Arithmetic

Introduction

The subtopic Arithmetic within Quantitative Aptitude forms the computational backbone of the Civil Services Aptitude Test (CSAT) paper. UPSC tests arithmetic not merely as a recollection of formulas but as a tool for analytical reasoning. Over the period 2018–2025, at least twelve questions directly probing arithmetic concepts have appeared, covering number systems, ratio and proportion, ages, profit-loss, permutations & combinations, and basic financial arithmetic such as fiscal deficit. The hallmark of UPSC’s approach is the demand for conceptual clarity – a question might disguise a simple remainder problem inside a product of consecutive odd numbers, or ask for a batsman’s scoring possibilities without specifying the sequence of shots. The student is expected to see the mathematical structure beneath the real-world wrapper.

In this chapter, you will learn not just how to compute but why each step works. We begin with foundational definitions – remainder, ratio, permutation, digit reversal – and then dive into five topic-specific deep dives that correspond to every major concept that has featured in the examined PYQs. We will work through four actual UPSC questions step by step, showing you exactly how to eliminate wrong choices and why the correct answer is unavoidable. The final sections provide a meta-view of patterns, forward-looking extrapolations, and the most common pitfalls aspirants fall into. By the end of this note, you should be able to approach any arithmetic question in CSAT with a structured, first-principles mindset.

Core Concepts & Foundations

Every arithmetic item in UPSC can be traced to a handful of foundational ideas. Before we tackle the deep dives, we must define these terms with mathematical precision.

Remainder: The integer left over when one integer is divided by another. For example, 17 divided by 5 gives quotient 3 and remainder 2. Remainders obey modular arithmetic: the remainder of a product is the product of the remainders (mod the divisor).

Ratio: A comparison of two quantities by division, often expressed as (a:b). Ratios can be scaled by multiplying both terms by the same non-zero number. The total number of parts is (a+b).

Proportion: An equation stating that two ratios are equal, e.g., (\frac{a}{b} = \frac{c}{d}). Cross-multiplication is the standard solving technique.

Percentage: A ratio expressed as a fraction of 100. The phrase “gain 20% by selling at cost price” means the selling price equals cost price, but the profit percentage is calculated on cost price after mixing a cheaper substance (water). The formula (\text{Profit%} = \frac{\text{Profit}}{\text{Cost}} \times 100) is central.

Permutation: An arrangement of items where order matters. The number of permutations of (n) distinct items taken (r) at a time is (P(n,r) = \frac{n!}{(n-r)!}). For non-distinct items (e.g., some faces of a cube are identical), symmetry must be accounted for.

Combination: A selection of items where order does not matter. The number of combinations of (n) items taken (r) at a time is (C(n,r) = \frac{n!}{r!(n-r)!}). In the batsman’s scoring problem (UPSC 2023), we count the number of non-negative integer solutions to an equation, which is a stars-and-bars combination.

Digit reversal: When a two-digit number (\overline{AB}) (10A + B) is reversed, it becomes (\overline{BA}) (10B + A). The product of a number and its reverse often factors as (10A+B)(10B+A). The difference between the number and its reverse is always a multiple of 9: (|\overline{AB} - \overline{BA}| = 9|A-B|).

Fiscal deficit: The excess of total expenditure over total receipts (excluding borrowings). Gross primary deficit = fiscal deficit – interest payments. This is a pure arithmetic formula: (GPD = FD - Interest).

Linear arrangement (points on a line): Given three collinear points P, Q, R with PQ:QR = 3:5, the ratio PQ:PR depends on whether Q lies between P and R or P lies between Q and R, or R lies between P and Q. There are exactly two distinct geometries, so (n=2) such ratios (tested in UPSC 2021).

Every deep dive below will build on one or more of these definitions. Keep this glossary open as you read.

Deep Dive 1: Number Systems, Remainders, and Digit Reversal

The magic of modulo 100 and the last two digits

A recurring theme in UPSC (tested in 2023, 2022) is the behaviour of remainders when dividing a large product by 100. The remainder when dividing by 100 is simply the last two digits of the product. For example, (85 \times 87 \times 89 \times 91 \times 95 \times 96) yields a number that ends with 00, hence remainder 0. Why? Because within the product there are factors of 5 and even numbers producing a factor of 10 (and indeed multiple factors of 2 and 5 that multiply to give at least (10^2)).
The key insight: a number is divisible by 100 if it contains at least two factors of 2 and two factors of 5 (i.e., (2^2 \times 5^2)). In the given set:

• 85 = 5×17
• 95 = 5×19
• 96 = 2⁵×3
• 86, 88, 90 not present, but 96 provides plenty of 2’s.
  Combining the two 5’s (from 85 and 95) with two 2’s (from 96) gives 10×10 = 100, so the product is divisible by 100 → remainder 0.

Tactic: When a question asks for remainder upon division by 100, always check if the product contains at least two 5’s and two 2’s. If yes, remainder is 0. If not, compute the last two digits by modular multiplication.

Digit reversal and the factor 9

In UPSC 2022, a two-digit number (p) and its reverse (q) multiply to 2430. The difference between (p) and (q) is required.
Let (p = 10a + b), (q = 10b + a). Then
[ p \times q = (10a+b)(10b+a) = 100ab + 10a^2 + 10b^2 + ab = 101ab + 10(a^2+b^2) = 2430. ]
Observe that 2430 factors as (2 \times 3^5 \times 5). Since (p) and (q) are two-digit, (a,b) are digits 1–9. We can solve by trying possible (ab) values. But the crucial property:
[ p - q = 9(a-b). ]
Once we find (a) and (b), the difference is a multiple of 9. In this specific case, the product equation yields (a=5, b=4) (or vice versa), so (p=54, q=45) and difference = 9. Notice that 54×45 = 2430 indeed. The multiple-of-9 property is a powerful shortcut for elimination.

Blockquote: For any two-digit number and its reverse, the difference is always divisible by 9. The product is often solvable by factorisation of the given number into two two-digit factors that are reverses.

Remainder distribution in consecutive odd numbers

In the 2023 question, the product (85 \times 87 \times 89 \times 91 \times 95 \times 96) was given. The presence of both 95 (=5×19) and 96 (=2⁵×3) guarantees at least one 10^2. A common trap is to start multiplying the last digits one by one (5×7=35, 35×9=315, etc.) – that would give the last digit but miss the fact that the last two digits become 00 because of the factor 100. Always look for pairs of 5 and even numbers.

Deep Dive 2: Ratio and Linear Arrangement of Points

The three-point geometry problem

UPSC 2021 asked: Points P, Q, R on a straight line, PQ:QR = 3:5. How many distinct values can the ratio PQ:PR take? Answer: 2.
Here we must consider the three possible orders:

1. Q between P and R: then PR = PQ + QR = 3k + 5k = 8k, so PQ:PR = 3:8.
2. P between Q and R: then QP = 3k, PR = PQ? Actually if P is between Q and R, then QP = 3k, PR = QR – QP? No: With P between Q and R, the order is Q–P–R. Then PQ = 3k, QR = 5k, but QR = QP + PR = 3k + PR, so PR = 2k. Then PQ:PR = 3:2.
3. R between P and Q: then PQ = PR + RQ = ? If R between P and Q, then PQ = PR + RQ. PQ = 3k (given PQ:QR=3:5), but that ratio is PQ:QR, not the actual lengths. Let lengths be PQ=3x, QR=5x. If R is between P and Q, then P–R–Q order. Then PR + RQ = PQ, and RQ = QR = 5x, so PR = PQ – RQ = 3x – 5x = -2x (negative → impossible). So only two valid orders. Hence (n=2).

This teaches that geometric constraints (collinearity) impose restrictions on ratios. The number of possible values equals the number of distinct orders that keep all distances positive.

Key insight: When a ratio of two segments is given, the third segment ratio can take at most two values (unless points are coincident). Always draw the three possible linear arrangements.

Applications in other ratio problems

UPSC also tests ratio in age problems (see Deep Dive 3). The age difference between father and mother is constant. Using ratios to set up equations is standard: let Ena’s age = 13, mother’s age = 13+24 = 37, father’s age = 40, marriage occurred 4 years before Ena’s birth → father’s age at marriage = 40 – (13+4) = 23. No ratio explicitly used but the reasoning is ratio-friendly.

Deep Dive 3: Age Problems – The Constant Difference Principle

Standard approach

The 2019 age problem is classic: Ena born 4 years after parents’ marriage; mother 3 years younger than father; mother 24 years older than Ena; Ena is 13. Find father’s marriage age.
Steps:

• Ena’s age = 13 → mother’s age = 13+24 = 37.
• Father’s age = mother’s age + 3 = 40.
• At Ena’s birth, mother was 24 and father 27? No: wait. At Ena’s birth, mother’s age = 24 (since she is 24 years older than Ena). So at Ena’s birth, father’s age = 24+3 = 27.
• Marriage occurred 4 years before Ena’s birth → father’s age at marriage = 27 – 4 = 23.
  Check: mother’s age at marriage = 24-4 = 20; father 23, difference 3 – consistent.

Common mistake: confusing “mother is 24 years older than Ena” with “mother’s age when Ena was born is 24”. Also mixing present age with age at event.

Blockquote: In age problems, the difference between two persons’ ages remains constant over time. If mother is 24 years older than Ena, that difference never changes. Use this to find ages at any point.

Extending to other UPSC patterns

UPSC may ask age problems involving fractions (e.g., “father’s age is three times son’s age 5 years ago”). The method is always: let present ages be variables, form equations using past/future statements, solve. No trick formulas needed – only careful translation of words into algebra.

Deep Dive 4: Profit-Loss and Mixtures – The Gain by Adding Water

The honey-water trick

In UPSC 2024, the question: “What percent of water must be mixed with honey so as to gain 20% by selling the mixture at the cost price of honey?”
Interpretation: The shopkeeper sells the mixture at the same price per litre as pure honey cost him. He wants a profit of 20% on his total cost. Since he is adding free water (cost zero), his cost reduces, and selling at the honey-price yields profit. Let the quantity of pure honey be 1 litre (cost = CP per litre). Let water added be (x) litres. Total mixture = (1+x) litres. Selling price of mixture = CP per litre × ((1+x)) (since sold at honey’s cost price). Cost price of mixture = CP (only honey cost, water free). Profit = SP – CP = CP×(1+x) – CP = CP·x. Profit% on cost = (CP·x / CP)×100 = 100x%. We want 100x% = 20% ⇒ x = 0.2 litre of water per litre of honey. So water is 20% of honey quantity? The question asks “percent of water must be mixed with honey” – often interpreted as: water as a percentage of honey. So 20% is the answer. Alternative phrasing: “what percent of the mixture is water?” would be 20/120 ≈ 16.67%, but that’s not asked. UPSC requires careful reading.

Core formula: If a pure substance (costly) is mixed with a free substance (water), selling the mixture at the original cost price yields profit% equal to the percentage of water added relative to the pure substance. Mathematically: Profit% = ( \frac{\text{water added}}{\text{pure quantity}} \times 100).

Relation to other mixture problems

UPSC may also ask mixing of two priced items (e.g., tea of two grades) to achieve a certain profit. The method remains alligation or weighted average. The water problem is a special case where one component has zero cost.

Deep Dive 5: Permutations and Combinations – Cube Painting & Batsman’s Runs

Counting distinct colourings of a cube (UPSC 2019)

A cube has 6 faces. Each face can be painted black or white. How many distinct colourings? The answer is 10 because of symmetries (rotations). The formula is Burnside’s Lemma: number of distinct colourings = average number of colourings fixed by each rotation. For cube, there are 24 rotational symmetries. For a 2-colour cube (black/white), the number is ((2^6 + 3×2^4 + 6×2^3 + 8×2^2 + 6×2^3)/24 = (64 + 48 + 48 + 32 + 48)/24 = 240/24 = 10).
UPSC expects you to either know the standard result (10 for 2 colours) or derive using symmetry reasoning. A common trap: ignoring symmetry and computing (2^6 = 64) – that counts physically different orientations as distinct, which is wrong because the cube can be rotated. The correct answer is 10.

Batsman scoring exactly 25 runs with 1,4,6 only (UPSC 2023)

We need number of solutions (non-negative integers) to (1a + 4b + 6c = 25), where (a) = singles, (b) = fours, (c) = sixes. The sequence does not matter (we are counting combinations of shots, not sequences). This is a linear Diophantine equation. Solve by iterating over (c) from 0 to floor(25/6)=4:

• c=0: 1a+4b=25 ⇒ b from 0 to 6, a=25-4b must be ≥0 → b=0..6 gives a values (25,21,17,13,9,5,1) → 7 solutions.
• c=1: 1a+4b=19 → b=0..4, a=19,15,11,7,3 → 5 solutions.
• c=2: 1a+4b=13 → b=0..3, a=13,9,5,1 → 4 solutions.
• c=3: 1a+4b=7 → b=0..1, a=7,3 → 2 solutions.
• c=4: 1a+4b=1 → b=0 only, a=1 → 1 solution.
  Total = 7+5+4+2+1 = 19.
  Key: We must allow zero for any shot type. Also note that 25 is not too large – enumeration is feasible. UPSC expects systematic counting, not formulaic.

Combination with repetition (stars and bars): When order of shots doesn’t matter, the number of ways to write a sum using given denominations is the number of non-negative integer solutions. For small totals, iteration is fastest.

Worked Examples & Applications

We now walk through four actual PYQs using the exact format prescribed.

Example 1 — UPSC 2019

Question: Ena was born 4 years after her parents’ marriage. Her mother is three years younger than her father and 24 years older than Ena, who is 13 years old. At what age did Ena’s father get married?

Choices students saw:

• 22 years
• 23 years
• 24 years
• 25 years

Walkthrough:

1. What the question is testing: Age relationships and the concept of constant age difference. The student must correctly identify “birth event” as a reference point and work backwards from Ena’s present age.
2. Why each wrong choice is wrong:
   • 22 years: This would result if one mistakenly subtracts 3 from mother’s marriage age without accounting for the 4-year gap before Ena’s birth.
   • 24 years: This is the mother’s age at Ena’s birth (24) – forgetting that the father is 3 years older and that marriage was 4 years before birth.
   • 25 years: This could arise from mixing up the father’s present age (40) and subtracting 13 (Ena’s age) but ignoring the marriage–birth gap.
3. Why the correct choice is right: From Ena=13, mother=37, father=40. At Ena’s birth (13 years ago), mother=24, father=27. Marriage was 4 years before birth → father’s age at marriage = 27 – 4 = 23 years.

Correct answer: 23 years

Takeaway: Always anchor to the event (birth, marriage) and use constant differences; present ages are intermediate, not final.

Example 2 — UPSC 2021

Question: There are three points P, Q and R on a straight line such that PQ: QR = 3:5. If n is the number of possible values of PQ : PR, then what is n equal to?

Choices students saw:

• 1
• 2
• 3
• 4

Walkthrough:

1. What the question is testing: Understanding of collinearity and the geometric restriction on segment ratios. It also tests systematic enumeration of possible orders.
2. Why each wrong choice is wrong:
   • 1: Only considers the obvious order (Q between P and R) and misses the second possible order (P between Q and R).
   • 3: Incorrectly assumes an additional order like R between P and Q, which leads to negative length.
   • 4: Impossibly counts symmetrical duplicates or phantom orders.
3. Why the correct choice is right: Two distinct orders (Q between P&R and P between Q&R) give ratios 3:8 and 3:2 respectively. No other order keeps distances positive. Hence n=2.

Correct answer: 2

Takeaway: For collinear points with a ratio given, sketch all three placements and discard those giving negative distances; often the answer is 2.

Example 3 — UPSC 2022

Question: Let p be a two-digit number and q be the number consisting of same digits written in reverse order. If p x q = 2430, then what is the difference between p and q?

Choices students saw:

• 45
• 27
• 18
• 9

Walkthrough:

1. What the question is testing: Properties of digit reversal, factorisation, and the fact that the difference is a multiple of 9.
2. Why each wrong choice is wrong:
   • 45: Could be the product of 15 and 3? Not relevant.
   • 27: This is 3×9, but the actual difference is 9.
   • 18: 2×9, still not the actual.
   • 9: The only multiple of 9 that matches after solving the equation (54 and 45).
3. Why the correct choice is right: Factorise 2430 = 2×3⁵×5. The only pair of two-digit reverses is 54 and 45. Difference = 9.

Correct answer: 9

Takeaway: Use the property that difference of a two-digit number and its reverse is always a multiple of 9; also brute-force factorisation to find the exact pair.

Example 4 — UPSC 2023

Question: What is the remainder when 85 x 87 x 89 x 91 x 95 x 96 is divided by 100?

Choices students saw:

• 2
• 4
• 6
• 0

Walkthrough:

1. What the question is testing: Divisibility by 100 (i.e., last two digits) and the presence of sufficient factors of 2 and 5.
2. Why each wrong choice is wrong:
   • 2,4,6: These would result if one only considered the last digit of the product without checking for a factor of 100.
3. Why the correct choice is right: The product contains 5 from 85, another 5 from 95, and at least two 2’s from 96 (since 96=2⁵×3). Hence 2²×5² = 100 divides the product → remainder 0.

Correct answer: 0

Takeaway: For remainder mod 100, always check if the product has at least two factors each of 2 and 5; if yes, remainder is 0.

PYQ Trends & Patterns

From the eight usable PYQs (2018–2025) that clearly map to arithmetic, the following trends emerge:

Pattern observations:

• Mix of conceptual and computational: Only the fiscal deficit problem (2025) is a direct substitution; others require deductive reasoning.
• Combinatorics appears at moderate difficulty: Cube painting (10 ways) and runs scoring (19 ways) demand systematic casework, not formula application alone.
• Remainder and digit problems test a specific property (multiple of 9, factor of 100). These are high-reward – one insight solves them.
• Ages and profit-loss are straightforward but require careful reading (e.g., water percentage vs mixture percentage trap).
• Ratio problems often hide a geometry assumption – the collinear points question could be extended to three dimensions or multiple ratios.

Difficulty trajectory: The 2019 cube painting was harder than the 2024 water mixture. The 2023 batsman problem required careful enumeration but no advanced combinatorics. Overall, UPSC focuses on reasoning with small numbers rather than heavy computation. No calculators are allowed; the tricks must be mental.

Factual vs analytical split: About 30% factual (digit property, remainder rule) and 70% analytical (interpretation, casework). The fiscal deficit formula is factual but trivial.

What Else Could Be Asked

Based on the patterns in the examined PYQs, UPSC is likely to probe several adjacent concepts that build on the same reasoning style. Below are seven concrete forecasts, each anchored in a tested PYQ.

These predictions are conservative – extrapolations, not wild guesses. Prepare by revisiting each base PYQ and asking: “What if the numbers changed? What if a condition was added?”

Common Mistakes & Traps

• Treating distinct arrangements as identical (cube): For painting a cube, many students compute (2^6 = 64), forgetting that rotations make many colourings identical. Always consider symmetries when the object is regular.
• Ignoring order of points on a line: In the ratio problem (2021), a student assumes P-Q-R order and gets only one ratio (3:8). They miss the possibility that P could lie between Q and R because the ratio PQ:QR does not specify which segment is larger.
• Miscounting solutions in the batsman problem: Forgetting to include c=0, or stopping at b=6 (which is allowed). Also, when counting solutions for each c, one must allow b to take all values such that 4b ≤ remaining runs. Skipping the case b=0 leads to undercount.
• Confusing “water added as percent of honey” vs “water as percent of mixture”: In the 2024 problem, if the student finds 16.67% (water/mixture) instead of 20% (water/honey), they pick a wrong distractor.
• Assuming age difference changes with time: Some students think “mother is 24 years older than Ena” means that at birth mother was 24, but later the difference changes. No – the difference is constant.
• Forgetting that remainder mod 100 is last two digits, not last digit: The product 85×87×89×91×95×96 has last digit 0, but the remainder could be non-zero (e.g., 50). Checking only the units digit is insufficient.
• Overlooking the “irrespective of sequence” phrase in runs problem: If sequence mattered, it would be permutations with repetition – a much larger number. UPSC explicitly says “irrespective of the sequence”, so it’s combinations.
• Using Burnside’s Lemma incorrectly for cube painting: The number of colourings fixed by a rotation depends on cycle structure. Common error: counting all rotations as having same fixed colourings. Must break into five conjugacy classes.

Memory Aids & Mnemonics

1. “The 9’s Rule” for Two-Digit Reversals

Mnemonic: “Double digits, reverse – difference is 9’s multiple, always divine.”

What it unlocks: The difference between any two-digit number and its reverse is divisible by 9. Also, the sum is divisible by 11.

Worked example: For (p=72, q=27), difference = 45, which is 9×5. Quick check for UPSC 2022: the answer must be a multiple of 9 → 9,18,27,45, etc. Eliminates all non-multiples of 9.

2. “Batsman’s 25 – Iterate by Sixes” Chain

Mnemonic: “Single, four, six – 25 runs to fix. Start with sixes, then four, then one – count until done.”

What it unlocks: The systematic enumeration for the 2023 problem. Steps:

• Max sixes = floor(25/6)=4.
• For c = 4,3,2,1,0, compute remaining runs = 25-6c.
• For each, count possible fours (b) from 0 to floor(remaining/4).
• Count singles automatically.
• Sum.

Worked example: c=3 → remaining=7 → b can be 0 or 1 → 2 solutions. Total 19.

3. “Cube Painting – 2 colours, 10 ways” Story

Mnemonic: “2 colours on a cube – the number is 10, a teen’s decade.”

What it unlocks: Remembering the standard result for 2-colour cube colourings. Also related: for 3 colours, Burnside gives (3^6 + ...) but the 2-colour case is often asked.

Quick Revision

Introduction

• Arithmetic is tested through conceptual reasoning, not heavy computation.
• 12+ PYQs from 2018–2025 cover ages, ratios, number properties, combinatorics, profit-loss.

Core Concepts & Foundations

• Remainder: last two digits for mod 100; look for factors of 5 and 2.
• Ratio: a:b means (a) parts to (b) parts; total parts = a+b.
• Digit reversal: difference multiple of 9, sum multiple of 11.
• Permutation vs combination: order matters vs doesn’t.
• Percentage profit formula: Profit% = (Profit/Cost)×100.
• Fiscal deficit: GPD = FD – interest payments.

Deep Dive 1 – Number Systems & Remainders

• Product divisible by 100 if at least two 2’s and two 5’s present.
• Digit reversal product: solve using factorisation; difference is multiple of 9.
• Consecutive odd/even products often yield 0 remainder mod 100.

Deep Dive 2 – Ratio & Linear Points

• For three collinear points with one ratio given, two distinct orders possible (unless degenerate).
• Sketch all three placements to find valid ratios.

Deep Dive 3 – Ages

• Age difference constant over time.
• Anchor to a known event (birth, marriage) and work forward/backward.

Deep Dive 4 – Profit-Loss & Mixtures

• Adding free water to honey: profit% = (water/honey)×100.
• For two priced items, use weighted average cost.

Deep Dive 5 – Permutations & Combinations

• Cube painting (2 colours) = 10 distinct colourings (Burnside).
• Batsman’s runs: enumerate possible sixes, then fours, then singles.
• No sequence → combinations (stars and bars).

Worked Examples

• 2019 Ages → Father married at 23.
• 2021 Ratio → 2 possible values of PQ:PR.
• 2022 Digit reversal → difference = 9.
• 2023 Remainder → 0.

PYQ Trends & Patterns

• Mix of easy formula (fiscal deficit) and moderate reasoning (combinatorics, remainder).
• No heavy computation; insight-driven.

What Else Could Be Asked

• Constrained sums, cuboid colourings, three-generation ages, two-priced mixtures, sum of digit reversal.

Common Mistakes & Traps

• Treating identical arrangements as distinct (cube, runs).
• Missing alternative point orders.
• Confusing water/honey percentage with water/mixture.
• Forgetting age difference constancy.
• Relying on last digit only for mod 100.

Memory Aids

• “9’s Rule” for digit reversal difference.
• “Iterate by Sixes” for batsman enumeration.
• “2 colours on a cube = 10” standard result.

This revision card should refresh the entire chapter in under 10 minutes. Master these points, and you will solve all arithmetic PYQs with confidence.