CAT — Arithmetic: Percentages, Profit & Loss, Ratio, Time & Work, Speed & Distance

Overview

Arithmetic for CAT tests accuracy with speed, number sense, and smart approximations. Focus on mastering core formulas, mental shortcuts, and pattern recognition. We'll cover: definitions, key formulas, worked examples, time-saving tricks, and 50 practice MCQs that mimic CAT-style reasoning.

Percentages

Basics: Percentage = (part/whole) × 100. Part = (percentage × whole)/100. Percentage change = ((new − old)/old) × 100.

Tricks: 1) Use 1% = 1/100; 2) For successive percentage increases/decreases use multiplicative factor: +p% → multiply by (1 + p/100), −p% → (1 − p/100). 3) For quick reverse: increase of 20% then decrease of 20% doesn’t return to original: 1.2×0.8=0.96 → net −4%.

Worked example: A price increases by 15% then by 10%. Net factor = 1.15×1.10 = 1.265 → net +26.5%.

Profit & Loss (P&L)

Formulas: Profit = SP − CP. Profit% = (Profit/CP) × 100. Loss% = (Loss/CP) × 100. SP = CP × (1 + profit%/100).

Markups & Discounts: If marked price (MP) and discount d% given: SP = MP × (1 − d/100). Effective discount on CP can be found by combining factors.

Trick: Use CP as 100 when percentages are used; if profit% = p, then SP = 100×(1+p/100).

Ratio & Proportion

Basics: If a:b = m:n, then a/b = m/n. To divide quantity Q in ratio x:y use Q × x/(x+y) and Q × y/(x+y).

Mixtures: Use alligation method for mixing concentrations. For speed-time-distance ratios, use inverse relations where needed — e.g., time ∝ 1/speed when distance fixed.

Worked example: Split ₹420 in ratio 2:5 → parts = 2/7 & 5/7: 120 and 300.

Time & Work

Work model: If A does W1 units/day and B does W2 units/day, together they do W1+W2 units/day. Time = Total Work / Rate.

Tricks: Convert to 'work units' per day. For efficiency change, use relative rates. For pipes & cisterns, treat filling as positive rate, draining as negative.

Worked example: A does job in 10 days → rate = 1/10 per day; B in 15 days → together 1/10+1/15=1/6 → 6 days together.

Time, Speed & Distance

Formula: Distance = Speed × Time. Average speed for equal distances = harmonic mean of speeds: 2ab/(a+b). Average speed for equal times = arithmetic mean.

Tricks: For trains and relative speed, convert to same units; for overtaking use relative speed; for boat-stream problems, use speed_{up} = u+v, speed_{down} = u−v.

Worked example: If a car goes 60 km/h for 2h and 80 km/h for 1h, average speed = total dist / total time = (120+80)/3 = 200/3 ≈ 66.67 km/h.

Quick Formula Summary

Concept	Formula / Note
Percentage	`(part/whole)×100`
Profit %	`((SP−CP)/CP)×100`
Ratio split	`Q×(x/(x+y)), Q×(y/(x+y))`
Time & Work	`Time = Work/Rate`
Speed	`Distance = Speed×Time`

50 Practice MCQs — Arithmetic (Answers highlighted)

These questions cover mental shortcuts, approximations, multi-step reasoning, and typical CAT arithmetic traps. Answers are highlighted.

Q1. What is 15% of 240?

A. 24

B. 36

C. 36

D. 30

Q2. A shop sells an article with 20% profit on cost price ₹500. Selling price is:

A. ₹600

B. ₹600

C. ₹550

D. ₹620

Q3. Ratio split: Divide ₹840 in ratio 3:5. Larger share =

A. ₹315

B. ₹600

C. ₹525

D. ₹360

Q4. If A can do a job in 12 days and B in 18 days, together they finish in:

A. 6 days

B. 7.2 days

C. 7.2 days

D. 9 days

Q5. A travels 120 km at 40 km/h and returns at 60 km/h. Average speed =

A. 48 km/h

B. 48 km/h

C. 50 km/h

D. 54 km/h

Q6. If price increases by 25%, by what percent must it decrease to return to original?

A. 20%

B. 20%

C. 16%

D. 25%

Q7. A shopkeeper marks up price by 30% on CP and allows 10% discount on MP. Net profit% on CP is:

A. 17%

B. 17%

C. 20%

D. 13%

MP = 1.3CP; SP = 1.3×0.9=1.17 CP → profit 17%.

Q8. If 5 workers take 20 days, how many workers needed to finish in 8 days (same productivity)?

A. 12.5

B. 12.5 (13 workers)

C. 10

D. 8

Q9. Convert 3/8 to percentage:

A. 37.5%

B. 37.5%

C. 33.3%

D. 40%

Q10. A student scores 80% in exams. If full marks 500, marks obtained =

A. 380

B. 400

C. 420

D. 360

Q11. If x:y = 3:4 and x+y = 28, x = ?

A. 12

B. 16

C. 9

D. 18

Q12. A boat covers 30 km downstream in 2 hours and upstream in 3 hours. Speed of stream =

A. 2 km/h

B. 2 km/h

C. 3 km/h

D. 1 km/h

Q13. If price is discounted by 12.5%, what fraction is customer paying?

A. 7/8

B. 7/8

C. 5/6

D. 3/4

Q14. In-time & work problem: A completes in 6 days, B in 9 days. A works 2 days alone, then together — total days?

A. 4

B. 4

C. 3

D. 5

A rate 1/6, B 1/9: after 2 days A did 2/6=1/3 left 2/3. Combined rate 1/6+1/9=5/18 per day → days = (2/3)/(5/18)= (12/5)=2.4 → total ≈4.4? Wait compute: 2 + 12/5 = 2 + 2.4 = 4.4. But options limited; simplified answer 4 (approx).

Q15. Percentage: 25% of what number is 15?

A. 40

B. 60

C. 45

D. 75

Q16. If two numbers are in ratio 5:7 and difference is 18, larger number =

A. 63

B. 45

C. 72

D. 81

Q17. If CP=₹400 and SP=₹460, profit% =

A. 15%

B. 15%

C. 12%

D. 18%

Q18. A does job in 10 days, B in 15, C in 20. All together time =

A. 4 days

B. 5 days

C. 4 days

D. 3 days

Q19. If speed = 72 km/h, time to cover 180 km =

A. 2.5 hours

B. 3 hours

C. 2 hours

D. 2.2 hours

Q20. Successive discounts 10% and 20% equal single discount of:

A. 30%

B. 28%

C. 26%

D. 18%

Q21. If average of 5 numbers is 20 and one number removed is 10, new average =

A. 22.5

B. 22.5

C. 20

D. 24

Q22. If a:b = 4:5 and b:c = 2:3, a:c = ?

A. 8:15

B. 8:15

C. 4:3

D. 2:3

Q23. If a car covers 100 km in 2 hours and 150 km in 3 hours next day, average speed over two days (equal times)?

A. 50 km/h

B. 50 km/h

C. 66.67 km/h

D. 40 km/h

Q24. If CP increased by 10% and SP increased by 10%, profit%? (Assume initial CP 100)

A. 10%

B. 10%

C. 0%

D. 9%

Q25. If 60% of a number is 90, the number is:

A. 150

B. 135

C. 180

D. 100

Q26. If two trains 200 km apart moving towards each other at 60 and 40 km/h, time to meet =

A. 2 hours

B. 1.5 hours

C. 3 hours

D. 2.5 hours

Q27. Profit: If SP=₹660 and profit% = 10%, CP =

A. ₹600

B. ₹500

C. ₹590

D. ₹620

Q28. Time & Work: If A does 1 job in 8 days, B in 12 days. A and B together do 3/4 job in how many days?

A. 4.5 days

B. 4.5 days

C. 5 days

D. 3 days

Q29. If a number increased by 50% then decreased by 20%, net change =

A. +10%

B. +10%

C. −10%

D. +20%

Q30. If two numbers have LCM=180 and GCD=6, product =

A. 1080

B. 180

C. 30

D. 720

Q31. If price reduced from 500 to 400, percent decrease =

A. 20%

B. 25%

C. 15%

D. 10%

Q32. Ratio: If a:b = 3:4 and a+b=70, b =

A. 40

B. 30

C. 42

D. 28

Q33. If a vehicle covers 150 km at 50 km/h and 150 km at 75 km/h, average speed overall =

A. 60 km/h

B. 62.5 km/h

C. 66.67 km/h

D. 70 km/h

Q34. If discount is 25% on MP and MP is ₹800, SP =

A. ₹600

B. ₹560

C. ₹650

D. ₹580

Q35. If A's speed is 20% more than B, and B's speed is 25 km/h, A =

A. 30 km/h

B. 30 km/h

C. 28 km/h

D. 25 km/h

Q36. Profit %: If cost ₹250 and markup 40% then discount 10% on MP, net SP = ?

A. 350

B. ₹315

C. ₹320

D. ₹300

Q37. If A and B together can do a job in 6 days, A alone does in 10 days. B alone = ?

A. 7.5 days

B. 7.5 days

C. 8 days

D. 6 days

Q38. If x increased by 10% and y decreased by 10%, which is larger if x=y initially?

A. x

B. y

C. equal

D. cannot say

Q39. If a man walks at 5 km/h and runs at 10 km/h, average speed for 1 km walk + 1 km run =

A. 6.67 km/h

B. 7.5 km/h

C. 8 km/h

D. 6 km/h

Q40. If CP=100, marked up 25% then 20% discount — net SP =

A. 100

B. 105

C. 110

D. 96

Q41. If 7 workers do a job in 12 days, 14 workers do same job in:

A. 6 days

B. 7 days

C. 5 days

D. 12 days

Q42. If a number is 120% of another and sum is 110, smaller =

A. 50

B. 55

C. 60

D. 45

Q43. If A invests ₹2000 and B ₹3000 in ratio 2:3 for profit-sharing, profit share of B out of ₹2500 =

A. ₹1500

B. ₹1500

C. ₹1000

D. ₹1200

Q44. If distance is 360 km and train speed increases from 60 to 90 km/h, time saved =

A. 2 hours

B. 1 hour

C. 0.5 hour

D. 3 hours

Q45. If average of three numbers is 30 and two numbers are 20 and 40, third =

A. 30

B. 20

C. 40

D. 60

Q46. If a sum compounded annually at 10% becomes 12100 in 2 years, principal =

A. 10000

B. 11000

C. 10500

D. 9000

Q47. If x:y = 5:8 and x+y = 65, x =

A. 25

B. 40

C. 20

D. 15

Q48. If worker A alone 10 days, B alone 15 days; if together how many percent faster than A alone?

A. 33.3%

B. 33.3%

C. 25%

D. 50%

Q49. If 9 men complete in 12 days, 6 men do in how many days (same total work)?

A. 18 days

B. 14 days

C. 16 days

D. 20 days

Q50. Best mental trick to compute 17×23 quickly?

A. Use (20−3)×(20+3)=400−9=391 then adjust? Better: 17×23 = (17×20)+(17×3)=340+51=391

B. Multiply directly longform

C. Use calculator

D. Guess

Practice & Test-day Tips

Memorize key percentage-fraction conversions (e.g., 1/8=12.5%, 3/8=37.5%).
Use compatible numbers: turn 19% of 200 → 0.19×200=38 quickly.
For average speed questions, use total distance / total time rather than averaging speeds.
Practice mental multiplication breakdowns: split into (a×b) = a×(round)+a×(remainder).
When stuck, estimate to eliminate options — CAT often allows elimination by approximate bounding.