Arithmetic — SSC CHSL Tier 1 Study Notes

Overview

Arithmetic forms the backbone of the Quantitative Aptitude section in SSC CHSL, typically accounting for 12–15 questions out of 25 total quant questions. This is the highest-weightage topic cluster, covering practical problems on percentage, profit-loss, ratio, average, interest, speed-time-distance, mixtures, ages, and time-work. Unlike pure number theory or geometry, arithmetic tests your ability to translate real-world scenarios into mathematical equations and solve them efficiently.

Mastery of arithmetic directly impacts your score because these questions are moderately paced — neither too fast (like simplification) nor too slow (like geometry proofs). Most arithmetic problems require 60–90 seconds if you recognize the pattern and apply the right formula. The key challenge is **not the complexity of math** but **understanding what the question asks** and avoiding calculation errors under time pressure.

Focus on building strong fundamentals in each sub-topic, memorizing core formulas, and practicing enough to recognize question types instantly. Arithmetic rewards consistent practice more than any other quant topic.

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Key Concepts

• **Percentage as a fraction**: Percent means "per hundred." Converting percentages to fractions (e.g., 25% = 1/4) or decimals (0.25) speeds up mental calculation and reduces errors.

• **Profit and Loss revolve around three prices**: Cost Price (CP), Selling Price (SP), and Marked Price (MP). Discount is on MP; profit/loss is on CP. Mixing these up is the most common mistake.

• **Ratio locks relative quantities**: A ratio a:b means for every a units of one quantity, there are b units of another. When the total is given, the actual values are a/(a+b) × Total and b/(a+b) × Total.

• **Average is total divided by count**: If you know two of {average, total, count}, you can find the third. Weighted averages come up in age and marks problems where different groups have different counts.

• **Simple Interest is linear; Compound Interest is exponential**: SI = (P × R × T)/100 grows the same amount each year. CI = P(1 + R/100)ⁿ – P grows on the previous year's total. Remember for half-yearly CI, halve the rate and double the time.

• **Speed = Distance/Time triangle**: Cover one variable, and the other two give you the answer. Relative speed is additive when moving toward each other, subtractive when in the same direction.

• **Mixture and Alligation use the cross-method**: When mixing two ingredients at different prices/concentrations, the ratio in which they're mixed is (C₂ – Mean):(Mean – C₁), where C₁ < Mean < C₂.

• **Time and Work use efficiency**: Work = Rate × Time. If A does a job in 'a' days, A's one-day work = 1/a. Combined work of A and B = 1/a + 1/b per day.

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Formulas / Key Facts

**Percentage**

• Increase/Decrease % = [(New – Old)/Old] × 100
• Successive changes: If two changes x% and y%, net = x + y + xy/100

**Profit, Loss, Discount**

• Profit = SP – CP; Loss = CP – SP
• Profit % = (Profit/CP) × 100; Loss % = (Loss/CP) × 100
• SP = CP × (100 + Profit%)100 or CP × (100 – Loss%)/100
• Discount = MP – SP; Discount % = (Discount/MP) × 100

**Ratio and Proportion**

• If a:b = c:d, then ad = bc (cross-multiplication)
• a:b:c with total T → parts are aT/(a+b+c), bT/(a+b+c), cT/(a+b+c)

**Average**

• Average = Sum of observations / Number of observations
• Weighted average of groups: (n₁A₁ + n₂A₂)/(n₁ + n₂)

**Simple Interest**

• SI = (P × R × T)/100
• Amount = Principal + SI

**Compound Interest**

• CI = P(1 + R/100)ⁿ – P
• For half-yearly: replace R with R/2, n with 2n
• For quarterly: R/4 and 4n

**Time, Speed, Distance**

• Distance = Speed × Time
• Average speed = Total distance / Total time (not the average of two speeds unless times are equal)
• Relative speed (opposite directions) = S₁ + S₂
• Relative speed (same direction) = |S₁ – S₂|

**Mixture and Alligation**

• Ratio of cheaper to dearer = (Dearer price – Mean price):(Mean price – Cheaper price)

**Problems on Ages**

• Present age ± years = past/future age
• Set up equations using ratios or differences given in the problem

**Time and Work**

• Work done = Efficiency × Time
• If A completes in 'a' days and B in 'b' days, together = ab/(a+b) days
• Pipe filling/emptying: inlet adds, outlet subtracts from total capacity per hour

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Worked Examples

**Example 1 (Profit-Loss):** A shopkeeper marks an article 40% above cost price and gives a 20% discount. Find profit %. *Solution:* Let CP = 100. MP = 140. SP = 140 × 80/100 = 112. Profit = 112 – 100 = 12. Profit % = 12%.

**Example 2 (SI vs CI):** On ₹8000 at 10% per annum for 2 years, find the difference between CI and SI. *Solution:* SI = (8000 × 10 × 2)/100 = ₹1600. CI = 8000(1.1² – 1) = 8000 × 0.21 = ₹1680. Difference = 1680 – 1600 = ₹80.

**Example 3 (Time-Speed):** A train 150 m long crosses a pole in 15 seconds. Find its speed in km/h. *Solution:* Speed = 150 m / 15 s = 10 m/s = 10 × 18/5 = 36 km/h.

**Example 4 (Mixture):** In what ratio must sugar at ₹30/kg be mixed with sugar at ₹45/kg so the mixture costs ₹36/kg? *Solution:* Use alligation: (45 – 36):(36 – 30) = 9:6 = 3:2. Ratio is 3:2 (cheaper:dearer).

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Common Mistakes

• **Confusing discount base with profit base**: Discount is always on Marked Price; profit/loss is always on Cost Price. Mixing these gives wrong SP or profit %.

• **Using wrong formula for successive percentage changes**: Students add 10% and 20% to get 30%, but the correct net = 10 + 20 + (10×20)/100 = 32%.

• **Average speed ≠ average of speeds**: When a person travels two equal distances at different speeds, average speed is 2S₁S₂/(S₁+S₂), not (S₁+S₂)/2.

• **Forgetting to convert SI/CI time and rate for compounding periods**: For half-yearly CI at 12% for 1 year, use R=6%, n=2, not R=12%, n=1.

• **Adding work rates instead of inverting**: If A does work in 10 days and B in 15 days, their combined time is NOT 10+15. Use 1/10 + 1/15 = 1/6, so 6 days together.

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Quick Reference

• **Profit % on CP, Discount % on MP** — never mix the base.
• **Successive % change = x + y + xy/100** — memorize this for two-step problems.
• **SI grows linearly; CI grows exponentially** — CI > SI after first period.
• **Alligation cross-rule** — (High – Mean):(Mean – Low) gives the ratio.
• **Combined work rate = sum of individual rates** — then invert to get time.
• **Average speed for equal distances** = 2S₁S₂/(S₁+S₂) — harmonic mean, not arithmetic.