Chapter 25
Arithmetic, Percentages & BODMAS
Order of operations, percentage change, profit/loss, simple & compound interest.
Full Chapter Notes
Source · FPSC Trap Decoder · CSS MPT Smart Notes (2026 Edition)
25.1 Context
Arithmetic and percentages appear in every single MPT paper. Past-paper analysis (2022–2025) confirms 8 direct questions from this cluster. These are the easiest marks in the entire test — but only for candidates who know the correct method.
| Metric | Detail |
|---|---|
| MPT Weightage | 5–8 Marks |
| Difficulty | Low to Medium |
| Confirmed Past Papers | 2022 · 2023 · 2024 · 2025 |
The three most-tested scenarios:
- Finding a percentage of a number
- Calculating percentage change
- Applying BODMAS to a multi-operation expression
Every question in this chapter should be solved mentally in under 20 seconds.
25.2 Concept Anchor
Percentages, fractions, and decimals are three different ways of writing the same thing. Once you see that 25% = 1/4 = 0.25, and that 10% simply means "move the decimal one place left," you stop calculating and start seeing answers.
That mental shift — from calculating to seeing — separates a 20-second solver from a 3-minute struggler under exam pressure.
25.3 Core Formulas
Finding a Percentage of a Number
Part = (Percentage ÷ 100) × Whole Number
"Per cent" means "per hundred." So 30% of 70 = (30/100) × 70 = 21.
The 10% Base Shortcut:
- 10% of any number = move the decimal one place left.
- 20% = double the 10% answer.
- 5% = half the 10% answer.
- 25% = divide the number by 4.
- 50% = divide the number by 2.
Essential Conversions to Memorise
These appear directly as FPSC questions — 1/8 as a percentage appeared in CSS MPT 2024.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.50 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/3 | 0.333… | 33.33% |
| 2/3 | 0.667… | 66.67% |
| 1/5 | 0.20 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.10 | 10% |
Percentage Change
% Change = (Difference ÷ Original Value) × 100
Always divide by the OLD value — the one you started with. Never the new value.
Finding the Original Number
Original = Given Number × (100 ÷ % it represents)
If you know a percentage of the unknown, divide by that fraction to work backwards.
Successive Percentage Change
Net % Change = x + y + (x × y / 100)
When a value changes twice, the changes do NOT simply add. If it rises 20% then falls 20%, the net result is NOT zero — it is always a small loss.
Net Loss Rule (equal percentages): Net loss = (x × x) / 100.
BODMAS — Order of Operations
Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction.
Critical Rule: Division and Multiplication have EQUAL priority. When both appear together without brackets, solve LEFT TO RIGHT — whichever comes first.
25.4 Solved Examples
Problem: Find 30% of 70.
Use the zero-cancel method. 30 and 70 each have one zero — cancel one zero from each. You are left with 3 × 7 = 21.
Verify: 10% of 70 = 7. So 30% = 3 × 7 = 21. ✓
Problem: A price rises from Rs. 80 to Rs. 100. What is the percentage increase?
- Difference: 100 − 80 = 20.
- Original (OLD) value = 80.
- (20 / 80) × 100 = 25%.
Why not 20%? Dividing by the new value (100) gives 20%. FPSC places 20% as Option A — it is wrong. The reference point is always where you started: 80, not 100.
Problem: A man spends 65% of his salary and saves Rs. 1,750. What is his total salary?
- Savings = 35% of salary (NOT 65% — that is spending).
- Original = 1,750 × (100 / 35) = 1,750 × 20/7 = Rs. 5,000.
Verify: 35% of 5,000 = 1,750 ✓.
Trap: Students use 65% as the denominator because the problem mentions spending first. Always identify which percentage the known number (savings) represents.
Problem: A salary increases by 20% and then decreases by 20%. Net change?
Common wrong answer: 0% (they cancel out). WRONG.
Equal-percentage Net Loss Rule: (20 × 20) / 100 = 4% net loss.
Verify with numbers: Start Rs. 1,000. After +20%: Rs. 1,200. After −20%: Rs. 1,200 × 0.80 = Rs. 960. Loss of Rs. 40 = 4% of original ✓.
Why a loss every time? The 20% decrease applies to the new higher value (1,200), so the absolute fall (240) exceeds the absolute rise (200).
Problem: Evaluate 36 ÷ 9 × 4.
Wrong approach: 9 × 4 = 36 first, then 36 ÷ 36 = 1. WRONG.
Correct — left to right (division and multiplication are equal priority):
- 36 ÷ 9 = 4.
- 4 × 4 = 16.
Rule: Never invent brackets that are not there. 36 ÷ 9 × 4 is NOT the same as 36 ÷ (9 × 4).
Problem: Price of sugar rises 25%. By what % must consumption fall to keep total expenditure unchanged?
Expenditure = Price × Quantity. If price ×1.25, quantity must × (1/1.25) = 0.80.
Reduction = 1 − 0.80 = 20%.
Formula shortcut: Reduction % = [x / (100 + x)] × 100 = [25 / 125] × 100 = 20%.
Verify: Old (100, 10) = 1,000. New (125, 8) = 1,000 ✓.
Trap: Most candidates write 25% (same as the rise). The reduction is always SMALLER than the rise.
25.5 Exam Strategies — Speed Techniques
Zero-Cancel Method for Percentages. When both the percentage and the base number end in zeros, cancel one zero from each before multiplying. 30% of 70 → 3 × 7 = 21. Works for 10%, 20%, 30%, … 90%.
Consumption Reduction Shortcut. Reduction % = [x / (100 + x)] × 100. When a price rises by x%, apply this directly to find the required consumption reduction. Result is always less than x%.
Reversal Rule for Percentage Comparisons. When A is x% MORE than B, B is less than A by [x / (100 + x)] × 100. Example: A is 20% more than B → B is 20/120 × 100 = 16.67% less than A. FPSC's most tested reversal trap.
25.6 Common Mistakes
Mistake 1: Dividing by the New Value
| Wrong | Right |
|---|---|
| Price 80 → 100. % increase = (20/100) × 100 = 20% (divides by new value) | % increase = (20/80) × 100 = 25% (divides by OLD value) |
The denominator is always where you STARTED.
Mistake 2: Successive Percentages Cancel Out
| Wrong | Right |
|---|---|
| Up 20%, down 20% = 0% change | Up 20%, down 20% = 4% NET LOSS = (20 × 20)/100 |
Mistake 3: BODMAS — Inventing Brackets
| Wrong | Right |
|---|---|
| 20 ÷ 5 × 2 = 20 ÷ (5 × 2) = 2 | 20 ÷ 5 × 2 → left to right → 4 × 2 = 8 |
Mistake 4: Wrong Percentage in 'Find the Original' Problems
| Wrong | Right |
|---|---|
| Saves Rs. 900; uses 70% as denominator → 900 × (100/70) = Rs. 1,286 | Savings = 30% of salary → 900 × (100/30) = Rs. 3,000 |
25.7 FPSC Trap Alerts
91 is Not Prime. 91 looks like a prime. It is not — 91 = 7 × 13. The prime between 90 and 100 is 97.
Square Root of a Negative Number. "What is the square root of −25?" Options will include 5, −5, and −2.5 — all wrong. The square root of a negative number is imaginary; correct answer is "None of these." Appeared word-for-word in CSS MPT 2022.
Percentage of Percentage. "What is 10% of 20% of 100?" Never add the percentages (10 + 20 = 30 is WRONG). Work step by step: 20% of 100 = 20 → 10% of 20 = 2.
The 1/8 Conversion. 1/8 expressed as a percentage = 12.5%. Pure memorisation question.
25.8 The 5-Minute Battle Card
| Topic | Key Rule / Shortcut |
|---|---|
| 10% of any number | Move decimal one place left |
| 20% / 5% | Double / halve the 10% answer |
| 25% / 50% | Divide by 4 / divide by 2 |
| Key conversions | 1/4 = 25%; 1/3 = 33.33%; 1/2 = 50%; 2/3 = 66.67%; 3/4 = 75%; 1/5 = 20%; 1/8 = 12.5% |
| % Change | (Difference ÷ OLD value) × 100. Always OLD value |
| Price 80 → 100 | Increase = 20/80 × 100 = 25%, NOT 20% |
| Successive % (up x%, down x%) | Always NET LOSS of (x × x)/100 |
| Finding original | Original = Known × (100 ÷ % it represents) |
| Consumption cut for price rise x% | Reduction = [x / (100 + x)] × 100 |
| Reversal (A is x% more than B) | B is less than A by [x / (100 + x)] × 100 |
| BODMAS | Brackets → Orders → D/M (left to right) → A/S |
| 36 ÷ 9 × 4 | 4 × 4 = 16 — NOT 1 |
| √(−25) | Imaginary → "None of these" |
| Square ending in 5 | 35² = (3 × 4) |
| Prime numbers near 100 | 97 prime; 91 = 7 × 13; 99 = 9 × 11 |
25.9 Practice MCQs (FPSC Level)
Part A — Basic Recall
Speed drills — every item under 20 seconds.
What is 20% of 150?
Show explanation
10% of 150 = 15. So 20% = 2 × 15 = 30. Zero-cancel: 2 × 15 = 30.
Trap: Distractors come from arithmetic miscount of zeros.
MPT 2022, 2024
A student scores 45 marks out of 60. What is her percentage?
Show explanation
45/60 = 3/4 = 75%. No calculation once you see the fraction.
Trap: Distractors near the answer trigger second-guessing.
MPT 2023
Evaluate: 36 ÷ 9 × 4
Show explanation
Equal priority → left to right. 36 ÷ 9 = 4, then 4 × 4 = 16.
Trap: 1 comes from doing 9 × 4 first (invented brackets).
MPT 2022, 2025
1/8 expressed as a percentage is:
Show explanation
1/8 × 100 = 100/8 = 12.5%. Pure memorisation.
Trap: 8% looks plausible because of the denominator.
MPT 2024
A price rises from Rs. 50 to Rs. 60. What is the percentage increase?
Show explanation
Difference = 10. Old = 50. % Increase = 10/50 × 100 = 20%.
Trap: 10% (just the difference); 16.67% (divides by new value 60).
MPT 2022, 2023
Part B — Trap-Based
Engineered around the cancel-out, sign, and wrong-denominator traps.
If 30% of a number is 90, what is the number?
Show explanation
X = 90 × (100/30) = 90 × 10/3 = 300. Verify 30% of 300 = 90.
Trap: 900 from multiplying without dividing; 270 from 90 × 3.
MPT 2023
A salary increases by 20% and then decreases by 20%. The net change is:
Show explanation
Net loss = (20 × 20)/100 = 4%. Verify: 1,000 → 1,200 → 960 = 4% loss.
Trap: 'They cancel out' is the most-marked wrong answer.
What is the square root of −25?
Show explanation
√(−25) is imaginary (5i). No real number squared gives −25.
Trap: A, B, C are all real numbers — all wrong.
MPT 2022
A man spends 65% of his salary and saves Rs. 1,750. What is his salary?
Show explanation
Saves 35% (not 65%). Salary = 1,750 × (100/35) = Rs. 5,000.
Trap: Rs. 2,692 = wrong denominator (65%).
What is 10% of 20% of 500?
Show explanation
20% of 500 = 100. Then 10% of 100 = 10. Never add percentages.
Trap: Adding 10 + 20 = 30%, then 30% of 500 = 150 (not listed); 50 is a similar lure.
MPT 2024
Part C — Elite Simulation
Multi-step and reversal items.
The price of sugar rises by 25%. By what percentage must a household reduce its consumption to keep total expenditure unchanged?
Show explanation
Reduction % = [25 / 125] × 100 = 20%. Verify: 125 × 8 = 1,000.
Trap: 25% (mirrors the rise); 16.67% (wrong reversal formula).
MPT 2023, 2025
If A's income is 20% more than B's, then B's income is what percentage LESS than A's?
Show explanation
A = 1.20B. B less than A by (0.20/1.20) × 100 = 16.67%.
Trap: 20% mirrors the forward direction — reversal is always smaller.
Evaluate: 50 − 10 + 6 × 2 ÷ 3
Show explanation
M/D first L→R: 6 × 2 = 12, 12 ÷ 3 = 4. Then 50 − 10 + 4 = 44.
Trap: Adding before multiplying gives 32 or 40.
A trader marks goods 40% above cost price and gives a 20% discount. What is his profit percentage?
Show explanation
CP = 100. MP = 140. After 20% discount: 140 × 0.80 = 112. Profit = 12 → 12%. Formula: 40 + (−20) + (40 × −20/100) = 12%.
Trap: 20% = ignoring the discount; 8% = sign error.
Consider: (1) Percentage change always uses the original (old) value as denominator. (2) A value that rises 30% then falls 30% results in a net loss of 9%. (3) 10% of 30% of 200 equals 6. Which are correct?
Show explanation
(1) ✓ always OLD; (2) ✓ (30 × 30)/100 = 9%; (3) ✓ 30% of 200 = 60, 10% of 60 = 6. All three correct.
Trap: Each partial option splits a fully-correct triple.
Answer Key & Full Solutions
Arithmetic, Percentages & BODMAS (Q1–15)
| Q | Correct | Type | Primary Trap | Why Others Fail |
|---|