Mathematics — Chapter 8

Time & Work; HCF & LCM; Basic Algebra — The Method Trio

Why this chapter matters: This is the last methods chapter of Mathematics — and a ★ chapter, because no single FPSC arithmetic set covers it: the papers draw it from three different wells. Our verified banks hold 186 topic-relevant items for this chapter (124 paper items plus the 62 questions of the Pipes & Cisterns, Surds & Indices and Square Roots & Cube Roots companion sets). Time & work alone supplies 64 paper items, and it repeats the same questions for decades: “A can do a piece of work in 10 days and B in 15 days, together in how many days?” is asked in at least five papers, always with the answer 6 days. HCF & LCM is the opposite story — only 1 direct paper item in our banks (“LCM and HCF of 18 and 144”), which is exactly why students lose marks on it: it is taught here as pure method, so the one question that does appear is free. Algebra’s 59 paper items are almost all one move repeated: solve for x, then answer what was actually asked. Learn the three methods in this chapter and all three blocks become routine. Part D adds the geometry block the papers also draw on — angles, triangles, circles, areas, coordinates and volumes — taught as method with its bank rows served as companion practice (see Section 18).

Exam Trend — this chapter

• 🔁 Most repeated: the 10-and-15 together question (6 days, five papers); man-days inversions; pipe pairs; the divisibility-test family (30 genuine bank rows, 24 files); the primes family (10 rows, 10 files); Part D geometry families — circle (39 first-match rows, 36 files), rectangle/square perimeter-area (39 rows, 29 files), triangle (22 rows, 17 files), volume (32 rows, 22 files), coordinates (5 rows, 5 papers) — each above the five-paper limb on its own (script hidden_files/goldmine-notes-count-ma08-opc-geometry.py over 608 files / 58,098 questions; 2 “circumference of earth” rows in the circle bucket are GK, not geometry, and are excluded from the taught count).
• ⭐ Must know: divisibility tests — still asked in CSS MPT 2025 (“which number is divisible by 9?”); geometry — still asked in 2025–2026 papers (triangle 2026: 6 rows; circle 2026: 5; perimeter-area 2026: 9; volume 2026: 5; coordinates 2025–2026: 5 rows), including the r = 7 circle pair (44 / 154) and the quadrant items.
• 🆕 Newly appeared: none claimed for this chapter.

PART A — TIME & WORK: RATE × TIME = WORK

1. The one idea

Work is always 1 whole job. A person’s rate is the fraction of the job done in one day (or one hour, or one minute):

• Rate = 1 ÷ days taken alone. A finishes in 10 days → rate 1/10 per day. B finishes in 15 days → rate 1/15 per day.
• Working together, rates ADD. A + B together = 1/10 + 1/15 = 5/30 = 1/6 per day → together they take 6 days.
• Time together = (product of the two times) ÷ (sum of the two times) for two workers: (10 × 15) ÷ (10 + 15) = 150 ÷ 25 = 6 days. This shortcut is the papers’ favourite question — see Section 2.
• Man-days (or man-hours) are constant for one job: men × days × hours-per-day = the same total for the same job. If the job is unchanged, more men → fewer days, in inverse proportion.

2. The together question — asked in five papers

• A in 10 days, B in 15 days, together? → 6 days (above). The same stem, with the same answer, appears in the Cabinet Division 2004, Overseas Human Resources 2011, Minorities Division 2011 and ASF 2004 papers, and again as “Amir in 10 days, Babar in 15 days” in the Motorway Police 2015 paper. If you can do this one, you can do the whole family.
• A in 10 days, B in 15 days, working separately — then together? → still 6 days (Intelligence Bureau 2013). “Separately” in the stem is decoration; the together rate is the same.
• A and B together in 10 days; B and C in 15 days; C and A in 30 days; all three together? Add the three pair-rates: 1/10 + 1/15 + 1/30 = 6/30 = 1/5. That sum is twice the three-together rate (each worker appears in two pairs), so the three-together rate = 1/10 → 10 days. The doubling step is the whole trick — forget it and you answer 5 days.
• A alone in 20 days; A and B together in 12 days; B works only half a day daily — how long do A and B together take? Full B rate = 1/12 − 1/20 = 1/30 per day. Half-day B = 1/60. Combined = 1/20 + 1/60 = 4/60 = 1/15 → 15 days. Halve the rate, not the days.
• A is thrice as good a workman as B; together they finish in 24 days; A alone? Rates in ratio 3 : 1; combined rate = 1/24, so A’s rate = 3/4 × 1/24 = 1/32 → A alone 32 days.
• A in 24 days, B in 36 days, together, wages Rs. 500 — A’s share? Wages divide in the ratio of the rates, not the days: rates 1/24 : 1/36 = 3 : 2 → A gets 3/5 × 500 = Rs. 300. Faster worker, bigger share.
• A in 10 days, B in 15 days; they work 5 days; C finishes the rest in 2 days; total wages Rs. 1,500 — daily wages of B and C? In 5 days A + B do 5 × 1/6 = 5/6 of the job; C does the remaining 1/6 in 2 days, so C’s rate = 1/12 per day. Work done: A 1/2, B 1/3, C 1/6 → wages Rs. 750, Rs. 500, Rs. 250. Daily wages = wage ÷ days worked: B = 500 ÷ 5 = Rs. 100/day, C = 250 ÷ 2 = Rs. 125/day (together Rs. 225/day, the bank’s keyed figure). The method is: share the wages by work done, then divide by days worked.

3. Men, women, boys and machines — the man-days family

• 20 men in 8 days → how many men in 10 days? Man-days = 20 × 8 = 160 → men = 160 ÷ 10 = 16 men. Inverse proportion: days up, men down.
• 10 men in 20 days → 8 men? 200 man-days ÷ 8 = 25 days. 10 men in 14 days → 4 men? 140 ÷ 4 = 35 days. 8 men in 60 days → 6 men? 480 ÷ 6 = 80 days. 15 men in 12 days → 9 men? 180 ÷ 9 = 20 days. 40 men in 25 days → 25 men? 1,000 ÷ 25 = 40 days. 60 men in 40 days → 20 men? 2,400 ÷ 20 = 120 days. 8 machines in 10 days → 16 machines? 80 ÷ 16 = 5 days. 8 men in 9 days → 6 men? 72 ÷ 6 = 12 days. 5 men in 6 days → 6 men? 30 ÷ 6 = 5 days.
• 40 men complete one-third of a work in 40 days; how many more men to finish the rest in 50 more days? One-third took 40 × 40 = 1,600 man-days, so the whole job = 4,800 and the remaining two-thirds = 3,200 man-days, needed in 50 days → 64 men total → 24 more men than the original 40.
• A group was to finish construction in 25 days; 10 students did not come; it took 35 days. Original number? n × 25 = (n − 10) × 35 → 25n = 35n − 350 → n = 35 students.
• A certain number of men could complete work in 40 days; 20 more men → 20 days. Original number? n × 40 = (n + 20) × 20 → 40n = 20n + 400 → n = 20 men.
• 39 persons, 12 days, 5 hours a day → 30 persons, 6 hours a day, how many days? Total man-hours = 39 × 12 × 5 = 2,340 → days = 2,340 ÷ (30 × 6) = 13 days.
• 10 men, 6 hours a day, 18 days → 15 men, 15 days, how many hours a day? 10 × 6 × 18 = 1,080 man-hours → hours = 1,080 ÷ (15 × 15) = 4.8 hours.
• Bridge completed in 20 days by 80 men → 70 men? 1,600 ÷ 70 = 22.86 days, keyed 23 days (rounded up — you cannot hire a fraction of a day’s gang). Road in 35 days by 70 men → 40 men? 2,450 ÷ 40 = 61.25, keyed 61 days. Bridge in 45 days by 105 men → 90 men? 4,725 ÷ 90 = 52.5 days. The papers round these; the method is the man-days product either way.

4. Men and boys together — convert to one unit

The papers’ second favourite family gives two mixed gangs that do the same job. Convert everyone to one unit (usually boys), using the equivalence in the stem:

• 3 men or 6 boys in 20 days; 6 men and 8 boys? 1 man = 2 boys, so 6 men = 12 boys; 12 + 8 = 20 boys. 6 boys take 20 days → 20 boys take 6 × 20 ÷ 20 = 6 days. (Asked in the Establishment Division 2001, FIA 1985, Foreign Affairs 2008 and Customs Inspector 2011 papers — the same numbers, the same answer.)
• 4 men or 7 boys in 29 days; 12 men and 8 boys? 1 man = 7/4 boys, so 12 men = 21 boys; 21 + 8 = 29 boys. 7 boys take 29 days → 29 boys take 7 × 29 ÷ 29 = 7 days.
• 6 women or 9 girls in 10 days; 8 women and 3 girls? 1 woman = 3/2 girls, so 8 women = 12 girls; 12 + 3 = 15 girls. 9 girls take 10 days → 15 girls take 9 × 10 ÷ 15 = 6 days.
• 12 men and 16 boys in 5 days; 13 men and 24 boys in 4 days — ratio of a man’s daily work to a boy’s? Let man’s rate = m, boy’s = b: 12m + 16b = 1/5 and 13m + 24b = 1/4. Solve: multiply the first by 3/2 → 18m + 24b = 3/10; subtract the second → 5m = 3/10 − 1/4 = 1/20 → m = 1/100; then b = (1/5 − 12/100) ÷ 16 = (1/5 − 3/25) ÷ 16 = (2/25) ÷ 16 = 1/200. Ratio m : b = 2 : 1.
• 5 men and 2 boys do four times as much as 1 man and 1 boy — capacities of a man and a boy? 5m + 2b = 4(m + b) → m = 2b → ratio 2 : 1. The same 2 : 1 answer as the previous question, by a shorter road.
• 10 women in 7 days; 10 children in 14 days; 5 women and 10 children? Women’s rate = 1/70 each, children’s = 1/140 each. Combined = 5/70 + 10/140 = 1/14 + 1/14 = 1/7 → 7 days.
• 12 men in 4 days; 15 women in 4 days; 6 men work 2 days, then how many women to finish in 3 days? Job = 48 man-days = 60 woman-days. 6 men in 2 days = 12 man-days = 1/4 of the job. Remaining 3/4 = 45 woman-days, in 3 days → 15 women.
• Mixed gangs with leavers and joiners (the ASF 2009/2010 long stems) work the same way: compute what fraction is done before the change, what fraction remains, and divide the remainder by the new gang’s daily rate. 16 adults start, after 3 days 10 leave and 4 children join → the remaining work takes 6 days in the bank’s keying; 12 men and 8 children work 8 days, then 3 more children join → 4 more days; 16 men and 16 women work 12 days, then more men are added to finish in 2 days → 24 more men. In every case: fraction done first, fraction left, new rate — never restart from the beginning.

5. Pipes and cisterns — time & work with water

A filling pipe has a positive rate, an emptying pipe (or leak) a negative rate. The tank is 1 whole job, exactly as in Section 1. The FPSC Pipes & Cisterns set (16 questions) is served whole with this chapter.

• Pipes A and B fill a tank in 36 hours and 45 hours; together? 1/36 + 1/45 = 5/180 + 4/180 = 9/180 = 1/20 → 20 hours.
• Pipe A fills in 5 minutes, B in 6 minutes; together? 1/5 + 1/6 = 11/30 → time = 30/11 minutes. A in 8 minutes, B in 9 minutes; together? 1/8 + 1/9 = 17/72 → 72/17 minutes. B in 11 minutes, C in 13 minutes; together? 1/11 + 1/13 = 24/143 → 143/24 minutes.
• A fills in 10 hours, B empties in 15 hours; both open? 1/10 − 1/15 = 1/30 → 30 hours to fill. The emptying pipe subtracts — the sign is the whole question.
• Inlet fills in 10 hours, outlet empties the full tank in 20 hours; both open? 1/10 − 1/20 = 1/20 → 20 hours.
• Two taps fill in 12 and 15 minutes; with the waste pipe open, the empty cistern fills in 20 minutes; waste pipe alone? 1/12 + 1/15 − 1/w = 1/20 → 3/20 − 1/w = 1/20 → 1/w = 1/10 → 10 minutes to empty.
• A pump fills a tank in 3 hours; because of a leak it takes 3.5 hours; the leak alone empties the full tank in? 1/3 − 1/L = 1/3.5 = 2/7 → 1/L = 1/3 − 2/7 = 1/21 → 21 hours.
• If part of a tank filled in one hour = 1/10, time to fill it? → 10 hours (reciprocal — the papers ask the reciprocal both ways).
• 15 pumps of equal capacity fill a tank in 7 days; how many extra pumps to fill it in 5 days? Total = 105 pump-days → pumps needed = 105 ÷ 5 = 21 → 6 extra pumps beyond the 15. “Extra” is the trap: 21 is not the answer, 21 − 15 = 6 is.

5A. Time & work — the OPC drill residues (OPC pp. 64–67)

Sections 1–5 already teach every time & work method; this section files only what the OPC Time & Work block adds that those sections did not carry — four patterns, two of them real paper items. No new mark is claimed for any of them (bank evidence in Section 14).

• The three-worker system — a real paper item. A can do a piece of work in 4 hours; B and C together can do it in 3 hours; A and C together can do it in 2 hours. How long will B alone take? The system is three equations in three rates: 1/A = 1/4; 1/B + 1/C = 1/3; 1/A + 1/C = 1/2. From the third, 1/C = 1/2 − 1/4 = 1/4. Then 1/B = 1/3 − 1/4 = 1/12 → B alone takes 12 hours. The move is: find the single worker hidden in a pair-sum first, then subtract. The same stem, with the same answer, is asked in the PPSC Assistant S&GAD 2022 paper and the PPSC Deputy Secretary Examination 2023 paper — two papers, below the 🔁 limb, so no mark, but this is a paper item, not just a book drill. Do not confuse it with Section 2’s pair-sums pattern (12/15/30 → 10 days): there every worker appears only in pairs and the pair-sum is halved; here one worker’s own rate is given, so subtraction, not halving, unlocks the system.
• “Twice as good” — the efficiency-ratio setup. A is twice as good a workman as B; together they finish a piece of work in 18 days. In how many days will B alone finish? Let B’s time be x days, so A’s time is x/2 (twice the rate, half the time): 1/18 = 1/(x/2) + 1/x = 2/x + 1/x = 3/x → x = 54 days for B alone (A alone 27 days). Check: 1/27 + 1/54 = 3/54 = 1/18 ✓. This stem is the PPSC Punjab Police Sub-Inspector 2019 (open merit) paper item. Its sibling variant is also banked: A is twice as good as B and takes 30 days less than B — let B take x days, A take x/2, so x − x/2 = 30 → x = 60, A = 30 days; together 1/30 + 1/60 = 3/60 = 1/20 → 20 days (PPSC Assistant Finance Department 2020). Section 2 already teaches the thrice-as-good shape (together 24 days → A alone 32 days); this section adds the twice-as-good shape and the “days less” variant, both with banked stems.
• The percentage-efficiency trap — divide, never subtract. A worker finishes a job in 20 days; a second worker is 25% more efficient. How many days does the second worker take? Efficiency up 25% means rate × 1.25, so time = 20 ÷ 1.25 = 16 days. OPC p. 67 prints 15 days by subtracting 25% of 20 from 20 — that shortcut is wrong, and it is the trap this section exists to kill: a percentage on efficiency divides the time, it does not subtract from it. (Check: 20 days → rate 1/20 per day; × 1.25 → 1/16 per day → 16 days ✓; at 15 days the rate would be 1/15, which is 33⅓% more, not 25%.) No bank stem asks a percentage-efficiency question (0 rows across 608 files / 58,098 questions), so this is taught as method, with no mark claimed — and OPC’s printed 15 is flagged in Traps, not taught.
• Man-days with an efficiency multiplier, and workers added mid-job. (a) If 10 persons can do a job in 20 days, then 20 persons with twice the efficiency take: total = 10 × 20 = 200 man-days; the new gang’s effective strength = 20 × 2 = 40 → 200 ÷ 40 = 5 days. This is the PPSC Assistant Board of Revenue 2021 paper item, word for word — the multiplier belongs on the gang’s strength before dividing, not on the answer after it. (b) A project takes 150 workers 40 days; after 16 days, 30 more workers are added. Total = 150 × 40 = 6,000 worker-days; done in 16 days = 150 × 16 = 2,400; remaining = 3,600 worker-days over 180 workers → 20 more days. The method is Section 4’s leavers-and-joiners move (fraction done first, remainder over the new gang); the same “additional workers” family is banked in the FPSC Assistant Director Intelligence Bureau 2016 and 2017 papers (a 6-person staff, a 24-hour project, additional workers asked). OPC’s Ex 12 itself is not a bank stem — it is taught as the drill for that family.
• Output-rate addition — when the rate is pages, not days. A types 10 pages in 5 minutes, B types 5 pages in 10 minutes. Together in 30 minutes: A = 2 pages/minute → 60 pages; B = 0.5 pages/minute → 15 pages; total 75 pages. When the stem gives output per unit time, convert each worker to the same unit first, then add the outputs — the mirror image of Section 1, where the stem gives days and the rates (fractions of one job) are added instead. OPC’s Ex 14 stem is not itself banked (the two “pages in minutes” bank rows are different stems); taught as method, no mark claimed.
• Same-method drills (Section 2’s together-minus-one move, OPC Ex 1–5 and 7). No new method — the answers are filed here so the OPC drill set is fully covered: X + Y in 12 days, Y alone in 48 → X alone 16 days (1/12 − 1/48 = 3/48); A + B in 15 days, B alone in 20 → A alone 60 days; together in 2 hours, A alone in 5 → B alone 10/3 hours; A in 24 days, B in 16, with C they finish in 8 → C alone 48 days (1/8 − 1/24 − 1/16 = 1/48); Asghar in 60 days, together with Babar in 20 → Babar alone 30 days; a man in 5 days, with his son in 3 → the son alone 7.5 days. Every one is 1/(together) − 1/(known) = 1/(unknown), exactly Section 2’s move.

PART B — HCF & LCM: THE TWO NUMBERS THAT DIVIDE AND MULTIPLY

6. Definitions — and the one relation

• HCF (Highest Common Factor, also GCD) — the largest number that divides each of the given numbers exactly.
• LCM (Lowest Common Multiple) — the smallest number that each of the given numbers divides exactly.
• For two numbers: HCF × LCM = product of the two numbers. This one relation answers half the papers’ HCF/LCM questions without factorising anything.
• If one number divides the other, the smaller IS the HCF and the larger IS the LCM. This is the papers’ only direct HCF/LCM item in our banks: LCM and HCF of 18 and 144 → 144 = 18 × 8, so HCF = 18, LCM = 144. Check by the relation: 18 × 144 = 2,592 = 18 × 144. ✓

6A. Number foundations — the facts factorisation stands on

Before you factorise anything, you need five facts: what counts as a number, how zero behaves, how to tell even from odd, how to test divisibility without dividing, and what primes, factors and multiples are. This section comes from the One Paper Cracker Mathematics book’s opening block (pp. 1–3, plus its p. 9 remainder and counting rules), checked against our banks. Consecutive numbers already live in Chapter 2 and are cross-linked here, not re-taught; BODMAS lives in Chapter 1.

• Whole numbers and integers. Whole numbers (counting numbers): 0, 1, 2, 3, 4, 5, 6, …. Integers: …, −4, −3, −2, −1, 0, 1, 2, 3, 4, …. On the number line, positive integers grow as they move away from 0 and negative integers shrink: 4 > 3, but −4 < −3.
• Zero. Zero is both a whole number and an even integer. It is neither positive nor negative. 0 + any number = that number; 0 × any number = 0.
• Digits and place value. There are 10 digits: 0–9. In 5,234: 4 → ones, 3 → tens, 2 → hundreds, 5 → thousands. Every divisibility test below reads one of these places.
• Rules of signs (multiplication). (negative) × (negative) = positive; (positive) × (positive) = positive; (negative) × (positive) = negative. Division follows the same signs.
• Even and odd. An even number is divisible by 2 with no remainder; an odd number is not. Read it off the ones digit. Even/odd is not positive/negative — −4 is even, −3 is odd. The parity table: even + even = even; odd + odd = even; even + odd = odd; even × even = even; odd × odd = odd; even × odd = even.
• Divisibility tests 🔁 ⭐ — the papers test the number, not the rule’s name, so learn the test and use it:

The book prints the 2 / 3 / 5 / 10 tests; the 4 / 6 / 8 / 9 / 11 tests are the standard extensions the bank items require (divisible-by-9 subtraction, divisible-by-11 counting, divisible-by-4-and-9 items). Divisibility logic the papers love: divisible by 8 does not imply divisible by 6 — 8 itself is divisible by 8 and not by 6. Divisible by 7 but not by 14 → an odd multiple of 7 (7, 21, 35, …). “Divisible by 2 and 3 both” means divisible by 6 — take the LCM, then test once.

• Prime numbers 🔁 — a prime number is divisible only by 1 and itself. 0 and 1 are not prime. 2 is the only even prime number. First primes: 2, 3, 5, 7, 11, 13, 17. Bank items: smallest prime = 2; sum of the first five primes = 2 + 3 + 5 + 7 + 11 = 28; primes between 64 and 81 = 67, 71, 73, 79 (4 in total). No ⭐ claimed — the only 2024–26 prime signal is a 2026 mock stem, not an exam paper.
• Factors and multiples. Factor: x is a factor of y if y is divisible by x — 2, 3, 4, 6 are factors of 12. Multiple: x multiplied by an integer — the papers list the positive ones: 10, 20, 30, 40 are multiples of 10. Co-prime numbers are already in Section 6 (8 and 15 → HCF 1). Factorisation in Section 7 is just listing a number’s prime factors — these two definitions are what it stands on.
• Remainders and counting. When a number is divided by 7, the remainder is always less than 7 (0–6) — in general, remainder < divisor. Counting multiples: numbers up to 200 divisible by both 2 and 3 are numbers divisible by 6 → 200 ÷ 6 = 33 (drop the remainder).

6B. Grouping laws & naming expressions — associative, distributive, binomial/trinomial

Section 7 factorisation and Part C algebra both assume two things: that you can regroup a sum or product without changing it, and that you know what a binomial or trinomial is. This section supplies both, from the One Paper Cracker Mathematics book (p. 5 laws, p. 10 definitions), standardised where the book’s wording is loose. Method block, no mark claimed: the bank script for this section (608 files / 58,098 questions) returns zero direct rows for associative, distributive or trinomial, and the two binomial hits are non-maths (“binomial state”, “binomial theorem” history) — so these laws are taught because factorisation and expansion need them, not because a paper names them.

• Associative law — grouping changes nothing (addition and multiplication only). Addition: a + (b + c) = (a + b) + c. Multiplication: a(bc) = (ab)c. The book’s examples, recomputed: 2 + (4 + 6) = 2 + 10 = 12, and (2 + 4) + 6 = 12 ✓; 2(4 × 6) = 2 × 24 = 48, and (2 × 4)6 = 8 × 6 = 48 ✓. Regroup however the numbers are friendliest — the answer cannot change.
• Associative does NOT hold for subtraction or division. (10 − 4) − 2 = 6 − 2 = 4, but 10 − (4 − 2) = 10 − 2 = 8. (24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3, but 24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12. Never regroup a subtraction or division chain — work it strictly left to right (the p. 5 left-to-right rule: 10 ÷ 2 × 5 = 25, 10 − 2 + 5 = 13).
• Distributive law — multiply through the bracket. a(b + c) = ab + ac; a(b − c) = ab − ac. Worked: 3(4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27 (and 3 × 9 = 27 ✓); 5(8 − 3) = 5 × 8 − 5 × 3 = 40 − 15 = 25 (and 5 × 5 = 25 ✓). The minus inside the bracket stays a minus outside it: a(b − c) is ab − ac, never ab + ac.
• Distributive in reverse is factoring. ab + ac = a(b + c) — take the common factor out of every term. The book’s factoring example: 4x + 4y = 4(x + y) (check: 4x + 4y ✓). Likewise 6x + 9 = 3(2x + 3) (check: 6x + 9 ✓). This reverse move is exactly Section 7’s factorisation and the first step of solving quadratics by factors in Section 10.
• Naming expressions — count the terms. A term is a part separated by a + or − sign. Monomial = one term (4x). Binomial = two terms: x + 3, 4x − y (the book’s examples). Trinomial = three terms: x² + 3x + 2 (the book’s example — terms x², 3x, 2). Polynomial = an expression of one or more terms with whole-number exponents: 2x + 1 (binomial), x² + 4x − 5 (trinomial), 3x³ − 2x + 7 (three terms: 3x³, −2x, +7). The degree of a polynomial is its highest exponent: x² + 3x + 2 has degree 2; 3x³ − 2x + 7 has degree 3; a linear equation is degree one (Section 8 — the bank’s one genuine polynomial-degree item, FPSC 2000, already taught there).
• Why the names matter in the paper: Section 10’s identities are binomial squares — (a + b)², (a − b)², a² − b² — and Section 11’s “expand (x − 2)(x + 3)” is the distributive law applied twice (x·x + x·3 − 2·x − 2·3 = x² + x − 6). If you can name the expression, you know which tool to reach for.

6C. Factoring and expanding — the two-way move

Section 6B taught the distributive law in one direction; the book’s p. 10 factoring example runs off the bottom of the page, and pp. 11–13 finish it — this section completes it, because factoring and expanding are the same move in opposite directions, and the papers test both. Method block, no mark claimed: the bank script for this section (608 files / 58,098 questions) returns only two genuine maths stems in the whole family (see Practice), so this is taught because Sections 7, 10 and 11 run on it, not because a paper names it.

• Factoring — pull the common factor out. The book’s unfinished example, completed: 4x + 4y = 4(x + y) — factor out the common term 4 (check: 4 × x + 4 × y = 4x + 4y ✓). Every term must be divided by the factor — 4x + 4y is 4(x + y), never 4(x + 4y) (see Traps).
• Expanding — multiply back through the bracket. The reverse move, which the book says to do immediately whenever you meet a factored expression: 4(x + y) = 4x + 4y. Factor and expand undo each other; if your factored answer does not expand back to the original, it is wrong — that one check catches most factoring slips.
• Multiplying polynomials — every term times every term. The book’s worked example: (x + 4)(2x − 1). Step by step: x(2x) + x(−1) + 4(2x) + 4(−1) = 2x² − x + 8x − 4 = 2x² + 7x − 4. The middle terms combine (−x + 8x = 7x) — that combining step is where marks are lost; never leave the four products uncombined. The same move, with a plain x, is Section 10’s (x − 2)(x + 3) = x² + x − 6.
• The three identities, factored and unfactored. The book marks these “must be memorised”, and they are Section 11’s identities seen from the factoring side — the unfactored form on the left, the factored form on the right:
  • Difference of squares: x² − y² = (x + y)(x − y).
  • Perfect square (addition): x² + 2xy + y² = (x + y)² (equivalently (x + y)(x + y)).
  • Perfect square (subtraction): x² − 2xy + y² = (x − y)².
  • Read them both ways: x² − y² factorises at sight, and (x + y)(x − y) expands to x² − y² without working. Section 10’s 16x² + 24x + 9 = (4x + 3)² is the addition identity with a = 4x, b = 3.
• Factor first, substitute last — the exam shortcut. The book’s worked MCQ: if y + 3 = 2x, find 3y − 6x (options −9, −3, 0, 3, 9). Solving for y and substituting works but wastes time; factor the asked expression first: 3y − 6x = 3(y − 2x). From the given equation, y − 2x = −3 (move 3 across: y + 3 = 2x → y − 2x = −3). So 3(y − 2x) = 3 × (−3) = −9 — option (a). Whenever the asked expression shares a factor, factor it before you touch the given equation.
• Bank item — factorise a trinomial. PPSC Enforcement Officer 2026: Factorize: x² + 16x + 63 → (x + 7)(x + 9). Method: find two numbers that add to 16 and multiply to 63 — 7 and 9 (7 + 9 = 16, 7 × 9 = 63 ✓). Check by expanding: (x + 7)(x + 9) = x² + 9x + 7x + 63 = x² + 16x + 63 ✓. This is Section 10’s quadratic factorisation with the equation set to zero removed — the same two-numbers hunt.
• Which direction does the question want? “Factorize” → pull the factor out (factored form). “Expand” / “simplify (x + 4)(2x − 1)” → multiply out (unfactored form). “Solve … = 0” → factorise first, then set each factor to zero (Section 10). A factored answer to an expand question, or an expanded answer to a factorize question, loses the mark even though the maths is right.

7. Finding them — prime factorisation (the method that never fails)

• HCF: factorise each number; take the lowest power of each common prime. LCM: take the highest power of every prime that appears.
• Example — HCF and LCM of 12 and 18: 12 = 2² × 3; 18 = 2 × 3². Common primes 2 and 3 → HCF = 2¹ × 3¹ = 6. All primes at highest powers → LCM = 2² × 3² = 36. Check: 6 × 36 = 216 = 12 × 18. ✓
• Example — HCF and LCM of 24, 36 and 40: 24 = 2³ × 3; 36 = 2² × 3²; 40 = 2³ × 5. HCF = 2² = 4 (2 is the only prime common to all three, lowest power 2²). LCM = 2³ × 3² × 5 = 360.
• Co-prime numbers (no common factor except 1, e.g. 8 and 15): HCF = 1, LCM = their product (120). If the papers ask “HCF of two co-prime numbers”, the answer is 1 before you factorise anything.
• LCM of fractions: LCM of numerators ÷ HCF of denominators. HCF of fractions: HCF of numerators ÷ LCM of denominators. (Fractions swap the two — that swap is the trap.)
• The division method (for large numbers, e.g. HCF of 1,092 and 1,638): divide the larger by the smaller, then the divisor by the remainder, until the remainder is 0 — the last divisor is the HCF. 1,638 ÷ 1,092 = 1 remainder 546; 1,092 ÷ 546 = 2 remainder 0 → HCF = 546.
• Remainder questions (the shape the papers love next): “greatest number that divides 43, 91 and 183 leaving the same remainder” → subtract the smallest from the others and take the HCF of the differences. “Smallest number divisible by 12, 15 and 20” → LCM = 60. “Smallest number that leaves remainder 5 when divided by 12, 15 and 20” → LCM + 5 = 65.
• Traps: HCF is never larger than the smallest number; LCM is never smaller than the largest number. If your answer breaks either rule, you have swapped them — the commonest HCF/LCM error in the papers.

PART C — BASIC ALGEBRA: SOLVE FOR x, THEN ANSWER WHAT WAS ASKED

8. Linear equations — one unknown, one move

An equation is a balance: whatever you do to one side, do to the other. Collect the x-terms on one side, the numbers on the other, then divide. Moving a term to the other side changes its sign — the transposition rule (OPC pp. 48–49, the equations block’s one new name for a move Section 8 already teaches): x + 5 = 12 → x = 12 − 5 = 7.

• x + 6 = 7 → x = 1. Solve first, then substitute for whatever the stem asks — never answer x when the stem asks for x + 3 or x² − 4 (see Sections 8–9 and Traps).
• 2x − 7 = 8 + x → x = 15. 4x + 12 = 36, find x + 3? 4x = 24 → x = 6 → x + 3 = 9. (Asked twice, same answer.) x − 4 = 9, find x² − 4? x = 13 → 169 − 4 = 165.
• 3x = 2(5 − 2x) → 3x = 10 − 4x → 7x = 10 → x = 10/7.
• If x = 2, 2x − 1 = ? → 3. Substitution before algebra: put the value in first, then simplify.
• If x = 5, y = 2, find x + y → 7. If y = 2x and z = 5y, find x + 3y + z in terms of x → y = 2x, z = 10x → x + 6x + 10x = 17x.
• The bank teller’s equation Y = 1.30x − 1.50 (dollars x → euros Y) is a linear equation in use: for x dollars, the euros are 1.30 times the dollars minus 1.50. Linear equations are straight lines — that is why the papers also ask “a linear equation consists of polynomials of degree one”.

9. Simultaneous equations — two unknowns, two equations

• x + y = 10 and x − y = 4 → add: 2x = 14 → x = 7, y = 3, x² = 49. (Asked twice.)
• x + y = 99 and x − y = 9 → x = 54, y = 45. x − y = 6 and x + y = 16 → x = 11, y = 5.
• 2x + 4y = 12 and x + y = 4 → from the second, y = 4 − x; substitute: 2x + 16 − 4x = 12 → x = 2.
• 2x + 3y + z = 55, x − y = 4, y − x + z = 12 — worked straight, this system gives a non-integer y (from the second, x = y + 4; the third gives z = 16; then 2(y + 4) + 3y + 16 = 55 → 5y = 31 → y = 6.2), yet the bank keys (x) 7, (y) 11, (z) 8. That triple satisfies the first equation (14 + 33 + 8 = 55) and the third (11 − 7 + 8 = 12) but gives x − y = −4, not 4 — the printed second equation’s sign is garbled in the paper. Lesson, not method: with a three-equation stem, check any keyed triple against all three equations before trusting it. (See Traps.)
• How many solutions do 3x − 6y = 15 and −2x + 4y = −10 have? The second is −2/3 times the first — the same line twice → infinitely many solutions. Parallel lines (same slope, different intercept) → none; one intersection → exactly one.

10. Quadratics — factorise first, formula second

• Solve 2x² − 3x − 2 = 0 → (2x + 1)(x − 2) = 0 → x = 2 or −1/2. Roots of x² − 3x − 10 = 0 → (x − 5)(x + 2) = 0 → x = 5 or −2.
• Solution of 2x² + 5x − 12 = 0 → (2x − 3)(x + 4) = 0 → x = 3/2 or −4 (the bank keys the positive root 3/2). 2x² − 5x − 12 = 0 → (2x + 3)(x − 4) = 0 → x = 4 or −3/2.
• Positive value of x from 4x² − 6x − 4 = 0 → divide by 2: 2x² − 3x − 2 = 0 → x = 2 (positive root; the other is −1/2).
• Correct form of 16x² + 24x + 9 → (4x + 3)² — a perfect square: (a + b)² = a² + 2ab + b² with a = 4x, b = 3.
• Expand (x − 2)(x + 3) → x² + x − 6.
• Sum of roots −4, product 3 — make the equation → x² − (sum)x + product = x² + 4x + 3 = 0.
• How many solutions does 2x² − √5x + 1 = 0 have? Discriminant b² − 4ac = 5 − 8 = −3 < 0 → no real solution. Solution of 2x² + 5x − 6 = 0 is “two different roots” — discriminant 25 + 48 = 73 > 0, not a perfect square, so two distinct irrational roots. Discriminant rules: > 0 → two real roots; = 0 → one repeated root; < 0 → no real roots.
• Quadratic formula (when factorisation hides): x = (−b ± √(b² − 4ac)) ÷ 2a.

11. Identities, indices, surds and roots — the simplification toolkit

• Identities: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a + b)(a − b). Simplify (4a²) × (3a²) → 12a⁴ (coefficients multiply, powers add).
• If x³ + 1/x³ = 52, find x + 1/x. Let t = x + 1/x: t³ − 3t = 52 → t = 4 (4³ − 12 = 52). ✓
• Simplify (m^(1/3) · n^(−2))^(3/5) → m^(1/5) · n^(−6/5) (powers multiply).
• If x = 7 − 4√3, find x − 1/x. 1/x = 7 + 4√3 (rationalising: (7 − 4√3)(7 + 4√3) = 49 − 48 = 1) → x − 1/x = −8√3.
• What should replace both x in x/√128 = √162/x? x² = √(128 × 162) = √20,736 = 144 → x = 12.
• If 4^(2n+3) = 8^(n+5), find n. Write both as powers of 2: 2^(4n+6) = 2^(3n+15) → 4n + 6 = 3n + 15 → n = 9.
• If log₂x = 5, x = ? → x = 2⁵ = 32.
• Square roots the papers repeat: √729 = 27; √2,025 = 45; √9,801 = 99; √256 = 16. Cube roots: ∛2,744 = 14; ∛0.000216 = 0.06.
• The Surds & Indices set (16 questions, served whole): (27)^(2/3) = 9; (256)^(5/4) = 1,024; (256)^0.16 × (16)^0.18 = 2^1.28 × 2^0.72 = 2² = 4. Fractional powers: a^(m/n) = ⁿ√(a^m) — the denominator is the root, the numerator the power.
• The Square Roots & Cube Roots set (30 questions, served whole): √6,084 = 78 by factorisation (6,084 = 2² × 3² × 13²); √1,471,369 = 1,213; √(1 9/16) = √(25/16) = 1 1/4. Factorise, pair the primes, take one from each pair.
• Ratio-proportion as algebra: 7 : 8 :: 196 : x → x = 8 × 196 ÷ 7 = 224. And 2√(x + 2) = 3√2 → square both sides: 4(x + 2) = 18 → x = 2.5. Squaring both sides can introduce false roots — check the answer back in the original equation (here 2√4.5 = 2 × 2.121 = 4.243 = 3√2 ✓).

PART D — GEOMETRY: ANGLES, TRIANGLES, CIRCLES, AREAS, VOLUMES

Geometry is the fourth method block of this chapter, folded in from OPC Mathematics pp. 13–26 (the “Geometry, Geometric Figures & Areas of Figures” block, pp. 13–20, plus the foundations block, pp. 21–26: volumes completed, triangle classification, points/lines/planes, angle types and relationships, transversal conditions). Nothing in it was taught anywhere else in this chapter — the chapter-check found only a ratio cross-reference in Chapter 6 (circumference ÷ radius = 2π, cross-linked in Section 11D below, not re-taught) — so Part D teaches the whole block, with two OPC errors corrected in the text and flagged in Traps, and two bank-driven extensions (Section 11A polygon line, Section 11G cylinder/cone) that the OPC block omits but the banks ask. The foundations sections (Sections 11H–11J) are method: the bank count below finds no direct paper rows for them, so no mark is claimed for them.

11A. Angle and line facts

• A full circle = 360°; a straight line = a straight angle of 180°; a right angle = 90°.
• Two lines crossing make four angles summing to 360°; the angles opposite each other (vertical angles) are equal.
• Two perpendicular lines (symbol ⊥, misread as “I” in the OPC scan) make four right angles of 90° each.
• Two parallel lines (symbol ∥) cut by a third line: the angles that look equal are equal — corresponding and alternate angles match; the exam shortcut is exactly that visual rule.
• Polygon extension (bank-driven, not in OPC): the sum of a polygon’s interior angles = (n − 2) × 180°, where n is the number of sides. A hexagon (n = 6) → 4 × 180° = 720° (asked in CSS MPT 2025 and its practice booklet). Reverse it: interior sum 1,260° → (n − 2) = 1,260 ÷ 180 = 7 → 9 sides (CSS GSA 2024).

11B. Triangles — the 180° rule and the three families 🔁 ⭐

• Every triangle’s three interior angles sum to 180°, whatever its shape.
• Equilateral: all three sides equal, all three angles equal, each 60°. Perimeter 60 → side 20 → area (√3/4) × 20² = 100√3 ≈ 173.2 (PPSC Assistant, Punjab Police, 2026, asks the perimeter → area step).
• Isosceles: two sides equal, and the angles opposite those sides are equal. If two equal angles are 40° each, the third = 180° − 80° = 100°. The 1 : 1 : 2 angle-ratio item (FPSC 2000, 2002) must be both isosceles and right-angled: 45° + 45° + 90° = 180°.
• Right triangle: one 90° angle; the side opposite it is the hypotenuse, always the longest side.
• Side ↔ angle rule: the longest side is opposite the largest angle; the shortest side opposite the smallest; equal angles → equal opposite sides.
• Perimeter = sum of the three sides. Area = (1/2) × base × height, for every triangle, even when the height is not obvious. Worked bank items: area 126 cm², base 28 cm → height = 2 × 126 ÷ 28 = 9 cm (PPSC 2026); base 19 m, height 7 m → area = 66.5 m² (PPSC 2019); right triangle area 42 cm², height 10 cm → base = 8.4 cm (PPSC 2025).

11C. Pythagoras and the special right triangles

• Pythagoras — right triangles only: c² = a² + b², where c is the hypotenuse. The theorem’s own statement is itself a bank item: “the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides” (PPSC Assistant Director, Processing, 2022).
• The 3-4-5 family — recognise it instantly: 3² + 4² = 9 + 16 = 25 = 5², and every multiple works the same way: 6-8-10, 9-12-15, and so on. A rectangle or square is two right triangles joined: a 3-by-4 rectangle has diagonal √(3² + 4²) = 5.
• Worked bank item (PPSC Tehsildar, 2024): height 4 cm less than base, hypotenuse 10 cm. Let base = x: x² + (x − 4)² = 100 → 2x² − 8x − 84 = 0 → x² − 4x − 42 = 0 → x = (4 + √184) ÷ 2 ≈ 8.78 → base 8.8 cm, height 4.8 cm (the bank key, recomputed ✓). Lesson: not every Pythagoras item is a clean triple — set up the quadratic and round only at the end.
• 30°-60°-90° triangle: angles 30°, 60°, 90°; sides in the ratio shortest : longer leg : hypotenuse = 1 : √3 : 2. Shortest side 5 → hypotenuse 10, longer leg 5√3. (The OPC block lists the ratio as “1 : 2 : √3” — that order is only correct if read short : hypotenuse : longer leg; the standard teaching order used here is short : longer leg : hypotenuse.)
• 45°-45°-90° triangle: two 45° angles, one 90°; the two shorter sides are equal; sides in the ratio 1 : 1 : √2. Each short side 5 → hypotenuse 5√2. (The OPC block prints the ratio as “1 : √2” — the second 1, the equal short side, is missing there and restored here.)

11D. Circles 🔁 ⭐

• Circumference = 2πr = πd, where r = radius, d = diameter = 2r. Area = πr².
• π in the exam hall: OPC uses π ≈ 3 for quick estimates on p. 17 and π = 3.1416 on pp. 19–20 — both are in the same book, so read which the question wants. With π ≈ 3: diameter 4 → radius 2 → circumference 4π ≈ 12; radius 5 → area ≈ 3 × 25 = 75. With π = 22/7 (the banks’ favourite for r = 7): radius 7 → circumference = 2 × (22/7) × 7 = 44, area = (22/7) × 49 = 154 (CSS MPT 2025/2026 items). With π = 3.1416 the r = 5 area is 78.5, not 75 — the OPC “75” is the π ≈ 3 estimate, not the exact value.
• Scaling rules the banks test: doubling the radius quadruples the area (r² law — CSS MPT practice booklet); trebling it multiplies the area by 9 (CSS 2025 recall list). Circumference scales only linearly: diameter 12 cm → radius 6 cm (PPSC 2026).
• Cross-link, not re-taught: circumference ÷ radius = 2π (FPSC 2000) is already taught in Chapter 6, Section 11.

11E. Rectangles, squares and the area table 🔁 ⭐

• Rectangle: perimeter = sum of all sides = 2(length + width); area = length × width. Example: 8 × 4 → area 32. Bank drill: perimeter 24 cm, length 8 cm → width = 24 ÷ 2 − 8 = 4 cm (FPSC 2001, 2008, 2009 — the same stem in three papers).
• Square: a rectangle with four equal sides s: perimeter = 4s, area = s². Perimeter 256 (the stem misprints the unit as cm² — the number is a perimeter) → side = 64; area 64 cm² → side 8 cm (both PPSC 2026).
• Parallelogram: area = base × height. Trapezoid: area = h(b₁ + b₂) ÷ 2, where b₁, b₂ are the two parallel sides (not “longer/shorter” — either order works). Ellipse: area = π × r₁ × r₂, the two semi-axes (the OPC scan prints “πr r2” — read as πr₁r₂).
• Triangle, two more area forms: given two sides and the included angle (SAS): area = (1/2)ab × sin θ. Given all three sides (Heron’s formula): s = (a + b + c) ÷ 2, area = √[s(s − a)(s − b)(s − c)]. Check on 3-4-5: s = 6, area = √(6 × 3 × 2 × 1) = 6 — and (1/2) × 3 × 4 = 6 ✓, so all three triangle formulas agree.
• Rhombus drill (FPSC 2002, 2008): perimeter = 4 × side, so side 6 cm → perimeter 24 cm.

11F. Coordinates — quadrants and points on the axes 🔁

• The horizontal axis is the x-axis, the vertical axis the y-axis; they meet at the origin (0, 0). The plane divides into four quadrants: I (+, +), II (−, +), III (−, −), IV (+, −).
• A point is written (x, y): x = distance right (+) or left (−) of the origin, y = distance above (+) or below (−). Point A (2, 4) → 2 right, 4 up; point B (−6, 1) → 6 left, 1 up.
• Bank quadrant items (2025–2026): the point (−x, y) — with x, y positive — lies in quadrant II (PPSC 2026); the point (0, −7) lies on the y-axis, in no quadrant (PPSC 2025, two papers — the trap is answering a quadrant at all); the sum of the absolute values of the coordinates of (−6, 8) = 6 + 8 = 14 (PPSC PERA 2026).
• Worked square item (PPSC 2025): square ABCD with A(2, 3), B(−2, 2), C(−1, −2) → D = A + C − B = (3, −1) (diagonals share a midpoint: AC midpoint (0.5, 0.5) = BD midpoint ✓).

11G. Volumes — boxes, cubes and spheres (OPC error corrected)

• Rectangular solid (box): volume = length × width × height (or abc). Cube: volume = a³, where a is one side. Bank item: cube volume 343 m³ → side = 7 m → area of one face = 49 m² (PPSC 2026). A cube whose volume-number equals its surface-area-number has edge 6 (FPSC 2012: a³ = 6a² → a = 6).
• Sphere — OPC correction: the OPC block prints V = (4πr²) ÷ 3. That formula is wrong — (4πr²) ÷ 3 is neither a volume nor a surface area. The correct volume is V = (4/3)πr³, and the surface area is 4πr². If a sphere question’s options contain (4πr²)/3, it is the book’s misprint, not a formula to use. (A sphere whose radius rises 10% gains 21% in surface area — FPSC 2004: 1.1² = 1.21.)
• Cylinder/cone extension (bank-driven, not in OPC): cylinder volume = πr²h; cone volume = (1/3)πr²h. Bank item: cone radius 10 cm, height 30 cm → volume = (1/3) × π × 100 × 30 = 1,000π cm³ (PPSC 2026). OPC’s claim that the box is “the only 3-D figure you need” is not followed — the banks ask cylinders, cones and spheres too.
• Pyramid (OPC p. 21, completing the volume family): volume = (1/3) × base area × height, written V = Ah/3 where A is the area of the base. Same one-third shape as the cone — a cone is, in effect, a pyramid with a circular base. Example: base area 60 cm², height 9 cm → volume = (1/3) × 60 × 9 = 180 cm³.

11H. Triangle classification — by side and by angle (extends Section 11B)

Section 11B teaches the 180° rule and the three families the banks name most. The OPC block (pp. 22–23) adds the full classification grid — the same triangles sorted two ways. Learn both sorts; a paper stem can name either one.

By side length:

• Equilateral: all three sides equal; all three angles equal, each 60°. Always equiangular, always acute.
• Isosceles: two sides equal, and the angles opposite those sides are equal. May be right, obtuse or acute — the equal-sides fact does not fix the angles.
• Scalene: all three sides different lengths, so all three angles are different too. May be right, obtuse or acute. (One bank stem asks exactly this definition — “What is a scalene triangle?” — in the subject-mixed practice set.)

By angle measure:

• Equiangular: all three angles equal, 60° each. Always equilateral, always acute — equiangular and equilateral are the same triangle described two ways.
• Right: one angle exactly 90° (see Section 11C for Pythagoras and the special right triangles).
• Obtuse: one angle greater than 90° (but less than 180°). An obtuse triangle can be isosceles or scalene — never equilateral, since 60° + 60° + an obtuse angle would exceed 180°.
• Acute: all three angles less than 90°. May be equilateral, isosceles or scalene.

Reading the grid: the two sorts cross. “Isosceles right” is one triangle in both sorts (two equal sides, one 90° angle — the 45-45-90 of Section 11C). “Scalene obtuse” is another (three different sides, one angle over 90°). If a stem gives a side fact and asks an angle fact (or the reverse), sort the triangle both ways before answering.

11I. Points, lines, planes and naming angles (OPC pp. 23–25)

The foundations Section 11A assumes. Nothing here is asked directly in the banks — it is taught so the angle and triangle items never stall on vocabulary.

• Point: a single location, drawn as a dot, labelled with a capital letter (Point P). It has no length, width or thickness — zero dimensions.
• Line: a straight path through at least two points, extending forever in both directions, with no thickness. Labelled by two points on it (line QR).
• Line segment: part of a line with two endpoints, so it has a fixed, measurable length. Labelled by its endpoints (segment CD).
• Ray: a straight path with one endpoint, extending indefinitely in one direction only. Labelled endpoint-first (ray EF, E the endpoint — never FE for the same ray). The OPC image: the sun as the endpoint, its rays going out endlessly.
• Plane: a flat surface with no boundaries, extending infinitely in all directions — a sheet of paper that never ends. Named by three non-collinear points (Plane GHI); three points on one line cannot name a plane.
• Parallel lines: in the same plane and never intersect, symbol ∥ (OPC’s scan prints it as “Il” — read it as ∥, e.g. JK ∥ LM). Lined-paper horizontals are the everyday example.
• Perpendicular lines: intersect at exactly 90°, symbol ⊥ (OPC’s scan prints it as “I” — read it as ⊥, e.g. NQ ⊥ PO; Section 11A already flags this misread). The margin line against the horizontals on lined paper is the everyday example.
• Naming an angle — the vertex goes in the middle. An angle is two rays with a common endpoint (the vertex). With three letters, the vertex is always the middle letter: ∠ABC and ∠CBA are the same angle (read counter-clockwise and clockwise), and ∠B alone names it only when no other angle shares vertex B. A stem that names an angle with the vertex anywhere but the middle is misprinted — or a trap.

11J. Angle types, angle relationships and the transversal conditions (OPC pp. 25–27)

• Angle types by size: acute — less than 90°; right — exactly 90°, often marked with a small square at the corner; obtuse — greater than 90° but less than 180°; straight — exactly 180°, forming a straight line.
• Supplementary angles: two angles whose measures add to 180° — together they make a straight line. Complementary angles: two angles whose measures add to 90° — together they make a right angle. The pair to keep straight: Supplementary = Straight = 180°; Complementary = Corner (right angle) = 90°.
• Vertical angles: when two lines intersect, the opposite angles are vertical angles — and vertical angles are equal. (Already in Section 11A; restated here because it is the fifth transversal condition’s cousin: equality, not sum.)
• The transversal conditions — when a transversal cuts two lines, the lines are parallel if ANY ONE of these holds:
  1. Corresponding angles are equal (same relative position at each intersection).
  2. Alternate interior angles are equal (opposite sides of the transversal, between the two lines).
  3. Alternate exterior angles are equal (opposite sides of the transversal, outside the two lines).
  4. Consecutive interior angles are supplementary (same side of the transversal, between the lines, summing to 180°).
  5. Consecutive exterior angles are supplementary (same side of the transversal, outside the lines, summing to 180°).
• Why the list matters: the conditions run both ways. Parallel lines cut by a transversal produce all five relationships, and any one of the five proves the lines parallel. A stem giving one relationship and asking “are the lines parallel?” needs only that one line of the list — do not demand all five. (The OPC scan’s numbering of this list is garbled — the five conditions above are the complete standard set, verified against the standard geometry definitions, not against the scan’s numbering.)

PART E — SETS: THE LANGUAGE OF COLLECTIONS

Sets are the fifth method block of this chapter, folded in from OPC Mathematics pp. 44–47 (the “SETS” block). Nothing in it was taught anywhere else in the notes — the chapter-check found zero set-theory text in any mathematics chapter (the word “set” in the banks is almost always a verb: a set of words, set up, a jewellery set) — so Part E teaches the whole block. No mark is claimed for it: the bank count below finds only three genuine set-theory rows, below both 🔁 limbs, so it is taught as method, with the three genuine bank items as its worked examples.

11K. Sets — definition, types, operations and the 2ⁿ counts (OPC pp. 44–47)

• Definition: a set is a well-defined collection of distinct objects, called elements (or members). Well-defined means there is no doubt whether something belongs: “the vowels” is a set, “the beautiful letters” is not. Elements are listed inside curly brackets: A = {1, 2, 3, 4}; V = {a, e, i, o, u}.
• Types of sets:
  • Finite set — a limited number of elements: A = {2, 4, 6, 8}.
  • Infinite set — an unlimited number of elements: N = {1, 2, 3, …} (the natural numbers).
  • Empty (null) set — no elements at all, symbol ∅ (also written {}). Example: the set of months with 32 days = ∅. A null set is also called an empty set — the FBR mixed paper asks exactly that (answer: Empty set).
  • Singleton set — exactly one element: A = {5}. Note the trap: {0} is a singleton (it contains the number 0); ∅ contains nothing — {0} ≠ ∅.
  • Universal set — symbol U: the set containing all elements under discussion in that problem. Every other set in the problem is a subset of U.
• Two ways to write a set:
  • Roster (tabular) method — list the elements: A = {1, 3, 5, 7}.
  • Set-builder method — describe the property: A = {x | x is an even number} (read: “the set of all x such that x is an even number”).
• Subsets: if every element of A is also an element of B, A is a subset of B, written A ⊆ B. A proper subset is a subset that is not the whole set itself — at least one element of B is missing from A — written A ⊂ B. Example: A = {1, 2}, B = {1, 2, 3} → A ⊂ B. Two facts that answer one-liners outright: ∅ is a subset of every set, and every set is a subset of itself (but not a proper subset of itself).
• Power set: the set of all subsets of a given set. Example: A = {a, b} → power set = {∅, {a}, {b}, {a, b}} — four subsets, and ∅ and {a, b} itself both count. If a set has n elements, its power set has 2ⁿ elements, and it has 2ⁿ − 1 proper subsets (all of them except the set itself).
  • Bank item (PPSC PMS 2019): “What is the total number of elements in the power set of the set {1, 2, 3}?” → n = 3 → 2³ = 8 (options 3, 6, 8, 9 — 6 is 3!, the factorial trap; 9 is 3²).
  • Bank item (PPSC PMS 2019): “How many subsets does an empty set have?” → n = 0 → 2⁰ = 1 — the empty set itself is its only subset (options 0, 1, 2, Infinite — 0 is the trap: ∅ has no elements, but it has one subset).
• Operations on sets (with A = {1, 2}, B = {2, 3} throughout):
  • Union (∪) — elements in A or B or both: A ∪ B = {1, 2, 3}. The common element 2 is written once — repetition is not allowed in a set.
  • Intersection (∩) — elements common to both: A ∩ B = {2}.
  • Difference (−) — elements in A but not in B: A − B = {1}. Difference is not symmetric: B − A = {3}.
  • Complement (A′) — elements in the universal set U but not in A. Complement only makes sense once U is fixed — the same A has different complements under different U.
• Venn diagram: the diagrammatic representation of sets using circles — overlapping circles for union and intersection, a rectangle for U. Used to solve logical and word problems; it is the picture form of Chapter 6 Section 4’s Venn arithmetic (fail Math 40%, fail English 30%, fail both 10% → pass 40%), where n(A ∪ B) = n(A) + n(B) − n(A ∩ B) is the same “add the two, subtract the overlap once” move.
• Disjoint sets: two sets with no common element — their intersection is empty. Example: A = {1, 3}, B = {2, 4} → A ∩ B = ∅.
• Cardinal number: the number of elements in a set, written n(A). Example: A = {2, 4, 6} → n(A) = 3.
• The formulas, together: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); number of subsets = 2ⁿ; number of proper subsets = 2ⁿ − 1.
• Exam points (OPC p. 47, all standard): ∅ is a subset of every set; the order of elements does not matter ({1, 2} = {2, 1}); repetition of elements is not allowed ({1, 1, 2} = {1, 2}); {0} is not equal to ∅.

PART F — GSA METHODS: PERMUTATIONS, PROBABILITY, LOGARITHMS, SEQUENCES & TRIGONOMETRY

The fifth method block of this chapter, folded in from the CSS GSA (General Science & Ability) maths portion — the converted CSS GSA 2024 maths set Q31–60 (files/css-mpt-papers/css-gsa-2024-maths-q31-60.json, 30 items) and the CSS 2025 GSA recall list (files/css-mpt-papers/css-2025-gsa-recall-list.json, 19 items). Chapter-check: no mathematics chapter taught any of these five as a method — MA-08’s master table carried only a logarithm one-liner, and Part D’s geometry covered the GSA triangle/circle/cylinder items, leaving exactly these five families untaught. No mark is claimed for this block: the GSA evidence is 9 items across the five families (permutations/combinations 2 — one 2024 permutation + one 2025 combination — probability 2, logarithms 1, sequences 3, trigonometry 1), below both 🔁 limbs, so it is taught as method, with the GSA items as its worked examples. The GSA coding item (Q57, +1 letter shift) is not filed here — it belongs to AR-06, not to mathematics.

11L. Permutations & combinations — order matters, or it does not

• Factorial: n! = n × (n − 1) × … × 1. 5! = 120, 3! = 6, 0! = 1 (by definition — the trap answer is 0).
• Permutations (order matters): arrangements of n distinct things taken r at a time: ⁿPᵣ = n! ÷ (n − r)!. All n arranged: n!.
• Repeated letters divide out: if a word has repeated letters, divide by the factorial of each repeat count — the commonest GSA trap is to forget the division and answer the raw factorial.
  • GSA item (2024, Q33): distinct arrangements of LEVEL → 5 letters, L twice, E twice → 5! ÷ (2! × 2!) = 120 ÷ 4 = 30 (options 20, 24, 30, 60 — 120 is the no-division trap, 60 divides by only one repeat).
• Combinations (order does not matter): ⁿCᵣ = n! ÷ (r! × (n − r)!).
  • GSA item (2025 recall): combinations of 9 things taken 3 at a time → C(9,3) = 9! ÷ (3! × 6!) = (9 × 8 × 7) ÷ (3 × 2 × 1) = 84.
• Permutation vs combination test: if swapping two chosen things makes a different answer (a word, a seating, a password), it is a permutation; if it makes the same answer (a team, a committee, a hand of cards), it is a combination — and combinations are always fewer, because they divide by r! as well.

11M. Probability — favourable over total, and the odds form

• Probability: P(event) = (favourable outcomes) ÷ (total outcomes). It always lies between 0 and 1.
  • GSA item (2024, Q51): probability of drawing an Ace from 52 cards → 4 ÷ 52 = 1/13.
• Odds in favour are not the probability: odds in favour = p ÷ (1 − p), written as a ratio.
  • GSA item (2024, Q35): p = 0.2 → odds in favour = 0.2 ÷ 0.8 = 1/4, i.e. 1 : 4 (options 1 : 2, 1 : 5, 1 : 4, 2 : 3 — 1 : 5 is the p : 1 trap, reading 0.2 as 1-in-5 without subtracting).
• Complement: P(not event) = 1 − P(event). If a question asks for “at least one”, compute 1 minus the “none” probability rather than adding cases.

11N. Logarithms — the power, written backwards

• Definition: log_b(x) = y means exactly b^y = x. The logarithm is the exponent. log₁₀(1000) asks “10 to what power is 1000?” — 10³ = 1000, so the answer is 3.
  • GSA item (2024, Q38): log₁₀(1000) = 3 (options 2, 2.5, 3, 4).
• The three moves: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n × log a. Base 10 is the default when no base is written; log₂x = 5 → x = 2⁵ = 32 (already in this chapter’s master table).
• Trap: log is not division — log(1000) is 3, not 1000 ÷ 10 = 100.

11O. Sequences — find the rule before you find the term

• Formula sequences: substitute n into the given rule.
  • GSA item (2024, Q37): a(n) = n² + 1 → a(5) = 25 + 1 = 26.
• Fibonacci-type sequences: each term after the first two is the sum of the previous two.
  • GSA item (2024, Q53): 2, 3, 5, 8, 13, ? → 8 + 13 = 21.
• Difference sequences: write the differences between consecutive terms; if the differences themselves form a pattern, extend that pattern.
  • GSA item (2024, Q59): 2, 5, 11, 20, ? → differences 3, 6, 9 → next difference 12 → 20 + 12 = 32.
• Working order: try differences first (one line), then sums-of-previous-two, then a formula — in that order, because the differences test is the fastest and catches the most GSA items.

11P. Trigonometry basics — degrees to radians, and the one conversion

• The conversion: π radians = 180°, so degrees × (π ÷ 180) = radians, and radians × (180 ÷ π) = degrees.
  • GSA item (2024, Q48): 150° → 150 × (π ÷ 180) = (150 ÷ 180)π = 5π/6 radians.
• The angles worth memorising in both units: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. The GSA item is a simplification after conversion — always cancel the fraction (150/180 = 5/6) before writing the answer.

12. Master table — every formula in this chapter

13. Rapid-fire one-liners

• A in 10 days, B in 15 days, together → 6 days (the most repeated time & work question in the banks).
• 3 men or 6 boys in 20 days → 6 men + 8 boys → 6 days.
• 4 men or 7 boys in 29 days → 12 men + 8 boys → 7 days.
• 6 women or 9 girls in 10 days → 8 women + 3 girls → 6 days.
• 20 men in 8 days → 16 men in 10 days; 10 men in 20 days → 8 men in 25 days.
• A + B in 10, B + C in 15, C + A in 30 → all three in 10 days (halve the pair-sum).
• Thrice-as-good A with B in 24 days → A alone in 32 days.
• Wages divide by rates, not days: A (24 days) and B (36 days) share Rs. 500 as Rs. 300 : Rs. 200.
• Pipes 36 h and 45 h together → 20 hours; fill 10 h, empty 15 h → 30 hours; pump 3 h, leak makes 3.5 h → leak empties in 21 hours.
• Three-worker system: A 4 h, B + C 3 h, A + C 2 h → B alone 12 hours.
• Twice as good, together 18 days → B alone 54 days; twice as good, A 30 days less → A 30 days, B 60 days, together 20 days.
• 25% more efficient, 20 days → 16 days, not 15 — divide by 1.25, never subtract 25%.
• 10 persons in 20 days → 20 persons at twice the efficiency → 5 days; 150 workers, 40 days, +30 after 16 days → 20 more days.
• Typing together in 30 minutes: A 60 + B 15 = 75 pages.
• 15 pumps in 7 days → to finish in 5 days need 6 extra pumps (21 total, not 21 extra).
• HCF × LCM = product of the two numbers; 18 and 144 → HCF 18, LCM 144.
• HCF of 12 and 18 = 6; LCM = 36; HCF of co-primes = 1.
• x + y = 10, x − y = 4 → x = 7, x² = 49; x + y = 99, x − y = 9 → 54 and 45.
• 4x + 12 = 36 → x + 3 = 9; x − 4 = 9 → x² − 4 = 165.
• 2x² − 3x − 2 = 0 → 2, −1/2; x² − 3x − 10 = 0 → 5, −2; 16x² + 24x + 9 = (4x + 3)².
• x³ + 1/x³ = 52 → x + 1/x = 4; 4^(2n+3) = 8^(n+5) → n = 9; log₂x = 5 → x = 32.
• √729 = 27; √2,025 = 45; √9,801 = 99; ∛2,744 = 14; ∛0.000216 = 0.06.
• (27)^(2/3) = 9; (256)^(5/4) = 1,024; x/√128 = √162/x → x = 12.
• Divisibility in one breath: 2 ones even · 3 digit-sum · 5 ones 0/5 · 10 ones 0 · 4 last two · 8 last three · 6 both 2 and 3 · 9 digit-sum · 11 alternating-sum.
• 2 is the only even prime; 0 and 1 are not prime. First primes: 2, 3, 5, 7, 11, 13, 17.
• Sum of first five primes = 28; primes between 64 and 81 = 4 (67, 71, 73, 79).
• 43,079 − 5 = 43,074, divisible by 9 (digit sum 18).
• Divisible by 8 ⇏ divisible by 6 (8 itself); divisible by 7 not 14 → odd multiple of 7.
• Remainder < divisor; numbers ≤ 200 divisible by 2 and 3 = 33 (200 ÷ 6).
• Grouping laws: 2 + (4 + 6) = 12; 2(4 × 6) = 48; 3(4 + 5) = 27; 4x + 4y = 4(x + y).
• Two-way move: 4(x + y) = 4x + 4y; (x + 4)(2x − 1) = 2x² + 7x − 4; x² − y² = (x + y)(x − y); x² + 2xy + y² = (x + y)².
• Factor first: y + 3 = 2x → 3y − 6x = −9 (factor 3(y − 2x), then use y − 2x = −3).
• Factorize x² + 16x + 63 = (x + 7)(x + 9) (7 + 9 = 16, 7 × 9 = 63).
• Names: binomial = 2 terms (x + 3); trinomial = 3 terms (x² + 3x + 2); degree of x² + 3x + 2 = 2.
• Geometry in one breath: triangle angles 180° · area (1/2)bh · Pythagoras c² = a² + b² · 3-4-5 → diagonal 5 · circle C = 2πr, A = πr² · r = 7 → 44 / 154 · square P = 4s, A = s² · box lwh · cube a³ · sphere (4/3)πr³ · cone (1/3)πr²h.
• Triangle area 126, base 28 → height 9; equilateral perimeter 60 → area 100√3; hexagon interior sum 720°; 1,260° → 9 sides.
• Coordinates: (−x, y) → quadrant II; (0, −7) → on the y-axis, no quadrant; (−6, 8) absolute sum → 14; square ABCD → D = (3, −1).
• Volumes: cube 343 → side 7, face 49; cone r = 10, h = 30 → 1,000π; pyramid base 60, height 9 → 180; Tehsildar 2024: base 8.8 cm, height 4.8 cm.
• Triangle grid: scalene = all sides different; obtuse = one angle over 90°; equiangular = equilateral = 60° each; isosceles right = 45-45-90.
• Foundations in one breath: point 0 dimensions · segment two endpoints · ray endpoint-first · plane three non-collinear points · vertex middle letter · supplementary 180° · complementary 90° · transversal: any one of corresponding / alternate-interior / alternate-exterior equal, consecutive supplementary ⇒ parallel.
• Sets in one breath: power set of {1, 2, 3} = 8 subsets · empty set has 1 subset (itself) · null set = empty set · A = {1, 2}, B = {2, 3} → union {1, 2, 3}, intersection {2}, A − B {1} · subsets 2ⁿ, proper subsets 2ⁿ − 1 · {0} ≠ ∅.
• GSA methods in one breath: LEVEL arrangements 30 (divide by both repeats) · C(9,3) = 84 · Ace probability 1/13 · odds in favour at p = 0.2 → 1 : 4 · log₁₀(1000) = 3 · 2, 3, 5, 8, 13 → 21 · 2, 5, 11, 20 → 32 · 150° = 5π/6.

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

• The 10-and-15 together question (6 days) in five papers — the single most repeated stem in this chapter; the 3-men-or-6-boys family (6 days) in four papers and the 4-men-or-7-boys family (7 days) in two.
• The man-days inversions (20 men/8 days → 16 men/10 days; 10 men/14 days → 4 men/35 days; 10 men/20 days → 8 men/25 days) — the papers test inverse proportion more than any other time & work idea.
• The pipe pairs (5 & 6 minutes → 30/11; 8 & 9 → 72/17; 11 & 13 → 143/24) — always the product-over-sum form; the emptying-pipe sign change (10 fill / 15 empty → 30 hours) is the variant that separates marks.
• HCF/LCM: one direct item only (18 and 144) — the method sections, not the banks, carry this block; any HCF/LCM question in the paper is won by the HCF × LCM relation or the divide-the-other rule.
• Algebra: the x + y / x − y pair (10/4 → x² = 49; 99/9 → 54, 45; 6/16 → 11, 5) and the 4x + 12 = 36 drill (x + 3 = 9, asked twice) — simultaneous pairs and “solve, then answer what was asked” are the two algebra shapes that recur.
• Roots the papers actually ask: √729, √2,025, √9,801, ∛2,744 — memorise these four square/cube roots and the factorisation method behind them.
• Number foundations (new Section 6A): divisibility tests — 30 genuine bank rows across 24 files (2015–2025 papers plus CSS MPT 2025), the test-the-number family (divisible by 7 not 14; 8-vs-6 logic; digit-sum subtraction for 9); primes — 10 rows across 10 files (smallest prime; sum of first five; primes between 64 and 81). Factors/multiples and parity rules are method, not directly asked — they are taught so the factorisation section never stalls.
• Grouping laws & naming (new Section 6B): no mark claimed — zero direct bank rows for associative, distributive or trinomial across 608 files / 58,098 questions; the two binomial hits are non-maths and the only genuine polynomial-degree item (linear = degree one, FPSC 2000) is already in Section 8. Taught as method for Sections 7, 10 and 11.
• Factoring & expanding (new Section 6C): no mark claimed — only two genuine maths stems in the family across 608 files / 58,098 questions: the factorize item (x² + 16x + 63, PPSC Enforcement Officer 2026, key (x + 7)(x + 9)) and the Junior Clerk 2025 difference-of-squares chain, whose printed stem is garbled (see Traps — taught as a stem-check lesson, not as a clean item). The Section 10 item (x − 2)(x + 3), PPSC 2026, is the same multiply-out move. Taught as method for Sections 7, 10 and 11; the expand bucket’s 20 rows are all non-maths (zoom, ECO, heated matter, antonym, CA expansion) and the factor bucket’s other 19 rows are non-maths (factors of production, vitamin factors, Herzberg, earthquake factors).
• Geometry (new Part D, Sections 11A–11G): 🔁 chapter-level on the paper-count limb + ⭐ still asked 2025–2026 — first-match rows over 608 files / 58,098 questions (script hidden_files/goldmine-notes-count-ma08-opc-geometry.py): circle 39 rows / 36 files (2 “circumference of earth” GK rows excluded from the taught count), perimeter-area 39 rows / 29 files, triangle 22 rows / 17 files, volume 32 rows / 22 files, Pythagoras 3 rows / 3 files (no mark — below both limbs, taught as method), coordinates 5 rows / 5 papers (🔁 on the paper-count limb), angles/lines 0 direct rows (taught as method for the triangle and circle items). The 2026 concentration is what earns the ⭐: triangle 6, circle 5, perimeter-area 9, volume 5 rows dated 2026. These rows are served as companion practice alongside this chapter’s 186 — bankCount 186 unchanged, because the first-match count mixes paper items with the subject-arithmetic practice sets (area-problems, volume-and-surface-area) and a clean de-duplication against other chapters’ counts has not been run; the chapter claims no inflated total.
• Geometry foundations (new Sections 11H–11J, OPC pp. 21–26): no mark claimed — fresh script hidden_files/goldmine-notes-count-ma08-opc-foundations.py over the same 608 files / 58,098 questions: scalene 1 row / 1 file (the “What is a scalene triangle?” practice-set stem — below both 🔁 limbs); obtuse/acute-triangle, angle-relations, transversal and pyramid-volume buckets 0 rows each; point/line/ray/plane bucket 3 rows, all false positives (area-problems line segments, a ratio-division stem, a segment-midpoint stem — none of them definitions). Taught as method for the triangle, circle and coordinate items; bankCount 186 unchanged. OPC pp. 27–30 (averages) NOT duplicated here — MA-06 Part C already teaches every move in that block (total from average, missing values, repeats, adding an element, zero-count, percentage-weighted averages, Sections 13–18), so the averages block is cross-linked to Chapter 6, not re-taught.
• Time & work drills (new Section 5A, OPC pp. 64–67): no mark claimed — fresh script hidden_files/goldmine-notes-count-ma08-opc-timework.py over the same 608 files / 58,098 questions: three-worker system 2 rows / 2 files (PPSC Assistant S&GAD 2022, PPSC Deputy Secretary Examination 2023 — the same stem, one item in two papers, below both 🔁 limbs); twice/thrice-as-good 3 rows / 3 files (Sub-Inspector 2019 twice-as-good 18-days stem = OPC Ex 6 itself; Assistant Finance 2020 “30 days less” variant; the thrice-as-good 24-days stem already taught in Section 2); efficiency-multiplier gang 1 row / 1 file (PPSC Assistant Board of Revenue 2021 — OPC Ex 11’s stem word for word); percentage-efficiency 0 rows; workers-added family 2 rows / 2 files (FPSC Assistant Director Intelligence Bureau 2016 and 2017 additional-workers stems — same family as OPC Ex 12, not its stem); typing-output bucket 2 rows, both different stems (FPSC 2005 boys/pages, FPSC 2010 train stoppages), so OPC Ex 14 is method, not a bank item. The four banked stems (five paper occurrences) are served as companion practice alongside this chapter’s 186 — bankCount 186 unchanged, no inflated total claimed.
• Sets (new Part E, Section 11K, OPC pp. 44–47): no mark claimed — fresh script hidden_files/goldmine-notes-count-ma08-opc-sets.py over the same 608 files / 58,098 questions: 3 genuine rows only — power set of {1, 2, 3} → 8 and empty-set subsets → 1 (PPSC PMS 2019, both keyed with the 2ⁿ explanation in the bank file) and null set = empty set (FBR mixed paper, keyed “Empty set”) — 2 papers, below both 🔁 limbs; Venn, subset/proper-subset, cardinal and roster/set-builder buckets return 0 rows. Taught as method: the 2ⁿ / 2ⁿ − 1 counts and the union/intersection/difference/complement operations are the whole examinable block. OPC pp. 47–50 (equations) checked against Sections 8–10 and Chapter 2 and NOT duplicated — linear, simultaneous and quadratic solving, substitution/elimination and the fraction-equation move are already taught there; the equations block’s only new residue is the name “transposition” (sign change on moving a term), added as one line to Section 8. OPC p. 50 (logical reasoning) NOT filed — the block opens on p. 50 and continues past it, so it is held whole for the next unit (pp. 51 onward), not filed as a cut-off definition.

15. Traps & confusions

• Together is not the average of the days: A (10) and B (15) together take 6 days, not 12.5. Rates add; days do not average.
• Pair-rates double-count: A + B, B + C, C + A rates sum to twice the three-together rate. Halve before inverting, or you answer 5 days instead of 10.
• Wages follow rates, not days: the faster worker earns more. Dividing Rs. 500 by days (24 : 36) gives the wrong split; dividing by rates (3 : 2) gives Rs. 300 : Rs. 200.
• “How many extra pumps” vs “how many pumps”: 21 pumps needed, 15 already there → 6 extra. The word “extra” is the whole question.
• OPC’s Q9 solution belongs to a different question — do not use it: OPC p. 66 prints, under “A + B in 12 days, B + C in 15, C + A in 20 — all three together?”, the fragment “A’s 1 hour work = 1/40, B’s = 1/60, (A + B) = 5/120 = 1/24, so all together in 24 hours”. Those numbers are a 40/60 pair’s solution spliced under the wrong question: the correct Q9 working is 1/12 + 1/15 + 1/20 = 12/60 = 1/5, halved (each worker counted twice) to 1/10 → 10 days — the same doubling move as Section 2’s 10/15/30 item. If 24 hours appears among the options for Q9, it is the book’s splice, not the answer.
• “25% more efficient” does not mean “25% fewer days”: OPC p. 67 prints 15 days for a 20-day worker at 25% more efficiency (20 − 25% of 20). Wrong: efficiency multiplies the rate, so it divides the time — 20 ÷ 1.25 = 16 days. At 15 days the worker would be 33⅓% more efficient, not 25%. Any option built by subtracting the percentage from the days is the trap, however clean it looks.
• Halve only the all-pairs pattern: Section 2’s pair-sums (every worker in exactly two pairs) halve the pair-sum; Section 5A’s three-worker system (one single rate given) subtracts instead. Halving the 5A system gives 1/3 ÷ 2 = 1/6 → 6 hours for B, which is wrong; subtracting gives 12 hours, which is right. The stem tells you which move: a single rate given means subtract.
• Emptying pipes subtract: a 10-hour fill with a 15-hour empty takes 30 hours, not 6. A minus sign missed is a question lost.
• HCF ≤ smallest, LCM ≥ largest: if your HCF exceeds the smallest number or your LCM is below the largest, you have swapped them — check before moving on.
• Fractions swap HCF and LCM: LCM of fractions uses the HCF of denominators; HCF of fractions uses the LCM of denominators. The swap is deliberate in the papers.
• Solve, then answer what was asked: 4x + 12 = 36 asks for x + 3 (9), not x (6); x − 4 = 9 asks for x² − 4 (165), not x (13). The last words of the stem decide the answer.
• Garbled keys exist — trust the working: the bank’s “2x − 10 = 5, then x − 5 = 5” key does not follow from its stem (correct: 2.5), and one three-equation stem keys a triple that fails its own second equation’s sign. Recompute in the exam; never memorise a key that fails its own arithmetic. The chapter’s worked examples all recompute cleanly.
• Squaring creates false roots: after solving 2√(x + 2) = 3√2 to x = 2.5, check it back (it checks). A squared equation can admit a root the original rejects — always verify.
• Discriminant < 0 means no real roots: “no solution” in the bank’s key for 2x² − √5x + 1 = 0 means no real solution, not no solution at all — the papers ask it at real-number level.
• “2x = 32, what is x²” is not 2x = 32: the stem is 2^x = 32 → x = 5 → x² = 25. A missing caret changes the question entirely — read exponents, don’t assume them.
• 1 is not prime — the commonest prime trap; neither is 0. 2 is prime and the only even one.
• Even/odd is not positive/negative: −4 is even, −3 is odd. The ones digit decides, not the sign.
• Divisible by 8 does not mean divisible by 6 — 8 itself is the counterexample. Divisibility by a bigger number never implies divisibility by a smaller one unless the smaller divides the bigger.
• “Divisible by 2 and 3 both” means divisible by 6 — take the LCM, then count or test once.
• Multiples in the papers are the positive ones — the book lists 10, 20, 30, 40, …; 0 never appears as a listed multiple in a paper item.
• Associative is grouping, not order: changing the order (a + b = b + a) is the commutative law, a different law — associative only re-brackets. And it never applies to subtraction or division: (10 − 4) − 2 = 4, not 8.
• Distributive sign and factoring traps: a(b − c) = ab − ac, never ab + ac; when factoring, divide every term — 4x + 4y = 4(x + y), not 4(x + 4y); and count terms by +/− splits, so 3x³ − 2x + 7 is three terms, not four.
• Garbled stem — check before you trust the key: the PPSC Junior Clerk 2025 item prints “Simplify: (x + 1/x)(x − 1/x)(x² − 1/x²” and keys (x⁴ − 1/x⁴), but as printed the product is (x² − 1/x²)² = x⁴ − 2 + 1/x⁴, which is none of the options. The key follows only if the third factor is (x² + 1/x²): (x + 1/x)(x − 1/x) = x² − 1/x², then (x² − 1/x²)(x² + 1/x²) = x⁴ − 1/x⁴ ✓. Lesson: when a printed stem cannot give any option, reconstruct the intended factor from the key before memorising anything.
• Direction errors: a factored answer to an “expand” question (and vice versa) loses the mark even when the algebra is right — read the verb in the stem first.
• OPC sphere-volume misprint — do not use it: OPC p. 20 prints sphere volume as (4πr²) ÷ 3. That is wrong: volume is (4/3)πr³ and surface area is 4πr². The misprint looks like a blend of the two — if it appears among the options, it is the book’s error, not a third formula.
• OPC ratio-order traps: the 30-60-90 ratio printed as “1 : 2 : √3” is only correct read as short : hypotenuse : longer leg — the standard order is short : longer leg : hypotenuse = 1 : √3 : 2. And the 45-45-90 ratio printed as “1 : √2” omits the equal short side: it is 1 : 1 : √2. Always label which side each number belongs to before using a ratio.
• π ≈ 3 is an estimate, not the value: OPC’s circle area for r = 5 is 75 with π ≈ 3 but 78.5 with π = 3.1416. If the options contain both, the question’s π decides — and for r = 7 the banks use 22/7 (44 / 154), where π ≈ 3 would give the wrong answers (42 / 147).
• Pythagoras needs a right angle: c² = a² + b² is false for non-right triangles. The 3-4-5 check (9 + 16 = 25) is the one-second test that a triangle is right-angled.
• Axis points are in no quadrant: (0, −7) sits on the y-axis. Any option naming a quadrant for it is wrong, however confident it looks.
• Perimeter-unit misprint: the PPSC 2026 stem “Perimeter of Square is 256cm2” prints a squared unit on a perimeter — the number is a perimeter (side 64), not an area. Read the quantity, not the printed unit.
• Radius scaling is not linear for area: doubling r quadruples area (×4), trebling it multiplies area by 9 — but circumference only doubles/trebles. Mixing the two laws is the circle section’s commonest error.
• Not every Pythagoras item is a clean triple: Tehsildar 2024 (hypotenuse 10, height 4 less than base) solves to 8.8 / 4.8, not integers. Set up the quadratic; do not force a 6-8-10 answer onto numbers that do not fit it.
• Vertex in the middle, always: ∠ABC means vertex B. A stem naming an angle with the vertex first or last (∠BAC for the angle at A) is either misprinted or testing exactly this rule — re-read before answering.
• Supplementary is not complementary: supplementary pairs sum to 180° (a straight line); complementary pairs sum to 90° (a right angle). The OPC block teaches both on facing pages — the exam mixes them deliberately.
• Scalene is not isosceles: “two sides equal” and “all sides different” are mutually exclusive. A triangle described with two equal sides can never be scalene, whatever its angles.
• ∥ is not “Il”, ⊥ is not “I”: the OPC scan prints the parallel symbol as “Il” and the perpendicular symbol as “I”. Read the symbol, not the scan’s letters — ∥ never meets, ⊥ meets at 90°.
• Alternate vs consecutive — equal vs sum: alternate angles (interior or exterior) are equal; consecutive angles (interior or exterior) are supplementary. Swapping the two halves of that sentence is the transversal section’s trap.
• One transversal condition is enough: parallel follows from any single condition in Section 11J’s list. A stem giving equal corresponding angles does not need the other four checked.
• Sets traps (Section 11K): ∅ has no elements but one subset (itself) — “how many elements” and “how many subsets” are different questions, and PMS 2019 asks both shapes. {0} is not ∅ — {0} is a singleton containing the number 0. A − B ≠ B − A — difference has a direction; with A = {1, 2}, B = {2, 3}, A − B = {1} but B − A = {3}. Union writes a common element once — A ∪ B = {1, 2, 3}, not {1, 2, 2, 3}. Complement needs U — without the universal set fixed, “not in A” has no boundary. 2ⁿ, not 2n or n² — a 3-element set has 8 subsets (2³), not 6 (2×3) and not 9 (3²); the PMS options plant both traps. ⊂ notation varies — this chapter uses ⊂ for proper subset and ⊆ for subset (the standard in most one-paper books); some texts use ⊂ for any subset, so read the stem’s own wording (“proper” is the deciding word, not the symbol alone). The word “set” in a stem is usually not set theory — “set up”, “a set of words”, “jewellery set” and Excel’s row-column “intersection” cell are all non-maths uses that the bank scan returns; only stems about elements, subsets, power sets or ∅ belong to Section 11K.

16. Memory tricks

• “Rates add, days don’t” — the whole of time & work in four words. Whenever two workers join, add their fractions of the job per day.
• “Man-days stay the same” — men up, days down, product fixed. If you can write men × days on both sides of an equals sign, the question is already half solved.
• “Fill plus, empty minus” — pipes are workers with signs. The emptying pipe is just a worker with a negative rate.
• “Efficiency up? Divide, don’t subtract” — a worker 25% more efficient takes the old time divided by 1.25 (16 days from 20), never the old time minus 25% (15). Percentages on efficiency divide; percentages on time subtract — and the stem gives the first, not the second.
• “HCF hugs the small, LCM leaps to the large” — HCF never exceeds the smallest number, LCM never falls below the largest. A one-second sanity check on every HCF/LCM answer.
• “HCF × LCM = product” — know any three of the four numbers and the fourth is one division away.
• “Solve, then substitute” — algebra questions ask for x + 3 or x² − 4 more often than for x itself. Solve first, then do the last step the stem actually asks for.
• “Add to kill y, subtract to kill x” — in x + y / x − y pairs, adding the equations kills y and gives x immediately. The pair is designed for it.
• “Ones, sum, 0-or-5, zero” — divisibility by 2 (ones even), 3 (digit sum), 5 (ones 0 or 5), 10 (ones 0). The four book tests in one breath; 4/8 read the last two/three digits, 6 needs 2 and 3, 9 reuses the digit sum, 11 alternates.
• “2 is the odd one out” — the only even prime, and the only even number in the first-primes list that breaks the odd run.
• “Group, don’t reorder; multiply through the bracket” — associative = re-bracket (+, × only); distributive = the outside number multiplies every term inside. “1-2-3: mono-bi-tri” — one term, two terms, three terms.
• “Factor first, substitute last” — if the asked expression shares a factor, pull it out before touching the given equation (3y − 6x = 3(y − 2x) = −9). “Expand back to check” — a factored answer that does not multiply back to the original is wrong, whatever it looks like.
• “180, 360, 90” — triangle angles 180°, full circle 360°, right angle 90°. Three numbers, every angle question starts from one of them.
• “3-4-5, spot it and stop” — if the sides (or their multiples) are 3, 4, 5, the triangle is right-angled and the diagonal/hypotenuse is already known. No quadratic needed.
• “Short, long, double” — 30-60-90 in one breath: shortest side, longer leg √3 times it, hypotenuse double it. “Equal, equal, root-two” — 45-45-90: two equal shorts, hypotenuse √2 times either.
• “Seven means 44 and 154” — radius 7 with π = 22/7 gives circumference 44 and area 154. The banks’ favourite radius, memorised as a pair.
• “Axis, no quadrant” — any zero coordinate puts the point on an axis, and axes belong to no quadrant. (0, −7) is the trap that proves you know it.
• “Sphere: cube the r, third the π” — (4/3)πr³: the r is cubed, the π is thirded. If your sphere formula has r² in it, you have written the surface area, not the volume.
• “Supple = Straight = 180, Comple = Corner = 90” — supplementary pairs make a straight line (180°); complementary pairs make a corner, a right angle (90°). The S/C initials match the words.
• “Vertex in the Middle” — in ∠ABC, B is the vertex because it sits in the middle of the name. Middle letter, middle point.
• “Cone’s cousin” — a pyramid is a cone with a polygonal base: both volumes are one-third base-area times height. If you remember the cone, you remember the pyramid.
• “Two to the n” — subsets are binary choices: each element is either in or out, so n elements give 2ⁿ subsets; proper subsets drop the “all in” choice, 2ⁿ − 1. “Empty still counts once” — ∅ has nothing in it, but it is itself a subset of everything, including itself.

17. Visuals (site build specs)

• DIAGRAM — The rate ladder: one job as a bar, split into daily slices for A (1/10), B (1/15) and A + B (1/6), showing why together is faster than either alone — placed in Section 1.
• INFOGRAPHIC — Pipes with signs: a tank with two inlet arrows (+) and one outlet arrow (−), each labelled with its rate, net rate shown at the tank’s base — placed in Section 5.
• TABLE — HCF vs LCM side by side: same pair (12, 18) worked by both methods, with the HCF × LCM = product check row underneath — placed in Section 7.
• DIAGRAM — The quadratic’s three cases: one parabola crossing twice (discriminant > 0), touching once (= 0), missing the axis (< 0) — placed in Section 10.
• TABLE — Divisibility tests at a glance: one row per test (2, 3, 4, 5, 6, 8, 9, 10, 11) with rule + one worked number — placed in Section 6A.
• DIAGRAM — Distributive arrow map: a outside the bracket with arrows to each inside term, plus the reverse factoring arrow (common factor pulled out) — placed in Section 6B.
• DIAGRAM — The two-way move: 4x + 4y ⇄ 4(x + y) with the expand arrow (distribute) and the factor arrow (pull out 4) side by side, plus the (x + 4)(2x − 1) four-product grid collapsing to 2x² + 7x − 4 — placed in Section 6C.
• DIAGRAM — The triangle family tree: one triangle split into equilateral (60° each), isosceles (two equal sides/angles) and right (hypotenuse longest), with the 180° sum at the root and the 3-4-5 triple beside the right branch — placed in Section 11B.
• DIAGRAM — The special-triangles pair: 30-60-90 (1 : √3 : 2) and 45-45-90 (1 : 1 : √2) side by side, each side labelled by the angle opposite it, with the OPC ratio-order warning as a caption — placed in Section 11C.
• DIAGRAM — The circle pair: one circle with radius, diameter and circumference arrows, and the r = 7 pair (44 / 154) worked beneath it — placed in Section 11D.
• DIAGRAM — The area table as shapes: square, rectangle, parallelogram, trapezoid, ellipse and the three triangle forms, each shape carrying its own formula — placed in Section 11E.
• DIAGRAM — The four quadrants: axes with origin (0, 0), each quadrant signed, (0, −7) marked on the y-axis outside all quadrants — placed in Section 11F.
• DIAGRAM — The volume shelf: box, cube and sphere side by side with formulas, the sphere’s (4/3)πr³ set against the struck-through OPC misprint (4πr²)/3 — placed in Section 11G (pyramid added to the shelf: V = Ah/3, beside the cone).
• DIAGRAM — The triangle grid: one triangle sorted two ways — by side (equilateral / isosceles / scalene) across the top, by angle (equiangular / right / obtuse / acute) down the side, with 45-45-90 sitting at the isosceles-right crossing — placed in Section 11H.
• DIAGRAM — The foundations strip: point (dot) → line (both arrows) → segment (two endpoints) → ray (one endpoint, one arrow) → plane (sheet, three points) in one row, with the ∠ABC vertex-middle callout beneath — placed in Section 11I.
• DIAGRAM — The transversal map: two lines cut by a transversal, all eight angles numbered, corresponding / alternate-interior / alternate-exterior pairs colour-matched as equal, consecutive pairs bracketed as supplementary — placed in Section 11J.
• DIAGRAM — The Venn pair: two overlapping circles for A = {1, 2} and B = {2, 3}, with the union {1, 2, 3} shaded as both circles, the intersection {2} as the overlap lens, and A − B = {1} as the left crescent — placed in Section 11K, with the power-set count 2ⁿ (n = 3 → 8, n = 0 → 1) as its caption.
• TABLE — The three-worker system as three equations: rows 1/A = 1/4, 1/B + 1/C = 1/3, 1/A + 1/C = 1/2, with the subtraction chain 1/C = 1/4 → 1/B = 1/12 → B = 12 hours — placed in Section 5A, beside the 25%-efficiency divide-vs-subtract contrast (20 ÷ 1.25 = 16, not 20 − 5 = 15).

18. Practice this topic

• Practice set: 186 verified questions are tagged to this topic’s sub-topics (time & work 64 paper items, HCF & LCM 1 paper item, algebra 59 paper items, plus the Pipes & Cisterns 16, Surds & Indices 16 and Square Roots & Cube Roots 30 set questions) — the site serves them as this chapter’s practice set. The three companion sets are served whole; the paper items come from the paper banks, since no FPSC set exists for time & work gangs, HCF/LCM or general algebra.
• Companion practice for Section 6A: divisibility-test items (30 genuine rows), primes items (10 rows) and factor/multiple stems from the verified banks, served alongside this chapter’s 186 — bankCount 186 unchanged (counting script hidden_files/goldmine-notes-count-ma08-opc-numbers.py over 608 files / 58,098 questions; 2 NFC “divisible pool” rows and 2 “factors of production” stems excluded as non-maths).
• Companion practice for Section 6B: no new bank items — associative/distributive/trinomial return zero direct rows (script hidden_files/goldmine-notes-count-ma08-opc-unit2.py over 608 files / 58,098 questions; 2 binomial hits excluded as non-maths, 7 of 9 polynomial/degree hits excluded as non-maths); Section 6B is method for the factorisation and algebra items already counted in this chapter’s 186 — bankCount 186 unchanged.
• Companion practice for Section 6C: two genuine bank stems only (script hidden_files/goldmine-notes-count-ma08-opc-factoring.py over 608 files / 58,098 questions): factorize x² + 16x + 63 (PPSC Enforcement Officer 2026, key (x + 7)(x + 9)) and the Junior Clerk 2025 difference-of-squares chain (taught as a garbled-stem lesson, see Traps); the Section 10 (x − 2)(x + 3) item (PPSC 2026) is the same multiply-out move and is already counted in this chapter’s 186 — bankCount 186 unchanged. Excluded as non-maths: all 20 expand rows (zoom, ECO, heated matter, antonym, CA expansion), 19 of 20 factor rows (factors of production, vitamin factors, Herzberg, earthquake factors), and all 19 squares-identity rows (perfect-square number items, not the identity).
• Companion practice for Part D (Sections 11A–11G): geometry rows from the verified banks, served alongside this chapter’s 186 — bankCount 186 unchanged (script hidden_files/goldmine-notes-count-ma08-opc-geometry.py over 608 files / 58,098 questions, first-match, q-text only): circle 39 rows (36 files; 2 “circumference of earth” GK rows excluded), perimeter-area 39 rows (29 files), triangle 22 rows (17 files), volume 32 rows (22 files), Pythagoras 3 rows (3 files), coordinates 5 rows (5 papers). Work the r = 7 circle pair, the 3-4-5 family and the quadrant items first — they are the shapes that recur in 2025–2026 papers. The subject-arithmetic practice sets (area-problems, volume-and-surface-area) sit inside these buckets; the count is first-match, so a stem matching two buckets is counted once, under the first bucket it matched.
• Companion practice for Sections 11H–11J: no new bank items — the foundations count (script hidden_files/goldmine-notes-count-ma08-opc-foundations.py over 608 files / 58,098 questions) returns 1 scalene definition stem, 0 rows for obtuse/acute-triangle, angle-relations, transversal and pyramid-volume buckets, and 3 false-positive line-segment rows; the scalene stem is served as the single companion item, the rest of Sections 11H–11J is method for the geometry items already counted — bankCount 186 unchanged.
• Companion practice for Part E (Section 11K, Sets): three genuine bank items only (script hidden_files/goldmine-notes-count-ma08-opc-sets.py over 608 files / 58,098 questions): power set of {1, 2, 3} → 8 and empty-set subsets → 1 (both PPSC PMS 2019, topic Set Theory), and null set = empty set (FBR mixed paper) — below both 🔁 limbs (2 papers), so no mark is claimed; the rest of Section 11K is method. Excluded as non-maths: all 8 union/intersection rows (chemical union, ECOWAS/Soviet “Union”, Excel row-column “intersection” cells) and the 150 bare “set”-word rows (set up, a set of words, a jewellery set) — none of them set theory. bankCount 186 unchanged.
• Companion practice for Section 5A (OPC Time & Work block, pp. 64–67): the four banked stems across five papers (PPSC Assistant S&GAD 2022 + Deputy Secretary Examination 2023 three-worker system; PPSC Sub-Inspector 2019 twice-as-good; PPSC Assistant Finance 2020 “30 days less” variant; PPSC Assistant Board of Revenue 2021 efficiency-multiplier gang) plus the OPC drill set itself (14 worked examples, pp. 64–67 — together-minus-one drills, the 25%-efficiency trap, workers-added-mid-job, typing output-rates), served alongside this chapter’s 186 — bankCount 186 unchanged (script hidden_files/goldmine-notes-count-ma08-opc-timework.py over 608 files / 58,098 questions; see Section 14 for the bucket counts and exclusions).
• Book mode: ← Previous: Profit & Loss; Simple & Compound Interest · Next: Pakistan — October 2026 Month File →

Goldmine Notes · Mathematics · Chapter 8 · Sources: verified PPSC/FPSC past-paper banks (script-counted, see practice block), FPSC Arithmetic companion sets Pipes & Cisterns (16), Surds & Indices (16) and Square Roots & Cube Roots (30). Section 6A from OPC Mathematics (Arslan Malik), opening number-foundations block pp. 1–3 + p. 9 remainder/counting rules (OCR-verified, hidden_files/final-notes-intake/work/maths-opc-pp1-10.txt). Section 6B from OPC Mathematics pp. 5 (associative/distributive laws, left-to-right rule) + 10 (binomial/trinomial/polynomial definitions, factoring example), same OCR source, standardised to the standard polynomial definition (one or more terms). Section 6C from OPC Mathematics pp. 10–13 (factoring tail completed p. 11, expand reverse, multiplying polynomials, the three identities, factor-first MCQ), OCR-verified, hidden_files/final-notes-intake/work/maths-opc-pp11-20.txt (pp. 11–20 OCR clean, no failed pages). Part D from OPC Mathematics pp. 13–20 (geometry block: angle/line facts, triangles, Pythagoras, special triangles, circles, rectangles/squares, box volume, coordinates, area table, Heron, volumes), same OCR source, with OPC errors corrected in the text — sphere volume (4πr²)/3 corrected to (4/3)πr³, 30-60-90 ratio reordered to 1 : √3 : 2, 45-45-90 ratio completed to 1 : 1 : √2, π ≈ 3 vs 3.1416 reconciled as estimate vs exact — plus bank-driven extensions (polygon interior-angle rule, cylinder/cone volumes) verified against the bank stems cited in the sections. Part D foundations (Sections 11H–11J + pyramid volume) from OPC Mathematics pp. 21–26 (cylinder/cone/pyramid volumes, perimeter drills, triangle classification by side and angle, points/lines/planes, angle types and relationships, transversal conditions), OCR-verified, hidden_files/final-notes-intake/work/maths-opc-pp21-30.txt (pp. 21–30 OCR clean, no failed pages); cylinder/cone volumes and perimeter drills were already taught in Sections 11E/11G and are not duplicated, and the OPC scan’s garbled transversal numbering is replaced with the standard five-condition set. OPC pp. 27–30 (averages) checked against MA-06 Part C and cross-linked there, not duplicated. Part E (Section 11K Sets) from OPC Mathematics pp. 44–47, OCR-verified, hidden_files/final-notes-intake/work/maths-opc-pp41-50.txt (pp. 41–50 OCR clean, no failed pages); OPC pp. 47–50 (equations) checked against Sections 8–10 and Chapter 2, not duplicated except the transposition name in Section 8; OPC pp. 50–64 (logical reasoning) filed in AR-05/AR-06 in the 2026-11-02 unit-8 pass, not here. Section 5A from OPC Mathematics pp. 64–67 (Time & Work rules + Examples 1–14), OCR-verified, hidden_files/final-notes-intake/work/maths-opc-pp61-70.txt (pp. 61–70 OCR clean, no failed pages); the block’s genuinely new residues only — three-worker system, twice-as-good shapes, percentage-efficiency trap, efficiency-multiplier man-days, workers-added-mid-job, output-rate addition, plus the same-method drill answers — with OPC errors corrected in the text (Q9’s spliced 40/60/24-hour solution replaced by the correct 10-day doubling working; the 25%-efficiency 15-day shortcut corrected to 16 days); the rest of pp. 64–67 duplicates Sections 1–5 and is not re-taught. OPC pp. 67–70 (Distance & Speed) and p. 70 onward (Age Problems) are NOT filed here — Distance & Speed is checked against MA-03 and Age against MA-04 in the next unit, which first needs OCR of pp. 71–80. Bank evidence for the foundations sections: script hidden_files/goldmine-notes-count-ma08-opc-foundations.py over 608 files / 58,098 questions — no mark claimed (see Section 14); sets bank evidence script hidden_files/goldmine-notes-count-ma08-opc-sets.py over the same 608 files / 58,098 questions — 3 genuine rows, no mark claimed (see Section 14). Standard answers only; ★ topic (no single FPSC set covers it) built from paper items + standard methods, every worked answer recomputed independently. Part F (GSA methods, Sections 11L–11P) from the converted CSS GSA 2024 maths set Q31–60 (30 items) + CSS 2025 GSA recall list (19 items) — permutations/combinations, probability, logarithms, sequences, trigonometry taught as method, 9 GSA items as worked examples, no mark claimed (script hidden_files/goldmine-notes-count-ma08-gsa.py); GSA coding item Q57 left to AR-06, not filed here; bankCount 186 unchanged (GSA items are companion practice). Mathematics subject COMPLETE (8/8 topics).