Mathematics — Chapter 6

Percentage, Ratio & Average — The Big Three Shortcut Families

Why this chapter matters: No FPSC arithmetic set exists for these three families — that is exactly why they are a ★ chapter — yet the papers ask them constantly: our verified banks hold 354 topic-relevant paper items for this chapter (percentage 249, ratio and proportion 42, average 63), the largest paper-only block in the mathematics subject so far. Percentage alone outnumbers every other paper block in the subject. The three families are also the only arithmetic where the papers repeat the same numbers across years: “45% failed, 550 passed” appears in four papers, “10% income tax = Rs. 1,500” in five, “7% is 42” and “17% is 51” in four each, and the “first three average 15, last three average 16, last number 19” question in five papers. Learn the handful of methods below and these 354 items become the fastest marks in the paper.

PART A — PERCENTAGE: EVERYTHING IS “OUT OF 100”

1. The one idea

Percent means per hundred. x% of a quantity = (x ÷ 100) × quantity. Every percentage question is one of three questions, and the first skill is recognising which one is being asked:

• Find the part: “What is 75% of 4,560?” → 0.75 × 4,560 = 3,420.
• Find the percent: “178 marks out of 300 — what percentage?” → 178 ÷ 300 × 100 = 59.33%.
• Find the whole: “The number, whose 7% is 42, is…” → 42 ÷ 0.07 = 600.

The papers’ favourite whole-finding drill is a whole family of identical questions — learn the division once and all of them fall: 7% is 42 → 600; 17% is 51 → 300; 25% is 150 → 600; 9% is 63 → 700; 5% is 10 → 200; 13% is 52 → 400; 3% is 60 → 2,000; 5% is 35 → 700; 5% is 55 → 1,100; 25% is 75 → 300; 20% is 100 → 500; 7% is 14 → 200; 7% is 21 → 300. And the reverse drill: 6 is 24% of a number → the number is 25, so 40% of it is 10; 60% of a number whose 80% is 320 → the number is 400, so 60% is 240; 40% of 180 = 72; 150% of 48 = 72; 250% of 36 = 90; 60% of 70 = 42; 80% of 700 = 560; 45% of 600 = 270; 2% of 400 = 8; 7% of 42 = 2.94.

Conversions to learn cold (the papers test them as one-liners): 8% = 0.08; 0.02 = 2%; 0.036 = 3.60%; 25% = 1/4; 5% = 1/20; 12.5% = 1/8 (so reducing a number by 12.5% means multiplying by 7/8); 10% of 20% of 30% = 0.6%; 20% of 30% of 20% of Rs. 850 = Rs. 10.20; 5% of (25% of Rs. 1,600) = Rs. 20; 63% of 25/7 = 2.25; 218% of 1,674 = 2.03 × 1,800 (about 2.03); 35% of 30 = 25% of ? + 1 → ? = 38; ?% of 6,144 = 5/2 × 245.76 → 10%; (25/3)% of ? = 150 → ? = 1,800; 24% of 44 + 12% of 22 = 13.2; 4% of 200 × 25.5 = 204; 12% of 1,200 = 144.

The fraction table that saves time: 10% = 1/10, 20% = 1/5, 25% = 1/4, 33⅓% = 1/3, 50% = 1/2, 66⅔% = 2/3, 75% = 3/4, 80% = 4/5, 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8, 87.5% = 7/8. A question that says “37½% of a number is 45” is asking for 3/8 of the number = 45 → the number is 120, and 87½% of it is 105 — in fractions, not decimals.

1A. Percent ↔ fraction ↔ decimal — the conversion drills (OPC pp. 41–43)

Section 1 taught the table; this section teaches the two conversions as moves, with the book’s worked drills. No mark is claimed for it: the bank count over 608 files / 58,098 questions (script hidden_files/goldmine-notes-count-ma08-opc-sets.py, percent buckets) finds zero “x% of y / y% of x” symmetry rows and zero “11 1/9%” rows — the drills below are method, not a repeated-paper family — and the fraction table itself is already Section 1’s, so only the genuinely new drills are filed here.

• Percent → fraction: write the percent over 100, then reduce. 40% = 40/100 = 4/10 = 2/5.
• Fraction → percent: divide numerator by denominator, then move the decimal point two places right. 4/5 = 0.8 = 80%. 3/8 = 0.375 → 37.5% (3/8 × 100% = 300/8% = 37½% — the book’s worked drill, answer (c)).
• The decimal equivalents to hold with the fraction table: 0.01 = 1/100 = 1%; 0.1 = 1/10 = 10%; 0.2 = 1/5 = 20%; 0.25 = 1/4 = 25%; 0.5 = 1/2 = 50%; 0.75 = 3/4 = 75%. Read any column of the three and the other two follow — a stem giving a decimal and asking for a percent is a two-place shift, not a calculation.
• The fraction-percent drills the table makes one-step: 33⅓% of 36 = 1/3 × 36 = 12; 11 1/9% of 171 = 1/9 × 171 = 19 (11 1/9% is the percent form of 1/9 — the ninth family Section 1’s table does not list, so learn it as a pair: 1/9 ↔ 11 1/9%).
• The symmetry identity: x% of y = y% of x. Proof in one line: x% of y = (x/100) × y = xy/100, and y% of x = (y/100) × x = yx/100 — the same product. So if A = x% of y and B = y% of x, then A = B (the book’s drill, answer “None of these” — none of “A smaller”, “B smaller” or “cannot be determined” is true). Use it whenever the awkward percent sits on the easy number: 32% of 50 = 50% of 32 = 16, no multiplication needed.
• A decreased-number drill in the complement form (Section 4’s move, new numbers): a number decreased by 20% gives 720 → the number after the decrease is 80% → 1% = 720 ÷ 80 = 9 → the original = 900. Same move as Section 4’s “sells 40%, 420 left → 700” — the decreased percent is the complement, and the complement is the divisor.

2. “More than” and “less than” — the base is everything

“A is x% more than B” and “B is x% less than A” are NOT symmetric — the base changes, and the papers test exactly that:

• A’s income is 25% more than B’s → B’s income in terms of A’s = 80% (B = 100, A = 125, 100 ÷ 125 = 80%). The reverse: income of A is 25% less than B → B’s income is 33⅓% more than A’s (A = 75, B = 100, 25 ÷ 75 = 33⅓%).
• What number is 20% more than Rs. 9,000? → 10,800. What number is 60% less than 900? → 360.
• Nasir read 60 pages on day one — 20% more than day two → day two = 60 ÷ 1.20 = 50 pages.
• The number of shells in Javaid’s collection is 80% of Justin’s; Justin has 80 more shells than Javaid → Justin = 80 ÷ 0.20 = 400, Javaid = 320.
• Two numbers are respectively 20% and 50% more than a third number → their ratio is 120 : 150 = 4 : 5 (this is the bridge into Part B — a percentage comparison is a ratio).
• Two numbers X and Y are 20% and 28% less than Z; by what percentage is Y less than X? → X = 80, Y = 72, difference 8 on a base of 80 = 10%.
• The price of a table is Rs. 400 more than a chair; 6 chairs cost Rs. 4,800 → chair = 800, table = 1,200, so the chair is 400 ÷ 1,200 = 33⅓% less than the table. (A second paper’s variant — 6 tables and 6 chairs together cost Rs. 4,800 — gives chair 200, table 600, and the chair 66⅔% less; same method, different numbers. Always recompute from the stem, never from memory of the other variant.)
• A man saves Rs. 500, which is 15% of his annual income → income = 500 ÷ 0.15 = Rs. 3,333.33. A man pays 10% of his income as income tax; the tax is Rs. 1,500 → income = Rs. 15,000. Yousaf paid Rs. 18,350 tax at 40% → income = Rs. 45,875. An amount taxed at 3% yields Rs. 75 → the amount is Rs. 2,500. 15% of sales tax on an article is Rs. 1,500 → the price is Rs. 10,000. A person paid 2% commission on land worth Rs. 6 crore 75 lakh → commission = Rs. 13.5 lakh.

3. Increase then decrease — the net-change formula

Successive changes multiply, they do not add. A +x% change followed by a −y% change gives a net change of (x − y − xy/100)%:

• A number increased by 10% and then decreased by 10% → net = 10 − 10 − 100/100 = 1% decrease. (The papers’ most quoted percentage fact.)
• Water tax up 20%, consumption down 20% → expenditure change = 1.20 × 0.80 − 1 = 4% decrease.
• TV prices up 30%, sales down 20% → revenue change = 1.30 × 0.80 − 1 = 4% increase.
• Price of sugar up 20%, consumption down 25% → expenditure change = 1.20 × 0.75 − 1 = 10% decrease.
• Price of sugar up 32%, expenditure allowed up only 10% → consumption must change by 1.10 ÷ 1.32 − 1 = 16.67% decrease. (The same shape with 25% price rise and no expenditure increase: consumption down by 1 − 1/1.25 = 20%.)
• Length +20%, width −20% → area change = 1.20 × 0.80 − 1 = 4% decrease. Length and breadth of a square +40% and +30% → area exceeds by 1.40 × 1.30 − 1 = 82%. Length +60%, area unchanged → width must fall by 1 − 1/1.60 = 37.5%. Each side of a rectangular field diminished by 40% → area diminished by 1 − 0.60² = 64%. Radius of a sphere +10% → surface area up by 1.10² − 1 = 21%. A square’s sides trebled → area becomes 9 times, an increase of 800%. Width twice the length; length +30%, width +20% → area change = 1.30 × 1.20 − 1 = 56%. Area of a plot up 30%, breadth unchanged → new : old areas = 13 : 10.
• A machine bought for Rs. 32,000 falls by 5% every year; after 2 years → 32,000 × 0.95² = Rs. 28,880. Repeated percentage change is compound multiplication — the same idea as compound interest in Chapter 7, but the papers ask it here as pure percentage.
• A ball-pen factory reduces production by 10% each month for 3 months → after three months production is 0.90³ = 72.9% of the start — a 27.1% fall in total, not 30%.

4. Elections, exams and pass/fail — the complement and the Venn

The complement is the shortcut: if 45% failed, 55% passed; if 65% are rural, 35% are urban.

• In the exam 45% failed and 550 students were successful → total = 550 ÷ 0.55 = 1,000. (Four papers ask this exact question.)
• 60% of employees went to a meeting; 28 did not go → total = 28 ÷ 0.40 = 70.
• A fruit seller sells 40% of his apples and still has 420 → original = 420 ÷ 0.60 = 700.
• After spending 88% of his income a man has Rs. 2,160 left → income = 2,160 ÷ 0.12 = Rs. 18,000. 30% of the total amount is left after spending Rs. 280 → total = 280 ÷ 0.70 = Rs. 400. 10% deducted from a bill, Rs. 585 remains → bill = 585 ÷ 0.90 = Rs. 650. A motor cycle sold at Rs. 68,000, which is 15% off the original price → original = 68,000 ÷ 0.85 = Rs. 80,000.
• A student must secure 40% to pass; he gets 40 marks and fails by 40 → pass mark = 80 = 40% of the total → total = 200.
• In an election between two candidates, one got 55% of the valid votes and 20% of the votes were invalid; total votes 7,500 → valid votes = 6,000, winner = 3,300, loser = 2,700, margin = 600. The two-step trap is the invalid votes: 55% is of the valid votes, not of 7,500.
• 68 votes invalid; winner scores 52% and wins by 98 votes → margin = 4% of valid votes = 98 → valid votes = 2,450, total votes = 2,518.
• In an election, 75% of voters cast votes, 2% of those invalid, and a candidate’s 9,261 votes are 75% of the valid votes → valid votes = 9,261 ÷ 0.75 = 12,348; votes cast = 12,348 ÷ 0.98 = 12,600; total enrolled = 12,600 ÷ 0.75 = 16,800. (Work backwards from the percentage that is given, one step at a time — the invalid votes come off the cast votes, and the cast votes are 75% of the enrolled.)
• In an examination, 40% fail Mathematics, 30% fail English, 10% fail both → fail at least one = 40 + 30 − 10 = 60% → pass percentage = 40%. (A second paper’s variant with 5% failing both gives pass = 100 − (40 + 30 − 5) = 35% — the “both” is subtracted once, not twice.)
• 70% passed Paper I, 60% passed Paper II, 15% failed both → passed both = 70 + 60 − 85 = 45%.
• Every candidate took Physics or Mathematics or both; 65.8% took Physics, 50.2% took Mathematics, total 2,000 → both = 65.8 + 50.2 − 100 = 16% → 320 candidates.
• In a class of 200 students, 120 play soccer and 100 play cricket, every student plays at least one game → both = 20, cricket only = 80 → 40% play cricket but not soccer.
• In a town, 65% watch TV news, 40% read a newspaper, 25% do both → neither = 100 − (65 + 40 − 25) = 20%.
• In a class, 20% own two cars each, 40% of the remainder own three cars each, the rest own one car each → average cars per member = 0.20 × 2 + 0.32 × 3 + 0.48 × 1 = 1.84 (this is also a weighted average — see Section 18).
• A school’s four classes have 10, 20, 30, 40 students with pass percentages 20%, 30%, 60%, 100% → passes = 2 + 6 + 18 + 40 = 66 of 100 → overall pass percentage = 66% (weighted by class size, not a plain average of 20/30/60/100 = 52.5%).
• Three candidates received 1,136, 7,636 and 11,628 votes → total 20,400; shares = 5.57%, 37.43%, 57% — compute each share against the total you just found, never against a rounded total.

5. Percentage + ratio hybrids — where Parts A and B meet

• Salaries of A, B, C are in the ratio 2 : 3 : 5 with increments of 15%, 10%, 20% → new salaries = 2 × 1.15 : 3 × 1.10 : 5 × 1.20 = 2.3 : 3.3 : 6.0 = 23 : 33 : 60. Apply the percentage to each part, then re-read the ratio.
• Salaries of A, B, C are in the ratio 1 : 2 : 3; B and C together earn Rs. 60,000 → 5 parts = 60,000 → A = 12,000, C = 36,000 → C is (36,000 − 12,000) ÷ 12,000 = 200% more than A.
• Two persons A and B get the same salary; their allowances are 65% and 80% of their basic pays → basic₁ × 1.65 = basic₂ × 1.80 → basic₁ : basic₂ = 180 : 165 = 12 : 11.
• 80% of the applications were rejected → accepted : rejected = 20 : 80 = 1 : 4.
• In a club the ratio of boys to girls is 5 : 3 → girls are 3/8 = 37.5% of members. Team juniors : seniors = 3 : 5 → seniors are 5/8 = 62.5% of the team. A ratio converts to a percentage by dividing the part by the total parts.
• Two numbers A and B: 5% of A + 4% of B = ⅔ of (6% of A + 8% of B) → 15A + 12B = 12A + 16B → 3A = 4B → A : B = 4 : 3. Percentages inside an equation are just coefficients — clear the percent signs first.

6. Mixture by percentage — alligation in one line

When two mixtures combine, the final percentage is the weighted average of the two percentages, weighted by quantity:

• How many kg of pure salt must be added to 30 kg of a 2% salt solution to make a 10% solution? → (0.6 + x) ÷ (30 + x) = 0.10 → x = 8/3 kg. The salt already present (0.6 kg) counts — the classic error is adding salt as if the solution were pure water.
• A milk vendor has two cans: the first 25% water, the second 50% water. Mixing equal quantities gives (25 + 50) ÷ 2 = 37.5% water; mixing in any other ratio, weight each percentage by its quantity before averaging. The ratio version of the same idea is in Section 13.

PART B — RATIO & PROPORTION: DIVIDE FIRST, ASK QUESTIONS LATER

7. The one idea

A ratio is a division. a : b means a/b, and the two rules that solve every ratio question are: (1) both terms must be in the same unit before you divide, and (2) the total is the sum of the parts — a share = (its parts ÷ total parts) × the whole.

• The ratio of 3.5 kg to 280 g → 3,500 : 280 = 25 : 2. (Convert first: the papers’ most common ratio trap is mixed units.)
• The ratio of 50 ml to 2 litres → 50 : 2,000 = 1 : 40.
• The total number of boys and girls in a class is 16 and 12 → simplest ratio = 4 : 3.
• If 4a = 8b, the ratio a : b → a/b = 8/4 = 2 : 1. (From the equation, cross-multiply: a : b = coefficient of b : coefficient of a.)

8. Dividing a quantity in a ratio

• Rs. 1,581 divided among A, B, C in the ratio 10 : 15 : 6 → 31 parts, 1 part = 51, B’s share = 15 × 51 = Rs. 765.
• Rs. 60,000 distributed in the ratio 3 : 5 : 7 → 15 parts, 1 part = 4,000, difference between largest and smallest = 4 × 4,000 = Rs. 16,000.
• A sum distributed among A, B, C, D in the proportion 5 : 2 : 4 : 3; C gets Rs. 1,000 more than D → C − D = 1 part = 1,000 → B’s share = 2 × 1,000 = Rs. 2,000. When a difference is given, find one part from the difference of parts, not from the total.
• In a school, the ratio of male to female teachers is 5 : 9; females are 27 → 9 parts = 27 → 1 part = 3 → males = 15.
• Doctors and nurses in a hospital total 240, in the ratio 5 : 7 → 12 parts, 1 part = 20 → doctors = 100.
• If the ratio of rice to wheat is 3 : 4 and rice is 21 → wheat = 21 ÷ 3 × 4 = 28. Sugar : flour = 2 : 3, sugar 18 → flour = 27. Sugar : water = 5 : 7, sugar 35 → water = 49. One part first, then multiply — never set up a fraction against the unknown.
• An 80 cm elastic is increased in the ratio 9 : 8 → new length = 80 × 9/8 = 90 cm. “Increased in the ratio a : b” means multiply by a/b.
• A truck covers 640 km in 10 hours, a car the same distance in 8 hours → speeds 64 and 80 km/h → ratio of speeds = 4 : 5.

9. Ratio chains — one shared term, one combined ratio

When two ratios share a term, make that term’s number equal in both (the same skill as partnership’s ratio chains in Chapter 5):

• If x : y = 6 : 5 and z : y = 9 : 25 → make y = 25: x : y = 30 : 25, so x : z = 30 : 9 = 10 : 3.
• If a : b : c = 3 : 4 : 7, then (a + b + c) : c = 14 : 7 = 2 : 1.
• The ratio of three numbers is 1 : 2 : 3 and their product is 750 → 1x × 2x × 3x = 6x³ = 750 → x³ = 125 → x = 5 → largest = 15.
• The sum of the ratio 1 : x : 5 is 11 → x = 11 − 6 = 5.
• If the product of two numbers is 320 and their ratio is 1 : 5 → numbers are x and 5x, 5x² = 320 → x = 8 → numbers 8 and 40 → difference of squares = 1,600 − 64 = 1,536.
• The monthly incomes of Peter and Paul are in the ratio 4 : 3, their expenses in the ratio 3 : 2, and each saves Rs. 600 → incomes 4x and 3x; expenses (4x − 600) : (3x − 600) = 3 : 2 → 8x − 1,200 = 9x − 1,800 → x = 600 → incomes Rs. 2,400 and Rs. 1,800. Two ratios, two unknowns, one equation each — the savings are the bridge.
• The expenses of A and B are in the ratio 2 : 3; Rs. 2,800 is divided equally between them; A saves Rs. 600 → each gets 1,400; A spends 800; B spends 800 × 3/2 = 1,200 → B saves Rs. 200.

10. Proportion — the fourth term and the coin problem

a : b :: c : d means ad = bc. The fourth proportional to 2, 3, 6 is 9: 2 : 3 :: 6 : 9, since 2 × 9 = 3 × 6 = 18.

• Coins of Rs. 5, Rs. 2, Re. 1 and 50 paisa in the proportion (by count) 1 : 5 : 2 : 6 make Rs. 1,900 → value per unit set = 5×1 + 2×5 + 1×2 + 0.5×6 = 20 → 20x = 1,900 → x = 95 → 95 five-rupee coins. Count-proportions and value-proportions are different — convert counts to values first.

10A. Third and mean proportionals — the two proportionals Section 10 did not yet teach (OPC pp. 36–37)

Section 10 taught only the fourth proportional. Proportion has two more named forms, and the book’s worked examples below are the chapter’s first teaching of them. No mark is claimed for this section: the bank count finds zero “third proportional” rows, zero “fourth proportional” phrase rows, and only two “mean proportional” rows in the verified banks (both cited below) — below both 🔁 limbs — so this section is taught as method, not as a repeated-paper family.

• Third proportional: a : b :: b : c → c = b² ÷ a. The middle term appears twice. Third proportional to 63 and 29.9 → 29.9² ÷ 63 = 894.01 ÷ 63 = 14.19 (OPC p. 37, answer (c)). Check the square first: 29.9² = (30 − 0.1)² = 900 − 6 + 0.01 = 894.01 — if the square is wrong, everything after it is wrong.
• Mean proportional: a : x :: x : b → x = √(a × b). The answer sits between the two numbers. Mean proportional between 7,921 and 9,409 → √(7,921 × 9,409) = 89 × 97 = 8,633 (OPC p. 37, answer (b)) — because √7,921 = 89 (89² = 7,921) and √9,409 = 97 (97² = 9,409). When both numbers are perfect squares, take the two roots and multiply; do not multiply first and then root a seven-digit product.
• The banks’ own mean-proportional pair (the only two rows found): mean proportional of 4 and 16 = √64 = 8 (PPSC ASI Anti-Corruption paper 2020); mean proportional between 2 and 8 = √16 = 4 (PPSC Junior Clerk Board of Revenue paper 2021). Same move, smaller numbers — the papers ask the definition, not the big arithmetic.
• Fourth proportional, decimal drills (OPC pp. 36, 38 — same ad = bc move as Section 10, with decimals): fourth proportional to 2.4, 4.6, 7.6 → (4.6 × 7.6) ÷ 2.4 = 34.96 ÷ 2.4 = 14.57 to two decimal places; fourth proportional to 0.5, 1.12, 1.3 → (1.12 × 1.3) ÷ 0.5 = 1.456 ÷ 0.5 = 2.912 exactly. See the Traps note on the book’s printed 14.56 for the first drill.
• Which proportional is being asked? “Third proportional to a and b” gives the third term (b² ÷ a); “mean proportional between a and b” gives the middle term (√(ab)); “fourth proportional to a, b, c” gives the fourth term (bc ÷ a). Three names, three formulas — read the name, not just the numbers.

11. Ratio in geometry and measurement

• The three angles of a triangle are in the ratio 1 : 1 : 2 → angles 45°, 45°, 90° → the triangle is isosceles and right-angled. (Two equal parts → two equal angles → isosceles; the parts sum to 180°.)
• The length of a rectangular plot is reduced to 3/5 of its original length → original : reduced area = 5 : 3 (breadth unchanged, so area follows length).
• The diameters of two circular coins are in the ratio 1 : 3 → their circumferences are in the ratio 1 : 3, and the smaller coin rolling around the bigger makes 3 + 1 = 4 revolutions (the +1 is the coin’s own rotation around the centre — the papers’ coin-rolling trap).
• The curved surface area of a cylindrical pillar is 264 m² and its volume 924 m³; the ratio of diameter to height → volume ÷ curved surface = (πr²h) ÷ (2πrh) = r/2 = 924 ÷ 264 = 3.5 → r = 7, diameter 14; from curved surface, 2π × 7 × h = 264 → h = 6 → diameter : height = 14 : 6 = 7 : 3.
• The volumes of two cubes are in the ratio 8 : 1 → edges in the ratio 2 : 1 (cube root), and surface areas in the ratio 4 : 1 (square). Volume ratios cube, area ratios square, length ratios stay — never mix the powers.
• The area of a triangle is 216 cm² and its sides are in the ratio 3 : 4 : 5 → a 3-4-5 triangle is right-angled, area = ½ × 3x × 4x = 6x² = 216 → x = 6 → sides 18, 24, 30 → perimeter = 72 cm.
• The ratio of the circumference of a circle to its radius = 2π (circumference = 2πr, divided by r).

12. Mixture by ratio — alligation without the diagram

• Tea worth Rs. 126/kg and Rs. 135/kg mixed with a third variety in the ratio 1 : 1 : 2; the mixture is worth Rs. 153/kg → (126 + 135 + 2p) ÷ 4 = 153 → 2p = 351 → p = Rs. 175.50/kg. The ratio is the weighting: the third variety counts twice.
• One quality of wheat at Rs. 9.30/kg mixed with another in the ratio 8 : 7; the mixture is worth Rs. 10/kg → (8 × 9.30 + 7p) ÷ 15 = 10 → 74.4 + 7p = 150 → p = Rs. 10.80/kg. The mixture price (10) lies between the two prices (9.30 and 10.80) — if your computed price falls outside the two given prices, you have inverted the ratio.
• A mixture contains alcohol and water in the ratio 12 : 5; on adding 14 litres of water the ratio becomes 4 : 3 → alcohol unchanged: 12x ÷ (5x + 14) = 4/3 → 36x = 20x + 56 → x = 3.5 → original mixture = 17 × 3.5 = 59.5 litres. In “add water” problems the alcohol (or the dry quantity) is the anchor — it does not change.
• Two vessels A and B contain spirit and water in the ratios 5 : 2 and 7 : 6; in what ratio must they be mixed to get spirit : water = 8 : 5? → spirit fractions: A = 5/7, B = 7/13, target = 8/13. Alligation: (target − B) : (A − target) = (8/13 − 7/13) : (5/7 − 8/13) = (1/13) : (9/91) = 7 : 9. The alligation cross is just “how far is the target from each side, swapped” — the side the target is nearer to gets the larger share (here the target 8/13 is nearer B’s 7/13, so B’s share, 9, is the larger).

PART C — AVERAGE: TOTAL ÷ COUNT, AND THE BALANCING POINT

13. The one idea

Average = total ÷ count. Every average question is one of three moves: find the total (average × count), find the count (total ÷ average), or find the missing item (required total − known total). The second idea: the average is the balancing point — amounts above it and below it cancel. Both ideas together solve the whole family.

14. Find the missing item — the required-total method

• A student gets 75, 82, 86 on three tests; what must he score on the fourth for an average of 85? → required total = 340, known = 243 → 97.
• A physics student averages 60 in 4 tests; what must his fifth test be to bring the average to 65? → 325 − 240 = 85.
• If the average of 5, 6, 7 and w is 8 → w = 32 − 18 = 14; if the average is 10 → w = 40 − 18 = 22.
• The average of three numbers is V; one is Z, another Y → the remaining number = 3V − Z − Y.
• If 10a + 10b = 35, the average of a and b = 35 ÷ 20 = 1.75. (Divide the equation by the count, not by the coefficients.)
• If 3a + 5b = 10 and 5a + 3b = 30, the average of a and b → add the equations: 8(a + b) = 40 → a + b = 5 → average = 2.5. (Adding simultaneous equations is often faster than solving them.)
• Find the average of 2a, −3a, 4a, 6a, 11a, 13a → sum = 33a, count 6 → 5.5a. Average of 4a, −4a, 8a, −8a = 0. Average of 1/2, 1/4, 1/6, 1/8 = (25/24) ÷ 4 = 0.260 (approximately). Average of 3/2, 13/2, 23/2 = 6.5.

15. The batsman and the replacement — average as a balancing point

• A batsman in his 17th innings scores 85 and thereby increases his average by 3 → let the old average be x: 16x + 85 = 17(x + 3) → x = 34 → new average = 37. The shortcut: the new score exceeds the old average by 3 × 17 = 51, so the old average was 85 − 51 = 34. (The same question in three papers gives the average after the innings as 37.)
• The average of a group of men increases by 5 years when a man aged 18 is replaced by a man aged 38 → the total rose by 20 over the group, so the group has 20 ÷ 5 = 4 men. Replacement questions are always (new − old) ÷ change in average = count.
• The average weight of A, B, C is 45 kg; A and B average 40 kg; B and C average 43 kg → B = (40 × 2 + 43 × 2) − (45 × 3) = 166 − 135 = 31 kg. Overlapping-group questions are solved by adding the two group totals and subtracting the whole total — the overlap is counted twice, which is exactly what you want.

16. Series shortcuts — never add a list the formula already sums

• Average of the positive integers from 1 to 100 = (1 + 100) ÷ 2 = 50.5. First + last, divided by 2 — for any consecutive series.
• Average of 7 consecutive numbers is 33 → the middle number is 33, the largest is 33 + 3 = 36.
• Average of the first 10 multiples of 7 = (7 + 70) ÷ 2 = 38.5. Average of the first five multiples of 3 = 9. Average of multiples of 10 from 10 to 190 = (10 + 190) ÷ 2 = 100.
• Average of prime numbers between 30 and 50 → primes 31, 37, 41, 43, 47, sum 199 → average = 39.8. Average of all primes between 30 and 40 → (31 + 37) ÷ 2 = 34. Primes have no formula — list them, add, divide.
• Average of the first ten prime numbers → 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, sum 129 → 129 ÷ 10 = 12.9 (OPC pp. 30–31 worked example, answer (c)). This is the Section 16 rule applied, not a new rule: the list is the whole method, and the only error available is a missed prime (2 is prime — the list starts at 2, not 3) or a wrong sum. No bank row asks this exact example (bank count: zero “first ten prime” stems), so no mark is claimed for it.
• Average of 10 numbers is zero; at most how many may be greater than zero? → 9 (nine positive numbers can be balanced by one sufficiently negative number; ten positive numbers cannot average zero).
• Average of six numbers is X; average of three of them is Y, of the other three Z → 6X = 3Y + 3Z → 2X = Y + Z. (Group averages combine by their counts.)

17. Overlapping groups — the “first three, last three” family

• There are four numbers. The average of the first three is 15 and of the last three is 16. If the last number is 19, find the first number. → first three total 45, last three total 48; the last three = middle two + 19, so the middle two = 29; first = 45 − 29 = 16. (Five papers ask this exact question, one with first-three 20, last-three 19, last number 22 → first = 60 − (57 − 22) = 25 by the same steps.)
• The average of the first two numbers is 2, of the last two is 3, and of the first and last is… — the same structure: write each average as a total, then add and subtract totals until only the asked number survives. Never try to find all three numbers.
• The average monthly income of P and Q is Rs. 5,050; of Q and R Rs. 6,250; of P and R Rs. 5,200 → P + Q = 10,100; Q + R = 12,500; P + R = 10,400 → total of all three = (10,100 + 12,500 + 10,400) ÷ 2 = 16,500 → P = 16,500 − 12,500 = Rs. 4,000; Q = 6,100; R = 6,500. (A PPSC variant uses 50,500 / 62,500 / 52,000 — ten times larger, same method, P = 40,000.)

18. Weighted averages — the group sizes are the weights

• A grocer’s sales for five months are Rs. 6,435, 6,927, 6,855, 7,230, 6,562; what must the sixth month be for a six-month average of Rs. 6,500? → required total 39,000, known 34,009 → Rs. 4,991. (Required-total method again — read the target average from the stem, never assume it.)
• Nine persons went to a hotel; eight spent Rs. 12 each, the ninth spent Rs. 8 more than the average expenditure → let the average be x: (96 + x + 8) ÷ 9 = x → 104 = 8x → x = 13 → the ninth spent Rs. 21. The ninth person’s excess over the average is spread over all nine — the equation writes itself.
• The average salary of skilled employees is Rs. 520 and of unskilled Rs. 420; if there are 3 skilled and 7 unskilled, the combined average = (3 × 520 + 7 × 420) ÷ 10 = Rs. 450. Never average the averages (520 + 420) ÷ 2 = 470 — that is only right when the groups are equal.
• A bus averages 30 passengers on Sundays and 40 on other days; in a 30-day month starting with a Sunday → Sundays fall on days 1, 8, 15, 22, 29 — 5 Sundays, 25 other days → the month’s total = 5 × 30 + 25 × 40 = 1,150 passengers (a daily average of 38.33). Two traps in one stem: count the Sundays from the start day (a 30-day month starting Sunday has five, not four), and notice the stem asks for the month’s total (“average number … in a month”), not the daily average.
• Annual rent of a house is Rs. 102,600 → average monthly rent = Rs. 8,550. Average rent of seven shops is Rs. 1,850 → total rent = Rs. 12,950. (Average × count = total — the papers ask it in both directions.)
• In Ajmal’s opinion his weight is greater than 65 kg but less than 72 kg; his brother says greater than 60 kg but less than 70 kg; his mother says not greater than 68 kg. If all are correct, what is the average of the different probable weights? → the three opinions intersect at weights greater than 65 and at most 68 — the probable weights are 66, 67, 68 → average = 67 kg. Intersection first (all opinions must hold together), then average the survivors — never average the bounds.

18A. Basic Statistics — mean, median, mode: the three measures of central tendency (OPC pp. 80–81)

• The three names, one idea. Mean, median and mode are numbers that represent a whole set of data — together they are called the measures of central tendency. The papers ask them as one-line definitions and as tiny worked sets, and the bank’s own phrase question — “Which is the most common measure of central tendency?” — is keyed Mean (PPSC Assistant Director Revenue paper 2023, options Mean / Median / Mode / Range). Mean is the chapter’s average (Part C above); median and mode are new here, and they are the whole of this section.
• Mean = the average (already taught). Add the data, divide by the count. The book’s worked set: 56, 35, 45, 67, 12, 24, 48, 55, 58, 30 → sum = 430, count 10 → mean = 430 ÷ 10 = 43 (OPC p. 80, sum and answer both verified against the page image — the OCR text is clean on this set). No new rule: this is Section 13’s total ÷ count with the word “mean” on it.
• Median = the middle number of an ordered set. Sort first, then take the middle. Odd count: the middle item itself. The book’s set 1, 2, 3, 4, 5, 6, 7 (already ordered, 7 items) → the 4th item is the median → 4 (OPC p. 80). Even count: average the two middle items. The book’s set 1, 2, 3, 4, 5, 6 (6 items) → the two middle items are 3 and 4 → median = (3 + 4) ÷ 2 = 3.5 — the book prints the setup (3+4)/2 at the foot of p. 80 and leaves the result uncomputed on the page; 3.5 is the arithmetic of its own setup, not a new figure.
• Mode = the most common number in the set. Count frequencies, take the highest. The book’s set (OPC p. 81, digits verified against the page image — the OCR garble “2, 5.62) 2.2, 56,2” resolves to this): 2, 5, 6, 2, 2, 2, 5, 6, 2 → 2 appears five times, 5 twice, 6 twice → mode = 2. No sorting needed for mode — only the count matters.
• The banks’ own statistics stems (all keys verified, all worked here). Median: 4, 8, 10, 12, 16 → middle of five = 10 (CSS MPT 2025 General Abilities section); 3, 5, 7, 9, 11 → middle of five = 7 (CSS MPT 2026 full mock); median of 7 calls, values already sorted 11, 13, 14, 15, 16, 17, 18 → 4th of seven = 15 (PPSC PMS SGAD evening paper 2026). Mode: 2, 3, 3, 4, 5 → 3 appears twice, the rest once → 3 (CSS MPT 2025); “which digit is the mode in 3, 5, 2, 0, 0, 1, 6, 6, 3, 1, 0, 7, 9?” → counts: 0 three times, 3/1/6 twice each, the rest once → 0 (PPSC Assistant BS-16 Planning & Development paper 2019, keyed B = 0; the same stem recurs verbatim in the PPSC Assistant Planning paper 2019, keyed 0). That is the whole bank footprint for this section: 3 median stems, 3 mode rows (2 unique stems), 1 central-tendency phrase row — 7 rows across 5 files, below both 🔁 limbs (fewer than 5 papers for any one fact, far below 100 questions), so no 🔁/⭐/🆕 mark is claimed for this section. It is taught as method the chapter previously did not cover, not as a repeated-paper family.
• Which measure does what — the one-line version for the stem. Mean uses every value (so one very large or very small value pulls it); median uses position only (so it ignores extremes — the middle of the sorted list); mode uses frequency only (so it is the only measure that works for non-numerical answers like a favourite colour, and a set can have no mode or two modes if two values tie at the top). The papers’ stems so far ask only the computation, but the distinction is what stops the three names blurring into one.
• Held, not taught: the triangle-median stem (“medians of a triangle intersect at the centroid”) is geometry vocabulary, not data statistics, and is served by the geometry material, not counted here; “mode of transmission”, “mode of speech”, “à la mode” and “payment mode” rows are word-sense matches, excluded from the count; the “mean” word-scan’s 1,598 rows are mostly the verb “mean” (idiom meanings, “what does X mean”) and Greenwich Mean Time, not arithmetic means — the arithmetic-mean rows sit inside Part C’s average count, not double-counted here.

18B. Measures of spread — range, mean deviation, standard deviation (PMS 2026 Basic Mathematics mock)

• The other half of statistics. Section 18A’s three measures say where the centre of a data set is; these three say how far the values stray from it — together they are called the measures of dispersion (spread). The papers ask them as one-line definitions: “The standard deviation measures: → Dispersion” and “The mean deviation is a measure of: → Dispersion” (both PMS 2026 Basic Mathematics mock, Q50/Q87).
• Range = largest − smallest. The quickest spread, using only the two extremes. Mock item: “The range is difference between: → Max & min” (mock Q66). Worked: for 2, 6, 10 → range = 10 − 2 = 8. Because it uses only two values, one extreme moves it — the trap is treating the range as a typical spread.
• Mean deviation = average of the absolute deviations from the mean. Take each value’s distance from the mean (ignoring sign), then average them: Σ|x − mean| ÷ N. Worked set 2, 4, 4, 4, 5, 5, 7, 9: mean = 40 ÷ 8 = 5; absolute deviations 3, 1, 1, 1, 0, 0, 2, 4 → sum 12 → mean deviation = 12 ÷ 8 = 1.5.
• Variance = average of the squared deviations; standard deviation = √variance. Same set: squared deviations 9, 1, 1, 1, 0, 0, 4, 16 → sum 32 → variance = 32 ÷ 8 = 4 → standard deviation = √4 = 2. Squaring stops positive and negative deviations cancelling (they always sum to zero); the square root brings the answer back to the data’s own units — variance is in squared units, the standard deviation is not.
• The one bank-attested variance stem. “The square of standard deviation is called: → Variance” (PPSC PMS BS-17 paper 2020) — the reverse direction of the definition above, and the only exam-paper (non-mock) spread stem the bank scan finds.
• Bank footprint + no mark. Script hidden_files/goldmine-notes-count-ma06-spread.py over 652 files / 55,297 questions: “standard deviation” 2 rows / 2 files (the 2020 paper + the mock), “mean deviation” 1 row / 1 file (the mock alone), “dispersion” 5 rows of which 3 are light-dispersion (physics) false positives. Below both 🔁 limbs, so no 🔁/⭐/🆕 mark is claimed for this section; it is taught as method the chapter previously did not cover, with the mock’s three items as its worked definitions. “Range” was not bank-counted — the word is too generic (salary ranges, age ranges) — and is taught here from the mock’s definition only.
• Held, not taught: no sample-vs-population (N vs N−1) distinction — the papers’ stems ask definitions only, and the mock’s items do not test the divisor; interquartile range and coefficient of variation are in neither the mock nor the bank scan and are not added.

19. Master table — which family, which move

20. Rapid-fire one-liners

• 7% is 42 → 600; 17% is 51 → 300; 25% is 150 → 600; 9% is 63 → 700; 5% is 10 → 200; 13% is 52 → 400; 3% is 60 → 2,000.
• 75% of 4,560 = 3,420; 150% of 48 = 72; 250% of 36 = 90; 80% of 700 = 560; 45% of 600 = 270; 20% of 30% of 20% of 850 = 10.20.
• Conversions: 40% = 2/5; 4/5 = 80%; 3/8 = 37½%; 33⅓% of 36 = 12; 11 1/9% of 171 = 19; x% of y = y% of x (so A = B); decreased 20% → 720 means original 900.
• 10% tax = 1,500 → income 15,000; saves 500 = 15% → income 3,333.33; 40% tax = 18,350 → income 45,875; 3% tax yields 75 → amount 2,500.
• A 25% more than B → B is 80% of A; A 25% less than B → B is 33⅓% more than A.
• +10% then −10% → 1% decrease; tax +20%, consumption −20% → 4% decrease; price +30%, sales −20% → revenue +4%; sugar +20%, consumption −25% → spending −10%; sugar +32%, spending +10% → consumption −16.67%.
• Length +20%, width −20% → area −4%; +40%/+30% → area +82%; sides −40% → area −64%; sphere radius +10% → surface +21%; sides ×3 → area +800%; length +60%, area fixed → width −37.5%.
• 45% failed, 550 passed → total 1,000; 60% at meeting, 28 absent → total 70; sells 40%, 420 left → 700; spends 88%, 2,160 left → income 18,000; 10% off, 585 remains → bill 650; 15% off, 68,000 → original 80,000; 40% to pass, gets 40, fails by 40 → total 200.
• Election: 55% of valid, 20% invalid, 7,500 votes → winner 3,300; 68 invalid, 52%, margin 98 → valid 2,450.
• Venn: 40% fail Math, 30% English, 10% both → pass 40%; 70% pass I, 60% pass II, 15% fail both → pass both 45%; 65.8% Physics, 50.2% Math, 2,000 students → both 320.
• Machine 32,000, −5% a year, 2 years → 28,880; building 133,100 vs land 72,900 at ∓10% → equal in 3 years.
• Ratio: 3.5 kg : 280 g = 25 : 2; 50 ml : 2 l = 1 : 40; 1,581 in 10 : 15 : 6 → B 765; 60,000 in 3 : 5 : 7 → largest − smallest 16,000; C − D = 1,000 in 5 : 2 : 4 : 3 → B 2,000; rice : wheat 3 : 4, rice 21 → wheat 28; x : y = 6 : 5, z : y = 9 : 25 → x : z = 10 : 3; product 320, ratio 1 : 5 → squares differ by 1,536; triangle sides 3 : 4 : 5, area 216 → perimeter 72; cubes volume 8 : 1 → edges 2 : 1; triangle angles 1 : 1 : 2 → isosceles right-angled; fourth proportional to 2, 3, 6 = 9; third proportional to 63, 29.9 = 14.19; mean proportional of 7,921, 9,409 = 8,633; mean of 4, 16 = 8, of 2, 8 = 4; fourth proportional to 2.4, 4.6, 7.6 = 14.57, to 0.5, 1.12, 1.3 = 2.912; coins 1 : 5 : 2 : 6 totalling 1,900 → 95 five-rupee coins; incomes 4 : 3, expenses 3 : 2, save 600 each → 2,400 and 1,800; tea 126/135/third in 1 : 1 : 2, mixture 153 → third = 175.50; alcohol : water 12 : 5, add 14 l water → 4 : 3 → original 59.5 litres; salt 30 kg at 2% → 10% needs 8/3 kg pure salt.
• Average: tests 75/82/86, want 85 → 97; physics 60 × 4, want 65 → fifth test 85; 5/6/7/w avg 8 → w 14; avg 10 → w 22; batsman 17th innings 85, avg +3 → new avg 37; replace 18 by 38, avg +5 → 4 men; A/B/C avg 45, A&B 40, B&C 43 → B 31 kg; first-three 15, last-three 16, last 19 → first 16; P&Q 5,050, Q&R 6,250, P&R 5,200 → P 4,000; 1–100 avg 50.5; 7 consecutive avg 33 → largest 36; primes 30–50 avg 39.8; primes 30–40 avg 34; first ten primes avg 12.9; first 10 multiples of 7 avg 38.5; hotel ninth man: eight spent 12, ninth 8 above average → average 13, ninth spent 21; 10 numbers avg zero → at most 9 positive; 6 numbers avg X, halves Y and Z → 2X = Y + Z.
• Statistics (Section 18A): mean of 56/35/45/67/12/24/48/55/58/30 → 43; median of 1–7 → 4, of 1–6 → 3.5; mode of 2/5/6/2/2/2/5/6/2 → 2; bank stems: median of 4/8/10/12/16 → 10, of 3/5/7/9/11 → 7, of sorted 11/13/14/15/16/17/18 → 15; mode of 2/3/3/4/5 → 3, of 3/5/2/0/0/1/6/6/3/1/0/7/9 → 0; most common measure of central tendency → Mean.
• Statistics (Section 18B): range of 2/6/10 → 8; mean deviation of 2/4/4/4/5/5/7/9 → 1.5; variance of the same set → 4, standard deviation → 2; square of standard deviation → variance; standard deviation measures dispersion, mean deviation measures dispersion.

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

• Percentage dominates this chapter (249 of 354 items, 70%) — and inside percentage, the single biggest family is the “find the whole” drill: “the number whose x% is…” appears in dozens of papers with rotating numbers (7% → 42, 17% → 51, 25% → 150, 9% → 63 are the four most repeated). If you can do one division — given ÷ percent — you own the largest block in the chapter.
• The repeat champions: “45% failed, 550 passed” in four papers; “10% income tax = Rs. 1,500” in five papers; “20% of 30% of 20% of Rs. 850” in three papers; “the number whose 17% is 51” in four papers; “first three average 15, last three 16, last number 19” in five papers; “batsman, 17th innings, 85 runs, average up by 3” in three papers; “60% at the meeting, 28 absent” in three papers; “sells 40% of apples, 420 left” in two papers; “two numbers 20% and 50% more than a third” in two papers. A question that reappears across papers is the highest-yield item in the chapter — these ten are worth memorising as worked examples, not just methods.
• Ratio is small in the papers (42 items) but highly patterned: dividing a sum in a ratio (10 items), unit conversion before dividing (3.5 kg : 280 g appears twice), ratio chains (x : y, z : y), and mixture-by-ratio (tea, wheat, alcohol-water) are the whole family. The papers test ratio method, not ratio trivia — the fact-type “ratio of Punjabi-speaking population” items are demography facts, not arithmetic, and are excluded from this chapter’s count.
• Average (63 items) splits into missing-item/required-total (the largest family: tests, physics marks, grocer sales), batsman/replacement (6 items), series shortcuts (multiples, primes, 1–100), overlapping groups (the “first three, last three” family in five papers, P&Q&R incomes in two), and weighted averages (classes’ pass percentages, cars per member, bus passengers). The overlapping-group family is the one students leave blank — it is also the most mechanical once the totals are written down.
• Cross-chapter boundaries held in the count: 104 percentage/ratio/average-keyword items that are really profit, loss, discount or interest belong to Chapter 7 and are excluded here; 12 age-ratio/average-age items belong to Chapter 4; 7 average-speed items belong to Chapter 3; 2 men–boys equivalence items belong to Chapter 5. The practice set serves only this chapter’s 354 items — the excluded items are served by their own chapters, not double-counted.
• Set overlap note: the FPSC Arithmetic sets contain no percentage, ratio or average set — the closest set neighbours are set 134 (Problems on Numbers), which carries a few average/ratio-flavoured number problems (average of four consecutive even numbers; a number 20% of the larger number), and they are counted in Chapter 2’s bank, not here. This chapter’s practice set is paper items only, which is why its count is reported as paper items rather than sets + papers.
• Proportion’s quiet corner (new Section 10A): third and mean proportionals are method the chapter previously did not teach, not a repeated-paper family — the bank count over 608 files / 58,098 questions finds zero “third proportional” rows and only two “mean proportional” rows (4 & 16 → 8, PPSC ASI 2020; 2 & 8 → 4, Junior Clerk 2021), below both 🔁 limbs, so no mark is claimed. Learn the three names and their three formulas (Section 10A); the papers that do ask proportion ask the fourth term or the mean of two small numbers.
• Conversion drills (new Section 1A): likewise method, not a repeated-paper family — the same 608-file count (script hidden_files/goldmine-notes-count-ma08-opc-sets.py, percent buckets) finds zero symmetry rows and zero “11 1/9%” rows, so no mark is claimed. Learn the 1/9 ↔ 11 1/9% pair and the x% of y = y% of x swap; the rest of Section 1A is Section 1’s table restated as moves.
• Basic statistics (new Section 18A): likewise method, not a repeated-paper family — the same 608-file count (script hidden_files/goldmine-notes-count-ma06-opc-stats.py over 608 files / 58,098 questions) finds 3 genuine median stems, 3 mode rows (2 unique stems) and 1 central-tendency phrase row — 7 rows across 5 files, below both 🔁 limbs, so no mark is claimed. Learn the three definitions and the sort-first rule for the median; the papers that do ask statistics ask one of the five worked stems in Section 18A. The 7 rows are companion practice for this section and are not folded into the chapter’s 354 bankCount, which counts the three original families only.

22. Traps & confusions

• The base changes when the direction changes. “A is 25% more than B” (base B) vs “B is 25% less than A” (base A) give different answers — 80% vs 33⅓% in the worked examples. Always ask “percent of what?” before writing a number.
• Successive percents never add. +20% then −20% is not 0% — it is −4%. +10% then −10% is −1%, not 0%. The net-change formula (x − y − xy/100) exists precisely because addition feels right and is wrong.
• Invalid votes come off first. “55% of the valid votes” with “20% invalid” means 55% of 80% of the total — not 55% of the total. The same trap appears in “45% failed” (the 55% who passed are the complement — use them, don’t recompute).
• The “both” is subtracted once. Fail Math 40%, fail English 30%, fail both 10% → fail at least one = 60%, not 70% and not 80%. Venn arithmetic: add the two, subtract the overlap.
• Mixed units before dividing. 3.5 kg : 280 g is 25 : 2, not 3.5 : 280. 50 ml : 2 litres is 1 : 40, not 50 : 2. Convert first, every time.
• A ratio is not a share until you divide by the total parts. 10 : 15 : 6 does not mean B gets 15/100 of the money — B gets 15/31. The total parts (31) is the denominator, never 100.
• Ratio chains need a common number. x : y = 6 : 5 and z : y = 9 : 25 do not combine to 6 : 5 : 9 — scale y to 25 first (30 : 25, then 30 : 9 = 10 : 3). The same error as partnership’s chains in Chapter 5.
• Mixture ratios are fractions of their own totals. Vessels 5 : 2 and 7 : 6 mixed to reach 8 : 5 need the ratio 7 : 9 by alligation — convert each ratio to a fraction of its own total first; adding 5 : 2 and 7 : 6 directly to get 12 : 8 is the classic error.
• Never average the averages. Skilled 520 and unskilled 420 do not average to 470 unless the groups are equal. Weight by group size, always — the classes’ pass percentages (66%, not 52.5%) are the standing warning.
• The asked quantity in “how many more / how many left” is the last step. “Sells 40%, 420 left” asks for the original (700), not the sold amount (280). Read the last word of the stem.
• Average-speed and average-age questions live elsewhere. “Sarah read 100 pages at 60 pages/hour and 100 at 40” is a harmonic-mean speed question (Chapter 3); “average age” ratio questions belong to Chapter 4. If an average question’s stem is really about speed or age, it is served by that chapter.
• “Increased in the ratio 9 : 8” multiplies by 9/8. It does not mean “increased by 9/8” or “increased by 9%” — the ratio is the multiplier.
• OPC truncation in the fourth-proportional decimal drill (Section 10A). (4.6 × 7.6) ÷ 2.4 = 34.96 ÷ 2.4 = 14.5666… — to two decimal places that is 14.57, not the book’s printed 14.56, which truncates instead of rounding. The book’s keyed option (d) 14.56 is still the closest option, but a question asking for the value to two decimal places wants 14.57; do not copy the book’s truncation as the rule.
• Third vs mean proportional — the name decides the formula. “Third proportional to 63 and 29.9” (b² ÷ a = 14.19) is not “mean proportional between 63 and 29.9” (√(63 × 29.9) ≈ 43.40). One asks for the next term after the pair, the other for the term between them; swapping the formulas swaps the answer.
• Conversion direction (Section 1A). Percent → fraction divides by 100 and reduces (40% → 2/5); fraction → percent divides and shifts the decimal two places right (4/5 = 0.8 → 80%) — doing the shift the wrong way gives 8% or 800%, both planted traps. “None of these” can be the right answer: x% of y and y% of x are equal, so a stem offering “A smaller”, “B smaller” or “cannot be determined” is answered by none of them.
• Median without sorting is a guess (Section 18A). The median is defined on the ordered set: 11, 13, 14, 15, 16, 17, 18 gives 15 only because the stem already sorted the calls — an unsorted list must be sorted first, or the “middle” you pick is just the middle of the writing order. Even count: the median is the average of the two middle items (3.5 for 1–6), not one of them and not their sum.
• Mode is the commonest, not the largest (Section 18A). In 2, 5, 6, 2, 2, 2, 5, 6, 2 the mode is 2 (five appearances), not 6 (the largest value) and not 5. A set with two values tied at the top has two modes; a set with all values appearing once has no mode — do not force one.
• Centre vs spread (Section 18B). Mean/median/mode say where the data sits; range/mean deviation/standard deviation say how far it strays — a stem asking “measures dispersion” wants a spread name, never a centre name.
• Variance is squared, SD is not (Section 18B). Variance 4 vs SD 2 on the same set: the square root is what returns the answer to the data’s units. “Square of standard deviation” → variance, not SD itself.
• “Median” and “mode” have false friends (Section 18A). The triangle’s median (a line from a vertex to a midpoint) is geometry, not data statistics; “mode of transmission”, “mode of speech” and “payment mode” are word senses. If the stem has no data set, it is not asking for this section’s mode.

23. Memory tricks

• “Per hundred, out of a hundred” — say it before every percentage question. It tells you the division: part ÷ whole × 100, or given ÷ percent for the whole.
• “Of what?” — the first question for every “more than / less than” item. The base is the number after “than”.
• “Multiply, don’t add” — for successive changes. Two changes are two multiplications; the net-change formula is the shortcut, not the definition.
• “Complement first” — failed 45%? Passed 55%. Spent 88%? Left 12%. The complement is usually the number the question actually wants.
• “One part first” — for every ratio division: total ÷ total parts, then multiply. The same trick as Chapter 5’s partnership.
• “Equalise the shared term” — for ratio chains: make the common term’s number the same (LCM), then read across.
• “Required total minus known total” — for every missing-item average question. The average is never needed after the first multiplication.
• “New minus old, divided by the change” — for replacement questions: (38 − 18) ÷ 5 = 4 men. The change in average is per person.
• “First plus last, over two” — for any consecutive or multiples series. Never add the list.
• “Weight by size” — for group averages: multiply each average by its group size, add, divide by the total size.
• “Third is b-squared over a; mean is the root of the product” — third proportional: square the middle, divide by the first; mean proportional: multiply the pair, take the root. If the answer is bigger than both numbers it cannot be a mean; if it is smaller than the second number it cannot be a third (for b > a).
• “Swap the percent” — x% of y = y% of x: put the easy percent on the easy number (32% of 50 = 50% of 32 = 16). “Ninth is eleven-and-a-ninth” — 1/9 ↔ 11 1/9%, the pair Section 1’s table does not list.
• “Max minus min; average the distances; root the squares” — range, mean deviation, standard deviation in one breath (Section 18B).
• “Sort, then middle; count, then commonest” — the two moves for Section 18A: sort the list and take the middle for the median (average the two middles when the count is even); count frequencies and take the commonest for the mode. Mean needs neither move — it is Section 13’s total ÷ count under a new name.

24. Visuals (site build specs)

• DIAGRAM — The three percentage questions: one triangle graphic (Part at top, Percent and Whole at the base corners) with the three worked examples beside it — 75% of 4,560 = 3,420 (find the part), 178 of 300 = 59.33% (find the percent), 7% is 42 → 600 (find the whole) — placed in Section 1 as the chapter’s entry graphic.
• FLOWCHART — Successive changes: two-step arrows showing +20% then −20% landing at −4% (not 0%), with the net-change formula (x − y − xy/100) and the sugar/tax/TV worked examples as leaves — placed in Section 3.
• DIAGRAM — The Venn pair: two overlapping circles for the fail-Math/fail-English family (40/30/10 → pass 40%) beside the pass-Paper-I/pass-Paper-II family (70/60, 15 fail both → pass both 45%), with the “subtract the overlap once” annotation — placed in Section 4.
• TABLE — Master table as a reference card: the Section 19 table rendered colour-banded, with the percentage rows, ratio rows and average rows in three bands — placed in Section 19 as the chapter’s revision graphic.
• DIAGRAM — The balancing point: a see-saw for the batsman example (old average 34, new score 85, 17 innings, balance shift +3) and the replacement example (18 out, 38 in, +5 over 4 men) — placed in Section 15 to teach average-as-balance visually.
• DIAGRAM — The three proportionals: one row per form with its formula and its worked example — fourth (a : b :: c : d, ad = bc, 2.4/4.6/7.6 → 14.57), third (a : b :: b : c, b² ÷ a, 63/29.9 → 14.19), mean (a : x :: x : b, √(ab), 7,921/9,409 → 8,633) — placed in Section 10A as the section’s revision graphic, with the “name decides the formula” caption.
• TABLE — The three-column conversion strip: percent / fraction / decimal in one row each (40% = 2/5 = 0.4; 33⅓% = 1/3; 11 1/9% = 1/9; 37½% = 3/8 = 0.375), with the x% of y = y% of x swap arrow beneath — placed in Section 1A.
• DIAGRAM — Centre vs spread on one data set: the 2/4/4/4/5/5/7/9 set plotted as dots on a number line with mean 5 marked, mean deviation 1.5 as the average arrow length, SD 2 as the root-mean-square arrow — placed in Section 18B.
• DIAGRAM — The three measures on one data set: one row per measure over the book’s sets — mean (sum ÷ count, 430 ÷ 10 = 43), median (sorted middle, 1–7 → 4, 1–6 → 3.5), mode (frequency tally, 2 × 5 beats 5 × 2 and 6 × 2) — with the “sort, then middle; count, then commonest” caption — placed in Section 18A as the section’s revision graphic.

25. Practice this topic

• Practice set: 354 topic-relevant paper items in the verified banks serve this topic — percentage 249 (find-the-part/percent/whole drills, more/less comparisons, successive changes, elections and exams, Venn pass/fail, growth, percentage–ratio hybrids, mixture by percentage), ratio and proportion 42 (dividing sums, unit-conversion ratios, ratio chains, geometry ratios, fourth, third and mean proportionals and coins, mixture by ratio), average 63 (required-total missing items, batsman and replacement, series shortcuts, overlapping groups, weighted averages). The site serves them as this chapter’s practice set; work the “find the whole” percentage drill and the “first three, last three” average family first (they are the most repeated shapes in the papers), then the successive-change and Venn families for speed. Profit/loss/discount/interest items that merely contain a percent sign are excluded here and served by Chapter 7; average-speed items by Chapter 3; average-age items by Chapter 4; men–boys equivalence items by Chapter 5. Basic statistics (Section 18A) adds 7 companion rows — 3 median stems, 3 mode rows (2 unique stems), 1 central-tendency phrase row — served with this chapter’s set, not counted in the 354. Measures of spread (Section 18B) add 4 companion rows — the mock’s three definition items (standard deviation, range, mean deviation) plus the PPSC PMS 2020 variance stem — served with this chapter’s set, not counted in the 354.
• Book mode: ← Previous: Partnership & Chain Rule · Next: Profit & Loss; Simple & Compound Interest →

Goldmine Notes · Mathematics · Chapter 6 · Sources: verified PPSC/FPSC past-paper banks (354 tightly matched paper items, script-counted — counting script hidden_files/goldmine-notes-count-ma06.py, raw dump hidden_files/goldmine-notes-ma06-count.txt, strays cleaned manually); every worked answer above recomputed by hand against its bank key before filing; percentage, ratio and average methods as standard arithmetic (no dedicated FPSC set exists for this ★ topic — the chapter is built from the paper items themselves). Section 10A/Section 16 additions from OPC Mathematics pp. 30–31 and 36–38 (OCR hidden_files/final-notes-intake/work/maths-opc-pp31-40.txt, 10/10 pages clean; proportional bank evidence script hidden_files/goldmine-notes-count-ma06-opc-prop.py over 608 files / 58,098 questions); OPC’s printed 14.56 corrected to 14.57 in Section 10A and flagged in Traps. Section 1A from OPC Mathematics pp. 41–43 (OCR hidden_files/final-notes-intake/work/maths-opc-pp41-50.txt, 10/10 pages clean; percent bank evidence in script hidden_files/goldmine-notes-count-ma08-opc-sets.py over the same 608 files / 58,098 questions — zero symmetry / 11 1/9% rows, no mark claimed). Section 18A from OPC Mathematics pp. 80–81 (OCR hidden_files/final-notes-intake/work/maths-opc-pp71-80.txt + maths-opc-p81.txt; both pages verified against page images rendered from 5. Maths.pdf — p. 81 mode digits 2, 5, 6, 2, 2, 2, 5, 6, 2 confirmed, OCR garble resolved; p. 80 even-median result 3.5 computed from the book’s own (3+4)/2 setup, which the page leaves uncomputed; statistics bank evidence script hidden_files/goldmine-notes-count-ma06-opc-stats.py over 608 files / 58,098 questions — 7 genuine rows, no mark claimed; bankCount 354 unchanged). Section 18B from the PMS 2026 GK Basic Mathematics mock test (files/pms-papers/pms-2026-gk-basic-mathematics-mock-test.json, 99 items; mock Q50/Q66/Q87 definitions, Q76 set reused for the worked range example; spread bank evidence script hidden_files/goldmine-notes-count-ma06-spread.py over 652 files / 55,297 questions — standard deviation 2 rows, mean deviation 1 row, no mark claimed; spread definitions verified against a standard statistics reference; bankCount 354 unchanged). Standard answers only.