Study guide
14.2 Word Problems
Pakistan Air Force GD Pilot Initial Test Exam
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1 Chapter 1: Introduction & Exam Overview
1.1 Current Exam Facts 1.2 Eligibility & Registration 1.3 Test Format & Selection Pipeline
2 Verbal Intelligence: Word Relationships
2.1 Analogies 2.2 Synonyms & Antonyms 2.3 Vocabulary in Context 2.4 Sentence Completion
3 Verbal Intelligence: Series & Coding
3.1 Number Series 3.2 Letter & Alpha-Numeric Series 3.3 Coding-Decoding
4 Verbal Intelligence: Quantitative Logic
4.1 Direction Sense 4.2 Percentages 4.3 Age & Relationship Problems 4.4 Time, Speed & Distance
5 Non-Verbal Intelligence: Patterns
5.1 Pattern & Figure Series 5.2 Odd-One-Out 5.3 Classification
6 Non-Verbal Intelligence: Spatial Reasoning
6.1 Rotation 6.2 Mirror Images 6.3 Matrix Reasoning 6.4 Spatial Reasoning
7 English: Grammar Foundations
7.1 Grammar Essentials 7.2 Tenses 7.3 Prepositions
8 English: Voice, Reported Speech, Correction & Vocabulary
8.1 Active & Passive Voice 8.2 Sentence Correction 8.3 Spelling 8.4 Vocabulary & Comprehension 8.5 Direct & Indirect Speech
9 Physics: Mechanics & Fluids
9.1 Mechanics 9.2 Fluid Dynamics & Bernoulli
10 Physics: Waves, Optics & Modern
10.1 Waves & Sound 10.2 Optics & Light 10.3 Modern Physics
11 Physics: Electromagnetism & Thermodynamics
11.1 Electromagnetism 11.2 Thermodynamics
12 Mathematics: Algebra, Calculus & Logs
12.1 Algebra 12.2 Calculus Basics 12.3 Logarithms
13 Mathematics: Geometry, Trig, Matrices & Vectors
13.1 Geometry 13.2 Trigonometry 13.3 Matrices & Vectors
14 Mathematics: Applied Reasoning
14.1 Percentages & Ratio 14.2 Word Problems
15 General Knowledge: PAF & Aviation
15.1 PAF Organisation 15.2 PAF History 15.3 Aviation Fundamentals
16 General Knowledge: Pakistan & General Science
16.1 Pakistan Geography 16.2 Armed Forces Selection & Basic Science
17 Test-Day Strategy & Selection Pipeline
17.1 Test-Day Strategy 17.2 Later Selection Stages
14.2 Word Problems
Key Takeaways Explain This Topic AI
Read the problem, name the unknown x, and translate each English phrase into an equation before solving.
Combined work rate: 1/a + 1/b = 1/T, so time together T = ab/(a+b); for a drain, subtract its rate.
Simple interest SI = P x R x T / 100; compound interest CI = P(1 + R/100)^T - P.
Probability of an event = favourable outcomes / total outcomes; rolling a prime on a fair die (2, 3, 5) gives 3/6 = 1/2.
0! = 1 by definition; nPr = n!/(n-r)! when order matters, nCr = n!/(r!(n-r)!) when order does not.
Last updated: August 2026
Word Problems
Quick Answer: Translate English to an equation, assign a variable, set up the relationship, solve. Common templates: age, mixture, work, pipes, profit/loss, interest, probability.
Translating English to Equations
The core skill is reading the problem, naming the unknown x , and writing what the words say as mathematics. Many practice word problems fit one of a small set of templates; recognising the template is half the battle.
Age Problems
Worked example: Ali is 3 times as old as Bilal. In 5 years, Ali will be twice as old as Bilal. Find their ages now.
Let Bilal = x now, so Ali = 3x now.
In 5 years: Ali = 3x + 5, Bilal = x + 5.
- Equation: 3x + 5 = 2(x + 5)
- 3x + 5 = 2x + 10
- x = 5
So Bilal is 5 and Ali is 15.
Mixture Problems
Worked example: How much water must be added to 40 L of a 30% salt solution to dilute it to a 20% solution?
The amount of salt stays constant. Salt = 40 x 0.30 = 12 L.
Let water added = w. New total volume = 40 + w.
- 0.20 x (40 + w) = 12
- 40 + w = 60
w = 20 L.
Work Problems (Combined Rate)
If worker A finishes a job in a days and worker B in b days, their combined rate is 1/a + 1/b (the fraction of the job done per day). Time working together is:
T = 1 / (1/a + 1/b) = ab / (a + b)
Worked example: A fills a tank in 6 hours, B in 3 hours. Together: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2. So T = 1 / (1/2) = 2 hours.
Pipes and Cisterns
Identical to work problems, but a draining pipe has a negative rate.
Worked example: Pipe A fills a cistern in 4 h; pipe B empties it in 6 h. With both open, the net rate is 1/4 - 1/6 = 3/12 - 2/12 = 1/12 per hour. Time to fill = 12 h.
Profit, Loss, and Discount
Cost price (CP) is what the seller paid; selling price (SP) is what the buyer pays.
Profit = SP - CP (when positive); Loss = CP - SP (when SP < CP).
Profit % = Profit / CP x 100. Loss % = Loss / CP x 100. Both are computed on the cost price.
Marked price (MP) minus discount equals selling price. Discount % is computed on the marked price.
Worked example: A shopkeeper marks an item 40% above cost (Rs 500). MP = 500 x 1.40 = Rs 700. He gives a 10% discount on the marked price. SP = 700 x 0.90 = Rs 630. Profit = 630 - 500 = Rs 130. Profit % = 130/500 x 100 = 26%.
Simple and Compound Interest
Simple Interest (SI) = P x R x T / 100. Total amount = P + SI.
Compound Interest (CI) = P x (1 + R/100)^T - P. Total amount = P x (1 + R/100)^T.
Worked example: Rs 10,000 at 10% per year for 3 years.
SI = 10,000 x 10 x 3 / 100 = Rs 3,000. Total = Rs 13,000.
CI = 10,000 x (1.1)^3 - 10,000 = 10,000 x 1.331 - 10,000 = Rs 3,310. Total = Rs 13,310.
The CI exceeds the SI by Rs 310 because each year's interest earns interest in the next year.
Probability
Probability of an event = favourable outcomes / total outcomes.
Worked example: Probability of rolling a prime number on a fair six-sided die. The primes on a die are 2, 3, and 5 - three favourable outcomes out of six. P = 3/6 = 1/2.
Factorials, Permutations, Combinations
0! = 1 by definition (the empty product). This is a favourite test fact.
nPr = n! / (n - r)! - order matters (arrangements).
nCr = n! / (r! x (n - r)!) - order does not matter (selections).
Worked example: In how many ways can 3 cadets be selected from 5 for a flight? Order does not matter, so use combinations: 5C3 = 5! / (3! x 2!) = (120) / (6 x 2) = 10.
Common Word-Problem Templates
- Type Key relationship Equation
- Age Now: A = kB; later: A + t = m(B + t) linear in one variable
- Mixture Quantity of solute is constant c1 x V1 = c2 x (V1 + x)
- Work Rate = 1 / time 1/a + 1/b = 1/T
- Pipes Fill positive, drain negative 1/a - 1/b = 1/T
- Profit/Loss On cost price (SP - CP) / CP x 100
- Simple Interest On principal P x R x T / 100
- Compound Interest On principal, compounded P x (1 + R/100)^T - P
- Probability Favourable / total n(E) / n(S)
- Strategy Checklist
Read the whole problem once before writing anything.
Name the unknown with a single variable.
Write the equation that mirrors the English sentence.
Solve and check the answer makes sense in the story (a negative age is a red flag).