total 140. History = 140 ÷ 3.5 = 40, English = 100.
• Q24: a 70 cm wire cut so one piece is 2/5 as long as the other. Longer piece x: x + (2/5)x = 70, x = 50, shorter piece = 20 cm.
• Q4 (ratio chain): one bruised mango per 30 mangoes; 3 of every 4 bruised are unsalable; 12 unsalable. Bruised = 12 × 4/3 = 16, crate = 16 × 30 = 480 mangoes. Work the chain backwards from the known number.
• Q9 (inverse proportion): 30 children per column × 16 columns = 480 children total; at 24 per column, 480 ÷ 24 = 20 columns. When one quantity rises, the other falls — total stays fixed.
• Q8 (doubling): income doubles yearly from Rs. 4 lakh. After 5 years: 4 lakh × 2⁵ = 128 lakh = Rs. 1.28 crore.

4. Decimals — conversion, recurring decimals, and decimal arithmetic

4.1 Fraction ↔ decimal

Divide numerator by denominator. 27/100000 = 0.00027, so 101 27/100000 = 101.00027 (set Q1). 3/1000 = 0.003. 25% = 0.25. 1/0.04 = 100/4 = 25 — clearing the decimal from the divisor by multiplying top and bottom by 100 is the standard move (set Q19; likewise 4.036 ÷ 0.04 = 403.6 ÷ 4 = 100.9, Q18).

4.2 Recurring decimals

A repeating block of n digits goes over n nines: 0.3636… = 36/99 = 4/11 (numerator + denominator = 4 + 11 = 15, set Q2). 0.4747… = 47/99. 0.125125… = 125/999. One second as a decimal of an hour is 1/3600 = 0.000277…, given in the set as 0.00027.

4.3 Decimal arithmetic — line the points up, count the places

• Addition/subtraction: align decimal points. 337.62 + 8.591 + 34.4 = 380.611; 48.95 − 32.006 = 16.944; 892.7 − 573.07 − 95.007 = 224.623. Chained subtraction keeps its sign: 832.58 − 242.31 − 779.84 = −189.57 (the answer is negative — the subtractions exceed the starting number).
• Multiplication: multiply as whole numbers, then place the point so the total decimal places equal the sum of the factors’ places. 0.014 × 0.014: 14 × 14 = 196, with 3 + 3 = 6 places → 0.000196. 0.002 × 0.5 = 0.001. For the product of 95.75 (2 places) and 0.02554 (5 places): 7 places nominally, but the product ends in 0, so the last digit drops and 6 places remain — trailing zeros do not count.
• Division: 0.213 ÷ 0.00213 = 100 (the divisor goes into the dividend exactly 100 times — shift both points right by the divisor’s places: 213 ÷ 2.13 = 21300 ÷ 213 = 100).
• Powers of ten: 3.14 × 10⁶ = 3,140,000 — move the point six places right.
• Approximation: 0.333 × 0.25 × 0.499 × 0.125 × 24 ≈ (1/3) × (1/4) × (1/2) × (1/8) × 24 = 1/8. When a question asks for the closest approximation, round each factor to a friendly fraction first.
• Identity shortcuts: (4.2² − 1.9²) ÷ (2.3 × 6.1) = 1, because 4.2² − 1.9² = (4.2 − 1.9)(4.2 + 1.9) = 2.3 × 6.1 — the numerator and denominator are the same number. And (5.32 × 56 + 5.32 × 44) ÷ (7.66² − 2.34²) = 5.32 × 100 ÷ (5.32 × 10) = 10: factor the common 5.32 out of the numerator, use a² − b² = (a − b)(a + b) on the denominator, and cancel.

5. Simultaneous equations, counting, and spacing — set 133’s remaining methods

5.1 Two equations, two unknowns — add or subtract to kill one variable

• Q5: 2 pens + 3 pencils = Rs. 86; 4 pens + 1 pencil = Rs. 112. Doubling the first (4 pens + 6 pencils = 172) and subtracting the second gives 5 pencils = 60, pencil = Rs. 12, pen = Rs. 25.
• Q28: 2 tables + 3 chairs = 3,500; 3 tables + 2 chairs = 4,000. Adding: 5 tables + 5 chairs = 7,500 (one table + one chair = 1,500). Subtracting: 1 table − 1 chair = 500. So a table costs (1,500 + 500) ÷ 2 = Rs. 1,000.
• Q27: 10 chairs = 4 tables, so a table = 2.5 chairs. 15 chairs + 2 tables = 20 chairs = Rs. 4,000, so a chair is Rs. 200 and a table Rs. 500; 12 chairs + 3 tables = 2,400 + 1,500 = Rs. 3,900.
• Q13: 4 men + 2 women = Rs. 46,000, a woman earns Rs. 500 more than a man. With a man’s salary m: 4m + 2(m + 500) = 46,000, m = 7,500, a woman earns Rs. 8,000.
• Q6 (distribution by difference): the same amount split among 14 boys gives each Rs. 80 more than among 18 boys. T/14 − T/18 = 80. 1/14 − 1/18 = 1/63, so T = 80 × 63 = Rs. 5,040.
• Q16 (chain of shares): A gets as much as B + C together, B gets Rs. 125 more than C, D gets as much as C. With C = x: (2x + 125) + (x + 125) + x + x = 5x + 250 = 750, x = 100, A’s share = 2x + 125 = Rs. 325.
• Q15 (trial by options): 7x + 5y = 38. Try x = 4: 28 + 10 = 38, so 4 pineapples and 2 watermelons. When the equation has two unknowns and one equation, test the small whole numbers.
• Q29 (heads and legs): buffaloes (4 legs) and ducks (2 legs); legs are 24 more than twice the heads. 4B + 2D = 2(B + D) + 24, so 2B = 24 and there are 12 buffaloes. The ducks cancel out entirely — that is the point of the question.

5.2 Counting digits, equal spacing, time intervals, unit conversion

• Counting digits (Q18): a 366-page book. Pages 1–9 use 9 digits, 10–99 use 90 × 2 = 180, 100–366 use 267 × 3 = 801. Total = 990 digits. Count by digit-length blocks, never page by page.
• Equal spacing (Q11): 26 trees along 225 m, one at each end. Trees at both ends mean 25 gaps, not 26: 225 ÷ 25 = 9 m between consecutive trees. The off-by-one is the whole question.
• Time intervals (Q10): a light seen every 13 seconds, first at 1:54:50 a.m., last at 3:17:49 a.m. Elapsed time = 4,979 seconds = 383 intervals of 13 seconds; with the first sighting counted, the light was seen 384 times. Sightings = intervals + 1 — the same off-by-one as the trees.
• Unit conversion (Q7): 42.5 m = 4,250 cm; 4,250 ÷ 85 = 50 pieces of 85 cm. Convert first, divide second — never divide mixed units.

6. Master table — fraction ↔ decimal ↔ percent equivalents

The conversions papers reuse most. Learn this table cold and half of Section 4 becomes instant:

BODMAS ladder (one line): Brackets (innermost first) → Of/powers → Division and multiplication left to right → Addition and subtraction left to right.

7. Rapid-fire one-liners

• BODMAS: Brackets, Of, Division, Multiplication, Addition, Subtraction — D/M and A/S are equals, worked left to right.
• “Of” always means multiply: 4/15 of 5/7 of a number = (4/15) × (5/7) × number.
• 16 × 5 − 2 + (84 − 3) ÷ 3² = 87 — brackets, then powers, then ×/÷, then +/−.
• 5 − 5 ÷ 5 × 5 + 5 = 5, not 25 — division and multiplication come before the subtraction.
• 100 ÷ 50 × 2 = 4 — left to right, never divide by the product that follows.
• 2/3 + 3/4 = 17/12; 1/3 × 3/2 ÷ 1/2 = 1.
• For fractions one less than their denominator (2/3, 3/4, 4/5, 5/6), the bigger the numbers, the bigger the fraction: 5/6 is biggest, 2/3 smallest, difference 1/6.
• 13/16 = 0.8125; 17/24 ≈ 0.708; 4/5 = 0.8 lies between 3/4 and 5/6.
• Recurring block over nines: 0.3636… = 4/11; 0.4747… = 47/99; 0.125125… = 125/999.
• One second = 1/3600 of an hour ≈ 0.00027 hour.
• 0.014 × 0.014 = 0.000196 (3 + 3 = 6 decimal places); 0.213 ÷ 0.00213 = 100; 1 ÷ 0.04 = 25; 3.14 × 10⁶ = 3,140,000.
• Wrong-minus-right times the number equals the error: (52 − 25) × N = 324 → N = 12; (3/4 − 3/14) × N = 150 → N = 280.
• Remainder trap: 1/2 of the remaining is not 1/2 of the whole — the pencil is 8 cm.
• Distribution: set the smallest share to x and express every other share in x — Rs. 8,600 ÷ 43 parts gives a daughter Rs. 800.
• Simultaneous equations: add to get the sum, subtract to get the difference — tables Rs. 1,000, chairs Rs. 500.
• Heads and legs: legs − 2 × heads, halved, is the four-legged count — 12 buffaloes.
• 366 pages need 990 digits (9 + 180 + 801).
• Trees and sightings: gaps = posts − 1 (26 trees → 25 gaps → 9 m); sightings = intervals + 1 (383 intervals → 384 sightings).
• Doubling five times multiplies by 32: Rs. 4 lakh → Rs. 1.28 crore.
• a² − b² = (a − b)(a + b) turns (4.2² − 1.9²) ÷ (2.3 × 6.1) into 1 and the 5.32 question into 10.

8. Most-asked facts (from our verified past-paper banks)

• Set 133 is the topic’s anchor: 29 questions (book pp. 740–742) — fractions 9, ratio and proportion 5, simultaneous equations 4, distribution 2, plus one each of unit conversion, doubling, inverse proportion, time intervals, equal spacing, wrong multiplication, counting digits, cost equations and heads-and-legs. Learn its methods, not its answers: the same ten methods recur across the papers.
• 167 fraction items in the papers — the largest block — are mostly short computations (2/3 + 3/4, 1/3 × 3/2 ÷ 1/2, 16/4 + 9 − 6) plus “of a number” and rebound word problems. Speed on the four operations is the highest-yield skill in this chapter.
• 23 BODMAS/simplification items in the papers, including the two most-set traps: 5 − 5 ÷ 5 × 5 + 5 and 100 ÷ 50 × 2 — both test the left-to-right rule for equal operations, and both are answered wrong by working left to right as written.
• The decimal-fractions set’s 25 questions split into conversion/recurring (Q1–Q8), decimal arithmetic (Q9–Q13, Q17–Q19, Q24–Q25), powers of ten and approximation (Q14–Q16) and identity shortcuts (Q22–Q23). The approximation question (Q16) is the only one that rewards rounding before calculating.
• 21 decimal items in the papers, led by “3/1000 written in decimal form” (asked twice, two different FPSC papers) and “evaluate” decimal sums — the papers ask decimals as conversion or straight computation, rarely as stories.
• The set’s own shape: set 133’s questions are almost all word problems, while the papers’ fraction items are almost all bare computations. Practise both forms — the method is the same, but the translation step (words → equation) is what set 133 trains.

9. Traps & confusions

• Working strictly left to right. BODMAS is not “do it in the order written” — 5 − 5 ÷ 5 × 5 + 5 = 5, and 100 ÷ 50 × 2 = 4. Division/multiplication first, then addition/subtraction, each group left to right.
• Multiplying where you should divide, and vice versa, with “of”. “4/15 of 5/7 of a number” is multiplication all the way down; “what fraction of an hour is a second” is division (1 ÷ 3600). Read whether the question asks for a part of something (multiply) or a part as compared to something (divide).
• Half of the remainder vs half of the whole. In the pencil question, the white part is 1/2 of the 7/8 that remains (7/16), not 1/2 of the pencil. “Of the remaining” changes the base — always re-base after each step.
• Trees, gaps, sightings, intervals. 26 trees give 25 gaps; 383 intervals give 384 sightings. Whenever a question counts posts, trees, poles or sightings, ask whether the count starts at zero or at one before dividing.
• Chained subtraction can go negative. 832.58 − 242.31 − 779.84 = −189.57. A negative result is a valid answer, not a signal to drop the sign.
• “Between” questions: check both ends. In the “does not lie between 4/5 and 7/13” item, the interval runs 0.538 to 0.8 — 1/2 = 0.5 fails because it is below the lower bound, which is 7/13, not 4/5. Order the bounds first.
• Unit-difference fractions reverse intuition. 5/6 looks “more reduced” than 2/3 but is the larger fraction — each is one piece short of a whole, and the piece it lacks (1/6) is the smallest.
• Trailing zeros in decimal multiplication. 95.75 × 0.02554 nominally needs 7 decimal places, but the product ends in 0, so 6 places remain. Count places after multiplying, not before.
• The wrong-multiplier family. (Wrong − right) × number = error works for multiplication errors (52 vs 25), fraction errors (3/4 vs 3/14) and distribution differences (T/14 − T/18 = 80) alike — one method, three question shapes.
• Simultaneous equations: the answer asked may not be the variable you solved first. Q5’s options contain only the pencil’s price (Rs. 12), not the pen’s (Rs. 25). Solve both, then read which one the question asks for.

10. Memory tricks

• BODMAS ladder: Brackets, Of, Divide, Multiply, Add, Subtract — and the twins rule: D and M walk together, A and S walk together, left to right, never one before the other.
• “Of means ×”: whenever you see “of” between numbers, replace it with a multiplication sign before doing anything else.
• Nines for repeaters: a repeating block of 2 digits sits over 99, of 3 digits over 999 — 36/99, 47/99, 125/999. The number of nines equals the number of repeating digits.
• Gaps minus one, sightings plus one: trees and poles lose one (gaps = posts − 1); flashes and sightings gain one (sightings = intervals + 1). “Fence-post” either way — count the posts, then adjust.
• Heads-and-legs in one line: (legs − 2 × heads) ÷ 2 = the four-legged animals. 24 extra legs ÷ 2 = 12 buffaloes.
• Digit blocks: 1-digit pages use 9, 2-digit pages use 180, then 3 per page after 99. 9 + 180 + (pages past 99) × 3.
• Doubling table: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32 — five doublings multiply by 32, so Rs. 4 lakh × 32 = Rs. 128 lakh = Rs. 1.28 crore.
• a² − b² = (a − b)(a + b): the difference of squares turns subtraction of squares into a multiplication you can cancel — the whole trick behind the set’s two identity questions.

11. Visuals (site build specs)

• DIAGRAM — The BODMAS ladder: a vertical ladder with six rungs (Brackets → Of/powers → Division/Multiplication side by side → Addition/Subtraction side by side), with the worked example 16 × 5 − 2 + (84 − 3) ÷ 3² = 87 descending rung by rung — placed at the head of Section 1.
• TABLE — Fraction ↔ decimal ↔ percent equivalents: the Section 6 master table rendered as a colour-banded reference card, with the unit-difference family (2/3, 3/4, 4/5, 5/6) highlighted — placed in Section 6 as the chapter’s revision graphic.
• DIAGRAM — Fence-post off-by-one: 26 trees with 25 gaps marked, beside a timeline of first/last light sightings with 383 intervals and 384 sightings — placed in Section 5.2, the chapter’s most-missed concept.
• FLOWCHART — Word problem → equation: story → name the unknown x → translate each clause (“of” = ×, “more than” = +, “times as much” = ×) → solve → check against the question asked — placed at the head of Section 5, using Q2 (Rs. 8,600 distribution) as the worked path.

12. Practice this topic

• Practice set: 265 topic-relevant items in the verified banks serve this topic — set 133’s 29 simplification questions, the decimal-fractions set’s 25 questions, and 211 paper items (167 fractions, 23 BODMAS/simplification, 21 decimals). The site serves them as this chapter’s practice set; work set 133 first for word-problem methods, then the papers’ bare computations for speed.
• Book mode: ← Previous: One-Word Substitution & Spellings · Next: Problems on Numbers →

Goldmine Notes · Mathematics · Chapter 1 · Sources: FPSC Edition 32, Arithmetic set 133 — Simplification (book pp. 740–742; 29 published questions, 6 printed items dropped in the bank as unanswerable as printed) and Arithmetic: Decimal Fractions (25 questions); verified PPSC/FPSC past-paper banks (265 topic-relevant items, script-counted — counting script hidden_files/goldmine-notes-count-ma01.py); BODMAS, fraction and decimal rules as standard arithmetic. Standard answers only.