Mathematics — Chapter 2

Problems on Numbers — Equation Setup Patterns

Why this chapter matters: Problems on numbers are the purest test of translation — turning English into an equation. The method is always the same (name the unknown, write what the sentence says, solve), but the question shapes repeat: one number with fractions, two or three numbers with a sum and a difference, consecutive numbers, two-digit numbers, fractions whose numerator and denominator change, and numbers with their reciprocals or squares. Our verified banks hold 87 topic-relevant items for this chapter: set 134’s 48 questions plus 39 tightly matched paper items. Set 134 is the anchor — learn its patterns, and the paper items fall out as the same patterns with different numbers.

1. The setup method — four steps for every question

1. Name the unknown. One number: call it x. Two numbers: call them x and y, or better, x and (sum − x) so there is only one unknown. Three numbers in a ratio: call them ax, bx, cx on one part x.
2. Translate clause by clause. “More than” = +, “less than” = −, “times” = ×, “of” = ×, “is” = =. “Exceeds … by 8” means the first quantity minus the second equals 8.
3. Solve, then check against the question asked. A question that asks for the smaller number, the largest number, the sum of the digits, or the product wants a final step after solving — read the last line before circling an answer.
4. Test with the options when stuck. If the equation looks messy, substitute the options back into the sentence. Set 134’s questions are built for this: most have small whole-number answers.

2. One number, one equation — the linear family

This is the largest block of set 134 (Q2, Q9–Q21). Every item is a single linear equation in x.

2.1 “Times, plus, minus” sentences

• Q2: 7x − 15 = 2x + 10 → 5x = 25 → x = 5.
• Q12: 3(2x + 9) = 75 → 2x + 9 = 25 → x = 8. Undo the outer operation first: divide by 3 before subtracting 9.
• Q10 (two-stage): (x − 4) ÷ 6 = 8 → x = 6 × 8 + 4 = 52; then (52 − 2) ÷ 5 = 10. Find the number first, then answer the second question about it.
• Q20: 2x + 20 = 8x − 4 → 6x = 24 → x = 4. Bring x-terms to one side, numbers to the other.
• Q15: 15x − x = 196 → 14x = 196 → x = 14. “Multiplied by 15 is increased by 196” means the increase (15x − x) equals 196, not 15x = 196.

2.2 Fraction-of-the-number sentences

• Q9: x − (3/5)x = 50 → (2/5)x = 50 → x = 125. “The difference between a number and its three-fifth” is the remaining two-fifths.
• Q13: (3/4)x − (1/3)x = 60 → (5/12)x = 60 → x = 144. Subtract fractions by a common denominator (12) first.
• Q14: x − 24 = (4/7)x → (3/7)x = 24 → x = 56; the sum of its digits is 5 + 6 = 11. Two answers in one question: the number, then its digit sum.
• Q16: x/5 + 4 = x/4 − 10 → x/20 = 14 → x = 280. Move the fraction terms together: x/4 − x/5 = 14, and 1/4 − 1/5 = 1/20.
• Q18: x/7 − x/11 = 100 → (4/77)x = 100 → x = 1,925. 1/7 − 1/11 = 4/77 — the numerator is the difference of the denominators when the denominators are coprime and the numerators are 1.
• Q19: (1/2 + 1/5 − 1/3)x = 7⅓ → (11/30)x = 22/3 → x = 20. Combine the fractions first (common denominator 30), then divide.
• Q21: (2/3)x − 50 = 40 + (1/4)x → (5/12)x = 90 → x = 216.
• Q11: (1/3)(1/4)x = 15 → x/12 = 15 → x = 180; then (3/10) × 180 = 54. “One-third of one-fourth” is a product of fractions, not a sum.
• Q17 (percent form): the difference of the two numbers is 20% of the larger, and the smaller is 12. So 80% of the larger = 12 → larger = 15. Percentages are fractions with denominator 100 — the same method.

2.3 The midpoint shortcut

• Q1: a number “as much greater than 36 as is less than 86” sits exactly midway: (36 + 86) ÷ 2 = 61. Whenever a number is equidistant from two numbers, it is their average. No equation needed.

3. Two and three numbers — sum, difference, product, ratio

Set 134’s second block (Q3–Q4, Q30–Q37, Q47–Q48). The universal move: express everything in one unknown, or use the sum-and-difference pair.

3.1 Sum and difference — the pair formulas

If two numbers sum to S and differ by D, then:

• Larger = (S + D) ÷ 2 · Smaller = (S − D) ÷ 2

• Q34: S = 33, D = 15 → smaller = (33 − 15) ÷ 2 = 9.

• Q33: S = 25, D = 13 → numbers 19 and 6 → product = 114. Note the question asks for the product, not the numbers.

• Q3: sum 184, and one-third of one exceeds one-seventh of the other by 8. With x and 184 − x: x/3 − (184 − x)/7 = 8 → x = 72, the smaller of 72 and 112 is 72. The fraction condition replaces the plain difference, but the one-unknown setup is identical.

• Q32: sum 22, and five times one equals six times the other. 5x = 6(22 − x) → x = 12, the bigger number is 12.

3.2 Sum and product — the difference-of-squares identity

If two numbers sum to S and multiply to P, then (difference)² = S² − 4P.

• Q4: S = 42, P = 437 → difference² = 42² − 4 × 437 = 1,764 − 1,748 = 16 → difference = 4. The numbers themselves are 23 and 19 (check: 23 + 19 = 42, 23 × 19 = 437).
• Q35: difference 5, product 500 → the integers are 20 and 25 (25 − 20 = 5, 20 × 25 = 500). Factor pairs of 500 that differ by 5: test 20 × 25 first.
• Q36: difference 5, product 336 → the pair is 16 and 21, sum 37.
• Q37: difference 3, difference of squares 39. Since a² − b² = (a − b)(a + b), 3(a + b) = 39 → a + b = 13 → larger = (13 + 3) ÷ 2 = 8. The identity a² − b² = (a − b)(a + b) is the whole question.
• Q8: 50 split into two parts whose reciprocals sum to 1/12. 1/x + 1/(50 − x) = 1/12 → x(50 − x) = 600 → parts 20 and 30 (20 × 30 = 600, 20 + 30 = 50). Reciprocals of a split always lead to a product equation.

3.3 Three numbers in a ratio

• Q30: ratio 4 : 5 : 6, average 25 → sum 75 → one part = 75 ÷ 15 = 5 → largest = 6 × 5 = 30.
• Q31: first = twice the second, third = one-third of the first. With the second = x: 2x + x + (2x/3) = 264 → (11x/3) = 264 → x = 72.
• Q47 (chained ratios): first : second = 2 : 3 and second : third = 5 : 3. Scale the second to a common value: 2 : 3 = 10 : 15 and 5 : 3 = 15 : 9, so the ratio is 10 : 15 : 9, total 34 parts = 136 → one part = 4 → second = 15 × 4 = 60. Chained ratios must share the middle term before adding.
• Q48 (three equations): a + b = 45, b + c = 55, c + 3a = 90. From the first two, c − a = 10; with c + 3a = 90, a = 20 and c = 30. Subtract pairs of equations to eliminate one variable at a time.

4. Consecutive numbers — and their averages

Set 134’s consecutive block (Q5, Q38–Q40) plus the paper items. Consecutive numbers are x, x + 1, x + 2; consecutive even numbers are x, x + 2, x + 4.

• Q38: three consecutive numbers sum to 87 → middle number = 87 ÷ 3 = 29 → greatest = 30. For an odd count of consecutive numbers, the average is the middle number.
• Q5: four consecutive even numbers average 27. Four numbers, even count: the average sits halfway between the middle two, so the numbers are 24, 26, 28, 30 → largest = 30.
• Q39: consecutive even numbers x and x + 2 with (x + 2)² − x² = 84 → 4x + 4 = 84 → x = 20 → numbers 20 and 22, sum 42. The difference of squares collapses to 4x + 4 — expand, don’t guess.
• Q40: (n − 1)² + n² + (n + 1)² = 2,030 → 3n² + 2 = 2,030 → n² = 676 → middle number = 26. Centring on the middle number (n − 1, n, n + 1) makes the squares collapse to 3n² + 2.

Average shortcut: for consecutive numbers, average = middle (odd count) or midpoint of the middle pair (even count). Working backwards from the average is faster than solving the sum equation.

5. Two-digit numbers — the 10t + u setup

A two-digit number with tens digit t and units digit u is 10t + u; reversing the digits gives 10u + t. Their difference is always 9(t − u) — a multiple of 9. This single fact solves most of the block (Q6–Q7, Q41).

• Q7: digit sum 9, and subtracting 63 reverses the digits. Difference 63 = 9(t − u) → t − u = 7; with t + u = 9, t = 8, u = 1 → number 81 (81 − 63 = 18, digits reversed).
• Q6: the number is 4 times its digit sum, and the units digit is 3 more than the tens digit. u = t + 3; 10t + u = 4(t + u) → 10t + t + 3 = 4(2t + 3) → 11t + 3 = 8t + 12 → t = 3, u = 6 → number 36 (digit sum 9, 36 = 4 × 9).
• Q41: the number exceeds its digit sum by 18, and the units digit is double the tens digit. u = 2t; (10t + u) − (t + u) = 9t = 18 → t = 2, u = 4 → number 24. Note the shortcut: a number minus its digit sum is always 9t, whatever u is.
• Paper pattern (the most repeated two-digit question in the banks): digit sum 15, difference of digits 3 → t = 9, u = 6 (or the reverse) → number 96 or 69 depending on which digit is larger; the banks’ standard answer is 96 when the tens digit is the larger one, and the question text must be read for which digit exceeds which. Always check the direction: “tens digit exceeds units digit” vs “units digit exceeds tens digit” swaps the answer.

6. Fractions whose numerator and denominator change

Set 134’s fraction block (Q42–Q46). Name the fraction n/d and write the changed fraction exactly as described.

• Q43: n + d = 11; 1 added to the numerator and 2 subtracted from the denominator gives 2/3. With d = 11 − n, the new denominator is (11 − n) − 2 = 9 − n, so (n + 1)/(9 − n) = 2/3 → 3n + 3 = 18 − 2n → n = 3 → fraction 3/8. Write the changed denominator exactly as printed — subtracting 2, not 1 — or the answer stops being a clean fraction.
• Q44: denominator is 3 more than numerator: d = n + 3; (n + 4)/(n + 7) = 4/5 → 5n + 20 = 4n + 28 → n = 8 → fraction 8/11.
• Q45: numerator exceeds denominator by 5: n = d + 5; adding 5 to the denominator decreases the fraction by 1¼: n/d − n/(d + 5) = 5/4 → d = 4, n = 9 → fraction 9/4 = 2¼.
• Q46 (the one-line version): increasing the numerator by 4 increases the fraction by 2/3 → 4/d = 2/3 → d = 6. The numerator’s increase, divided by the denominator, is the fraction’s increase.
• Q42: A = 2B and A × B = 2/25 → 2B² = 2/25 → B = 1/5 → A = 2/5.

7. Squares and reciprocals — the quadratic family

Set 134’s quadratic block (Q22–Q29). These are quadratic equations, but the papers only ever ask for the small positive root — factor or test whole numbers.

7.1 Number and its square

• Q22: x + x² = 182 → x² + x − 182 = 0 → (x + 14)(x − 13) = 0 → x = 13 (13 + 169 = 182).
• Q23: x² − 20x = 96 → x² − 20x − 96 = 0 → (x − 24)(x + 4) = 0 → x = 24 (576 − 480 = 96).
• Q24: 3x² − 4x = x + 50 → 3x² − 5x − 50 = 0 → (3x + 10)(x − 5) = 0 → x = 5.

Testing shortcut: when the equation is x² + … = a round-ish number, test the options’ squares first: 13² = 169, 24² = 576. The square dominates, so the answer is near √182 ≈ 13.5 or √96 ≈ 9.8 adjusted for the linear term.

7.2 Number and its reciprocal

Multiply through by x to clear the reciprocal — every reciprocal question becomes a quadratic:

• Q25: x + 1/x = 34/8 = 17/4 → 4x² − 17x + 4 = 0 → (4x − 1)(x − 4) = 0 → x = 4 (the positive whole root); the product of the number and its square root is 4 × 2 = 8.
• Q26: (2/3)x = (25/216)(1/x) → x² = 25/144 → x = 5/12.
• Q27: x + 17 = 60/x → x² + 17x − 60 = 0 → (x + 20)(x − 3) = 0 → x = 3.
• Q28: x − 4 = 21/x → x² − 4x − 21 = 0 → (x − 7)(x + 3) = 0 → x = 7.
• Q29: x + 1/x = 3(x − 1/x) → 4/x = 2x → x² = 2 → x = √2. Not every answer is whole — when the equation reduces to x² = 2, say so.

8. Master table — the setup patterns at a glance

9. Rapid-fire one-liners

• Midpoint: (36 + 86) ÷ 2 = 61.
• 7x − 15 = 2x + 10 → x = 5; 3(2x + 9) = 75 → x = 8; 2x + 20 = 8x − 4 → x = 4.
• x − (3/5)x = 50 → x = 125; (3/4 − 1/3)x = 60 → x = 144; x/7 − x/11 = 100 → x = 1,925.
• Sum 33, difference 15 → smaller = 9; sum 25, difference 13 → product = 114.
• Sum 42, product 437 → difference = 4 (numbers 23, 19); difference 3, squares differ by 39 → larger = 8.
• 50 split with reciprocals summing to 1/12 → parts 20 and 30.
• Ratio 4 : 5 : 6, average 25 → largest = 30; chained 2 : 3 and 5 : 3 → 10 : 15 : 9, second = 60.
• Three consecutive numbers sum to 87 → greatest = 30; four consecutive evens average 27 → largest = 30.
• Consecutive evens, squares differ by 84 → numbers 20, 22, sum 42; squares of three consecutive sum to 2,030 → middle = 26.
• Two-digit: digit sum 9, minus 63 reverses → 81; 4 × digit sum, units 3 more than tens → 36; exceeds digit sum by 18, units double tens → 24.
• Fraction: numerator + denominator 11, changed to 2/3 → 3/8; denominator 3 more, changed to 4/5 → 8/11; numerator exceeds denominator by 5, +5 to denominator drops it by 1¼ → 9/4.
• Numerator +4 raises the fraction by 2/3 → denominator = 6.
• x + x² = 182 → x = 13; x² − 20x = 96 → x = 24; 3x² − 4x = x + 50 → x = 5.
• x + 17 = 60/x → x = 3; x − 4 = 21/x → x = 7; (2/3)x = (25/216)(1/x) → x = 5/12; x + 1/x = 3(x − 1/x) → x = √2.

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

• Set 134 is the topic’s anchor: 48 questions — one-number linear equations 15, two/three-number problems 13, consecutive 4, two-digit 3, fractions 5, squares and reciprocals 8.
• The paper items (39) are led by two-digit numbers (12) — the digit-sum-15/difference-3 question recurs across FPSC papers (2016, 2017, 2006, 2008) — then sum-of-two-numbers items (9) and consecutive items (5). Two-digit and sum/difference are the highest-yield paper shapes after the set itself.
• The reciprocal question “increased by 17 = 60 times the reciprocal” (answer 3) appears in three separate FPSC papers (2016, 2017, 2005) — a verbatim repeater. Learn Q27’s equation, not just its answer.
• The sum-of-squares identity question (sum of squares 138, pairwise products 131 → sum?) appears twice (2016, 2017): (a + b + c)² = sum of squares + 2 × pairwise sum = 138 + 262 = 400 → sum = 20. It is the three-number version of Q4’s identity.
• The papers ask two-digit and sum/difference questions as bare computations (like set 134’s Q34), while set 134’s one-number items are word problems — practise the translation step on the set, then the speed step on the papers.

11. Traps & confusions

• Answering the wrong quantity. Q14 asks for the sum of the digits (11), not the number (56); Q33 asks for the product (114), not the numbers; Q36 asks for the sum (37), not the pair. Solve fully, then re-read the last line.
• “Increased by” vs “is”. Q15’s “multiplied by 15 is increased by 196” means 15x − x = 196, not 15x = 196. The increase is the difference between the new value and the old.
• “Of the remaining” and re-basing. Carried from Chapter 1: in fraction-of-a-number questions, each step’s base is what remains after the previous step — but in this chapter’s Q11, “one-third of one-fourth” is a product of fractions of the same original number, not a remainder chain. Read whether the second fraction applies to the original or to what is left.
• The changed-fraction number. Q43’s printed change is “2 subtracted from the denominator” — the set’s working uses 2, and a solver who assumes 1 gets a non-integer answer (17/5). When your answer is not a clean fraction, re-read the change numbers before re-solving.
• Two-digit direction. Digit sum 15, difference 3 gives two possible numbers (96 and 69). The question’s wording — which digit is larger — decides. Never assume the tens digit is the larger one.
• Consecutive even vs consecutive numbers. Consecutive even numbers step by 2 (x, x + 2, x + 4); consecutive numbers step by 1. Q5’s “four consecutive even numbers” averaging 27 gives 24, 26, 28, 30 — stepping by 1 would give non-even numbers and a wrong largest.
• Reciprocal questions have two roots. x + 17 = 60/x gives x = 3 and x = −20; the question asks for a positive number, so take 3. The negative root is algebra, not an answer.
• “Difference of squares” vs “square of the difference”. Q37 uses a² − b² = (a − b)(a + b); Q4’s identity uses (a − b)² = (a + b)² − 4ab. Mixing them up swaps 39 with 9 and 16 with 4.
• Ages and percentages are not this topic. Paper items about ages (sum of squares of ages) or percentages of a number were excluded from this chapter’s count — they are served by the Ages chapter (Topic 4) and the Percentage chapter (Topic 6). If a “find the number” question is really about ages, solve it with the ages method, not this chapter’s.

12. Memory tricks

• Sum-difference pair: bigger = (sum + difference) ÷ 2, smaller = (sum − difference) ÷ 2 — add for the bigger, subtract for the smaller, halve either way.
• 10t + u: a two-digit number is ten times the tens digit plus the units digit — write it before doing anything else, and reversal becomes 10u + t for free.
• Nine times the digit gap: reversing a two-digit number changes it by 9 × (tens − units) — 63 ÷ 9 = 7, so Q7’s digits differ by 7.
• Midpoint = average: “as much greater than a as less than b” is just the average of a and b.
• Reciprocal → multiply by x: the moment you see 1/x, multiply the whole equation by x and the fraction disappears into a quadratic.
• Consecutive average = middle: three consecutive numbers summing to 87 → middle is 87 ÷ 3 = 29, no equation needed.
• Chain the ratios: 2 : 3 and 5 : 3 share the second number — scale both until the middle terms match (15), then read off 10 : 15 : 9.

13. Visuals (site build specs)

• FLOWCHART — Words → equation: story → name x → translate each clause (“more than” = +, “times” = ×, “of” = ×, “is” = =) → solve → check the quantity asked — placed at the head of Section 1, using Q2 (7x − 15 = 2x + 10) as the worked path.
• DIAGRAM — The two-digit machine: a two-digit number shown as tens box × 10 + units box, with the reversed number beside it and the 9(t − u) difference arrow between them — placed in Section 5, using Q7 (81 → 18) as the example.
• TABLE — Setup patterns at a glance: the Section 8 master table rendered as a colour-banded reference card, with the sum-difference pair and the 10t + u setup highlighted — placed in Section 8 as the chapter’s revision graphic.
• DIAGRAM — Consecutive numbers on a number line: 24, 26, 28, 30 centred on their average 27, beside 28, 29, 30 centred on 29 — placed in Section 4, showing why the average is the middle.

14. Practice this topic

• Practice set: 87 topic-relevant items in the verified banks serve this topic — set 134’s 48 problems-on-numbers questions and 39 tightly matched paper items (digits 12, sum 9, consecutive 5, product 4, squares 2, reciprocal 3, linear-find 1, fractions 1, ratio 1, difference 1). The site serves them as this chapter’s practice set; work set 134 first for the setup patterns, then the papers’ two-digit and sum/difference items for speed. Paper items about ages, percentages, series and vocabulary that merely contain “find the number” are excluded from this count and served by their own chapters.
• Book mode: ← Previous: Numbers & Simplification · Next: Trains, Time & Distance →

Goldmine Notes · Mathematics · Chapter 2 · Sources: FPSC Edition 32, Arithmetic set 134 — Problems on Numbers (48 questions); verified PPSC/FPSC past-paper banks (39 tightly matched paper items, script-counted — counting script hidden_files/goldmine-notes-count-ma02.py); equation-setup, consecutive-number, two-digit and reciprocal methods as standard arithmetic. Standard answers only.