Arithmetic Questions for CDS

Number System & Divisibility

The foundation. CDS loves testing divisibility rules, prime factorization, and number classification. You must instantly recognize whether a number is divisible by 2, 3, 5, 9, 11 without a calculator. Prime factorization is the key to solving HCF/LCM problems. Remember: every integer can be uniquely expressed as a product of primes (Fundamental Theorem of Arithmetic). This is tested indirectly in almost every arithmetic section.

• Divisibility by 2: last digit is even. By 5: last digit is 0 or 5. By 10: ends in 0.
• Divisibility by 3: sum of digits divisible by 3. By 9: sum of digits divisible by 9.
• Divisibility by 11: alternating sum of digits (left to right) is divisible by 11.
• Prime factorization: break any number into prime factors — essential for HCF/LCM.
• Even/Odd: even + even = even; odd + odd = even; even + odd = odd. Multiply: even × anything = even.

HCF (GCD) & LCM

HCF = Highest Common Factor (largest number dividing both). LCM = Least Common Multiple (smallest number divisible by both). Prime factorization method is fastest: HCF takes lowest powers of common primes; LCM takes highest powers of all primes. Key relationship: HCF(a,b) × LCM(a,b) = a × b. CDS often asks word problems: 'largest number dividing both', 'smallest number divisible by all', or 'arrange in rows/columns equally'.

• HCF by prime factorization: take LOWEST power of COMMON primes only.
• LCM by prime factorization: take HIGHEST power of ALL primes present.
• HCF(a,b) × LCM(a,b) = a × b — use to verify or find unknown.
• Word cue 'largest/greatest' → HCF. Word cue 'smallest/least' → LCM.
• For 3+ numbers: apply HCF/LCM pairwise or use prime factorization of all.

Percentages, Profit-Loss & Discount

Percentage is 'per hundred'. Profit-Loss and Discount are the most tested arithmetic topics in CDS. Master the formulas and always define: Cost Price (CP), Selling Price (SP), Marked Price (MP). Profit = SP - CP. Loss = CP - SP. Discount is reduction on MP. These appear in almost every CDS paper, often combined with ratios or averages.

• Profit% = (Profit / CP) × 100. Loss% = (Loss / CP) × 100.
• SP = CP + Profit (or CP - Loss). Profit/Loss always calculated on CP, not SP.
• Discount = MP - SP. Discount% = (Discount / MP) × 100.
• Marked Price, Discount, Selling Price: MP - Discount = SP.
• Successive discounts: apply one, then apply next to the result (not additive).

Simple Interest & Compound Interest

Simple Interest (SI) is linear; Compound Interest (CI) is exponential. SI is rarely tested alone in CDS — usually combined with CI or as a comparison. CI is more common. Key: identify Principal (P), Rate (R% per annum), Time (T in years). SI grows by same amount each year; CI grows by increasing amounts (interest on interest). Always read carefully: 'per annum', 'quarterly', 'half-yearly', 'monthly' — this changes the formula.

• SI = (P × R × T) / 100. Amount = P + SI.
• CI: Amount = P(1 + R/100)^T. CI = Amount - P.
• For CI compounded half-yearly: use R/2 and T×2. For quarterly: R/4 and T×4.
• Difference between CI and SI grows with time — tested in 'compare' questions.
• Read 'per annum' vs 'per quarter' vs 'per month' carefully — changes denominator.

Ratio, Proportion & Averages

Ratio compares two quantities (a:b). Proportion states two ratios are equal (a:b = c:d). Averages (mean, median, mode) are tested in word problems. CDS loves 'dividing in ratio' questions and 'average speed/cost' problems. Remember: if a:b = 3:4, then a = 3k and b = 4k for some k. This substitution solves most ratio problems.

• Ratio a:b can be written as a/b. Simplify by dividing by GCD.
• If a:b = 3:4 and a + b = 70, then a = 30, b = 40 (use a = 3k, b = 4k).
• Proportion: a/b = c/d means a×d = b×c (cross-multiply).
• Average = Sum / Count. For average speed: total distance / total time (not average of speeds).
• Weighted average: (value1 × weight1 + value2 × weight2) / (weight1 + weight2).

• Confusing Profit% and Markup%. Profit% is always on CP, not on SP. If CP = 100, SP = 150, profit% = 50%, NOT 33%.
• Treating Successive Discounts as additive. 10% + 20% discount is NOT 30%. It is 100 × 0.9 × 0.8 = 72, so 28% total discount.
• Averaging speeds directly. If you travel 100 km at 50 km/h and 100 km at 100 km/h, average speed is NOT 75 km/h. Use total distance / total time = 200 / 3 = 66.67 km/h.
• Forgetting to adjust R and T for CI when compounding is not annual. If compounded half-yearly, use R/2 and 2T, not R and T.
• Mixing up HCF and LCM in word problems. 'Largest number dividing' = HCF. 'Smallest number divisible by' = LCM. Reverse these and you lose marks.

• PEMDAS for order: Parentheses, Exponents, Multiply/Divide, Add/Subtract. Always do division before subtraction in interest problems.
• For Profit-Loss: CP is the base. Profit% = (SP - CP) / CP × 100. Think 'profit relative to what you paid'.
• HCF = 'Highest' = take LOWEST powers. LCM = 'Least' = take HIGHEST powers. Opposite logic, easy to flip — write it down.
• SI vs CI: SI is 'Simple' (linear, boring). CI is 'Compound' (exponential, grows faster). CI always > SI for same P, R, T > 1 year.
• Discount on MP, Profit on CP. Two different bases. Mark them clearly in your working.

• CDS Arithmetic is 25-30% of Maths. Expect 8-10 questions. Most are word problems, not pure calculation. Read the question twice before solving.
• Profit-Loss and SI/CI dominate. Practice 'mixed' questions: e.g., 'cost price increases 10%, then 20% discount applied'. These are common in recent papers.
• Time management: Arithmetic questions should take 1.5-2 minutes each. If stuck, skip and return. Don't waste 5 minutes on one question.
• Divisibility and HCF/LCM are often hidden in longer problems. Recognize them early. E.g., 'arrange books in equal rows' = HCF problem.
• Calculator is NOT allowed in CDS. Practice mental math and approximation. For 1000 × 1.1^2, compute 1.1^2 = 1.21 mentally, then multiply.