Chapter 24

Algebra & Word Problems

Unit VIII · Mathematics & Logic. 16 confirmed questions across 2022–2025 — the most important maths topic.

Concept Anchor

Algebra is a translation skill. Every word problem is a story — convert it into one or two equations, then solve. The failure point is not the arithmetic; it is the translation. 'A father is 4 times his son's age' becomes F = 4S before any calculation begins.

Full Chapter Notes

Source · FPSC Trap Decoder · CSS MPT Smart Notes (2026 Edition)

24.1 Context

With 16 confirmed questions across four past papers (2022–2025), Algebra is the most important mathematics topic in the MPT — and one of the most important topics in the entire General Science & Ability paper.

Metric	Detail
MPT Weightage	6–10 Marks
Difficulty	Medium
Past Papers	16 questions (2022–2025)

FPSC tests algebra in three formats:

Linear equations (solve for x)
Word problems (age, heads-and-feet, sum-and-difference)
Sets / Venn diagrams

Every format follows a predictable pattern. A candidate who has practised the five core problem types in this chapter will recognise the structure of any FPSC algebra question within three seconds.

24.2 Concept Anchor

Algebra is a translation skill. Every word problem is a story — your job is to convert that story into one or two equations and solve.

The failure point for most candidates is not the arithmetic — it is the translation. "A father is 4 times his son's age" must be converted to F = 4S before any calculation can happen. Master the five translation patterns and every FPSC algebra question becomes a one-minute task.

24.3 Core Formulas

Linear Equation — One Variable

ax + b = c → x = (c − b) / a

Whatever you do to one side of the equation, do exactly the same to the other side until x stands alone.

Special case — clearing fractions. If the equation contains x/2, x/3, or similar, multiply every term on both sides by the LCM of all denominators first. This eliminates all fractions in one step.

Example: x/2 − x/3 = 16. LCM(2, 3) = 6. Multiply every term by 6: 3x − 2x = 96 → x = 96.

Two-Variable System — Heads and Feet

Equation 1: x + y = Total Heads
Equation 2: 2x + 4y = Total Feet (2 = two-legged; 4 = four-legged)

Multiply Equation 1 by 2, then subtract from Equation 2: 2y = Total Feet − 2 × Total Heads.

Every animal has one head. Two-legged animals (roosters, ducks, humans) contribute 2 to the feet count; four-legged animals (buffaloes, cows, horses) contribute 4.

Age Problems

Name current ages (Father = F, Son = S, etc.).
Present relationship → Equation 1.
Future or past relationship → Equation 2. Solve simultaneously.

Tense rule:

"In X years" → add X to every current age in that equation.
"X years ago" → subtract X from every current age in that equation.
"Born X years after marriage" → Father's marriage age = Father's age at child's birth − X.

Sum and Difference

Given Sum = S and Difference = D:

Larger number = (S + D) / 2
Smaller number = (S − D) / 2

Add the sum and difference and halve to get the bigger number. Subtract and halve to get the smaller.

Venn Diagram (Sets) — Neither Formula

n(Neither) = Total − n(A) − n(B) + n(Both)

Subtract both groups from the total — but this subtracts the "both" group twice. Add it back once. Missing this single step is the most common error in every Venn diagram question.

Work Rate — Pipes and Workers

Rate of A = 1 / (Time A takes alone)
Rate of B = 1 / (Time B takes alone)
Combined Rate = Rate A + Rate B
Time together = 1 / Combined Rate

If Pipe A fills a tank in 6 hours, it fills 1/6 of the tank per hour. Add the rates, then take the reciprocal to find the combined time.

Problem-Type Quick Reference

Problem Type	Method
Linear Equation (one variable)	Isolate x: same operation on both sides
Fractions in equation	Multiply all terms by LCM of denominators
Heads & Feet (two-variable)	Write two equations; multiply eq1 by 2; subtract
Age Problems	Name current ages; write present + future/past equations
Sum & Difference	Larger = (S+D)/2; Smaller = (S−D)/2
Venn Diagram / Sets	Neither = Total − A − B + Both
Work Rate (Pipes/Workers)	Rate = 1/Time; Combined rate = RA + RB; Time = 1/Combined

24.4 Solved Examples

Example 1 — Clearing Fractions

Problem: Find x: x/2 − x/3 = 16.

Denominators 2 and 3 → LCM = 6.
Multiply every term by 6: 3x − 2x = 96.
Solve: x = 96.

Verify: 96/2 − 96/3 = 48 − 32 = 16. ✓

Why candidates fail: they try to subtract x/2 − x/3 directly by finding a common denominator, getting confused mid-calculation. The LCM-multiply method eliminates all fractions in one step.

Example 2 — Heads and Feet (Simultaneous Equations)

Problem: A farmer has roosters and buffaloes. Heads = 48, Feet = 146. How many roosters?

Let roosters = r, buffaloes = b.
r + b = 48 (heads); 2r + 4b = 146 (feet).
Multiply heads eq by 2 → 2r + 2b = 96.
Subtract from feet eq: 2b = 50 → b = 25 buffaloes.
Back-substitute: r = 48 − 25 = 23 roosters.

Verify: 23 + 25 = 48 ✓; 23 × 2 + 25 × 4 = 46 + 100 = 146 ✓.

FPSC trap: This exact question appeared in CSS MPT 2023 Special. The answer 23 was NOT listed in options A, B, or C — making "None of these" the correct answer. Always calculate your own answer first.

Example 3 — Age Problem

Problem: Father is currently 4 times his son's age. In 6 years he will be exactly 3 times his son's age. Find the son's current age.

Son's current age = x; Father's = 4x.
In 6 years: Father = 4x + 6; Son = x + 6.
4x + 6 = 3(x + 6) → 4x + 6 = 3x + 18 → x = 12.

Verify: Son = 12, Father = 48. In 6 years: Son = 18, Father = 54. 54 = 3 × 18 ✓.

Trap: writing 4x + 6 = 3x + 6 (forgetting to multiply the bracket). Always expand 3(x + 6) = 3x + 18, not 3x + 6.

Example 4 — Sum and Difference

Problem: Sum of two numbers is 84; difference is 36. Find both.

Direct formula:

Larger = (84 + 36) / 2 = 60.
Smaller = (84 − 36) / 2 = 24.

Simultaneous-equation method also works (x + y = 84, x − y = 36 → 2x = 120 → x = 60, y = 24).

Verify: 60 + 24 = 84 ✓; 60 − 24 = 36 ✓.

Example 5 — Venn Diagram (Neither Group)

Problem: Village of 200 people. 90 like tea, 100 like coffee, 46 like both. How many like neither?

n(Neither) = 200 − 90 − 100 + 46 = 56.

Verify: Tea only = 44; Coffee only = 54; Both = 46; Neither = 56 → Total = 200 ✓.

The +Both error. Subtracting 90 and 100 from 200 removes the 46 "both" people twice. Adding them back once corrects this. Without it you get 200 − 190 = 10, wrong by exactly 46.

Example 6 — Work Rate (Pipes)

Problem: Pipe A fills a tank in 6 hours; Pipe B in 4 hours. Time together?

Rate A = 1/6 per hour; Rate B = 1/4 per hour.
Combined = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 per hour.
Time = 1 ÷ (5/12) = 12/5 = 2.4 hours.

Verify: in 2.4 h, A fills 0.40; B fills 0.60. Total = 1.00 ✓.

Work rate is never solved by averaging times. Always convert each individual time to a rate (1/Time), add the rates, then take the reciprocal.

Example 7 — Family Counting

Problem: Family: one man, his wife, 4 daughters, 3 sons. Each son is married with 3 sons and 1 daughter. How many female members in total?

Generation	Female Category	Count
1	Man's wife	1
1	Man's 4 daughters	4
2	3 sons' wives	3
3	Each son's 1 daughter × 3 sons	3
	Total female members	11

Trap: Each son's 3 sons = 9 grandsons are male; do NOT count. Drawing the tree prevents this error.

24.5 Common Mistakes

Mistake	Wrong Approach	Correct Approach
Negative sign in brackets	−(x − 4) = −x − 4	−(x − 4) = −x + 4
Age problems: adding years to only one person	4x + 6 = 3x	4x + 6 = 3(x + 6)
Venn diagrams: omitting the +Both	Neither = 200 − 90 − 100 = 10	Neither = 200 − 90 − 100 + 46 = 56
"A is 4 times B" vs "A is 4 more than B"	Treats them as identical	"4 times" → A = 4B; "4 more" → A = B + 4
Guessing "None of These" without calculating	Question looks hard → pick D	Always calculate first; pick D only if your answer is genuinely absent

Times vs More Than — Critical Translation Distinctions

Phrase	Correct Translation
"A is 4 times B"	A = 4B
"A is 4 more than B"	A = B + 4
"A is 4 times more than B"	A = 5B (the original + 4 times it)

24.6 FPSC Trap Alerts

The 91 Prime Trap. Which is prime — 91, 93, 97, 99? Answer: 97. Students mark 91 because it looks prime, but 91 = 7 × 13. Test every candidate by dividing by 2, 3, 5, and 7. Appeared in CSS MPT 2022 and 2023.

The "None of These" Pattern. FPSC intentionally places the correct answer outside options A, B, C in algebra questions. Confirmed examples: roosters = 23 (not listed), sum = 144 (not listed), x = 96 (not listed). Always calculate first.

The Tense Trap in Age Problems. "In 5 years" applies to both people in the equation, not just one. Father AND son each gain 5 years.

The Heads-Feet Setup Trap. Students sometimes write 4r + 2b = feet (swapping the coefficients). Roosters = 2 legs; buffaloes = 4 legs. Always confirm.

24.7 The 5-Minute Battle Card

Topic	Key Rule / Formula
Linear Equation	Apply identical operations to both sides until x = number.
Fractions in Equation	Multiply ALL terms by LCM of denominators. Example: x/2 − x/3 = 16 → LCM = 6 → 3x − 2x = 96 → x = 96.
Heads & Feet	r + b = heads; 2r + 4b = feet. Multiply eq1 by 2, subtract. 48 heads, 146 feet → 23 roosters, 25 buffaloes (CSS MPT 2023).
Age Problems	"In X years" → add X to ALL current ages. "X years ago" → subtract X from ALL. "4 times" ≠ "4 more than".
Sum & Difference	Larger = (S+D)/2; Smaller = (S−D)/2. Example 84/36 → 60 and 24.
Venn — Neither	Neither = Total − n(A) − n(B) + n(Both). Always add back 'both'.
Work Rate	Rate = 1/Time. Combined = RA + RB. Time = 1/Combined. NEVER average times.
Bracket Signs	−(x − 4) = −x + 4. Minus FLIPS every sign inside.
Prime Numbers	97 prime. 91 = 7 × 13; 93 = 3 × 31; 99 = 9 × 11.
Family Counting	Draw the generation tree. List each female category separately.
"None of These"	Choose only when your calculated answer is genuinely absent.

24.8 Practice MCQs (FPSC Level)

Part A — Basic Recall

Foundational equation and translation drills.

Solve for x: 3x + 7 = 22

Show explanation

3x = 22 − 7 = 15 → x = 5. Verify: 3(5) + 7 = 22 ✓.

Trap: Distractors come from arithmetic slips on subtraction or division.

MPT 2022, 2024

Which of the following is a prime number?

Show explanation

97 is divisible only by 1 and itself (√97 < 10; not divisible by 2, 3, 5, 7). 91 = 7×13; 93 = 3×31; 99 = 9×11.

Trap: 91 looks prime at a glance — most-marked wrong answer.

MPT 2022, 2023

Find the value of x: x/2 − x/3 = 16

Show explanation

LCM(2,3) = 6. Multiply all terms: 3x − 2x = 96 → x = 96.

Trap: 12/24/48 are deliberate distractors matching incorrect bracket or LCM moves.

MPT 2023

The sum of two numbers is 84 and their difference is 36. What is the larger number?

Show explanation

Larger = (84 + 36)/2 = 60. Smaller = (84 − 36)/2 = 24.

Trap: Distractors match averaging or skipping the difference step.

MPT 2022, 2024

In a group of 200 people, 90 like tea, 100 like coffee, and 46 like both. How many like NEITHER?

Show explanation

Neither = 200 − 90 − 100 + 46 = 56.

Trap: 10 = forgetting to add back n(Both).

MPT 2023 Special

Part B — Trap-Based

Items engineered around bracket expansion, swap, and 'None of these' lures.

A farmer has roosters and buffaloes. Heads = 50, feet = 140. How many buffaloes?

Show explanation

r + b = 50; 2r + 4b = 140. Subtract 2×(eq1): 2b = 40 → b = 20.

Trap: Coefficient swap (4r + 2b) gives roosters as buffaloes.

MPT 2023, 2024

A father is currently 4 times his son's age. In 6 years he will be 3 times his son's age. What is the son's current age?

Show explanation

4x + 6 = 3(x + 6) → 4x + 6 = 3x + 18 → x = 12.

Trap: Wrong answer 18 comes from writing 4x + 6 = 3x + 6 (bracket not expanded).

The difference of two integers is 36. The smaller is 60% of the larger. What is their sum?

Show explanation

x − 0.6x = 36 → 0.4x = 36 → x = 90 (larger). Smaller = 54. Sum = 144 — not listed.

Trap: A/B/C all look reasonable but the calculated value is genuinely absent.

MPT 2023 Special Q119

Family: man + wife + 4 daughters + 3 sons. Each son has 3 sons and 1 daughter. How many female members in total?

Show explanation

Wife (1) + 4 daughters + 3 sons' wives + 3 granddaughters = 11. The 9 grandsons are male.

Trap: Including grandsons gives 20; ignoring sons' wives gives 8.

MPT 2023 Special

Solve: 2(x + 3) − (x − 4) = 15

Show explanation

2x + 6 − x + 4 = 15 → x + 10 = 15 → x = 5. Note −(x − 4) = −x + 4.

Trap: Writing −(x − 4) as −x − 4 (sign error) produces 1, the trap option.

MPT 2024

Part C — Elite Simulation

Highest-difficulty FPSC-style items.

In a village of 300, 130 like mango, 150 like guava, and 70 like both. How many like exactly ONE fruit?

Show explanation

Exactly one = (130 − 70) + (150 − 70) = 60 + 80 = 140.

Trap: 210 = at-least-one (130 + 150 − 70); a different question.

The sum of three consecutive odd integers is 57. What is the largest?

Show explanation

n + (n+2) + (n+4) = 57 → 3n + 6 = 57 → n = 17. Integers: 17, 19, 21 → largest = 21.

Trap: 17 is the smallest; 19 is the middle.

MPT 2024

Pipe A fills a tank in 6 hours. Pipe B fills it in 4 hours. Working together, how long to fill the tank?

Show explanation

Combined rate = 1/6 + 1/4 = 5/12 per hour. Time = 12/5 = 2.4 hours.

Trap: 2 hrs = averaging (6+4)/2 — wrong approach.

A class has 40 students. 28 study Physics, 20 study Chemistry, 10 study both. How many study NEITHER?

Show explanation

Neither = 40 − 28 − 20 + 10 = 2.

Trap: 0 = forgetting to add back n(Both); 12 = subtracting Both instead of adding.

Statements: (1) To solve x/3 + x/4 = 7, multiply all terms by 12. (2) 'A is 5 times B' translates to A = B + 5. (3) In Venn diagrams, the both group must be added back when calculating neither. Which are correct?

Show explanation

(1) Correct — LCM(3,4) = 12. (2) FALSE — '5 times' = 5B, not B + 5. (3) Correct — n(Both) is added back.

Trap: Statement 2 is the planted error.

Answer Key & Full Solutions

Algebra & Word Problems (Q1–15)

Q	Correct	Type	Primary Trap	Why Others Fail