Arithmetic Reasoning
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Cheat Sheets
Study MaterialArithmetic Reasoning – Cheat Sheet
Arithmetic Reasoning is a fundamental topic in Quantitative Aptitude and Logical Reasoning, frequently tested in SSC, Banking, Railway, UPSC, Defence, Insurance, CAT, and other competitive examinations.
This cheat sheet covers all major arithmetic reasoning concepts including age problems, number problems, ratio and proportion, partnership, speed-time-distance, work-time, percentages, averages, profit-loss, interest, mixture and alligation, mensuration, and mathematical puzzles. Mastering these concepts enables quick and accurate problem-solving in exams.
About This Cheat Sheet
Complete guide to Arithmetic Reasoning for competitive exams:
Age Problems
- Present Age
- Past Age
- Future Age
- Ratio-Based Ages
Number Problems
- Finding Numbers
- Digit Problems
- Consecutive Numbers
- Number Properties
Ratio & Proportion
- Simple Ratio
- Compound Ratio
- Direct Proportion
- Inverse Proportion
Partnership
- Capital Ratio
- Time Ratio
- Profit Distribution
- Combined Ratios
Speed-Time-Distance
- Relative Speed
- Average Speed
- Train Problems
- Boat & Stream
Work-Time
- Individual Work
- Combined Work
- Pipes & Cisterns
- Efficiency Problems
Percentages & Averages
- Increase/Decrease
- Successive Percentage
- Population Problems
- Basic & Weighted Average
Profit, Loss & Interest
- Cost Price & Selling Price
- Discount & Marked Price
- Simple Interest
- Compound Interest
Mixture & Alligation
- Alligation Rule
- Replacement Problems
- Mixing Ratios
- Concentration Problems
Mensuration & Puzzles
- Perimeter & Area
- Volume Problems
- BODMAS Rules
- Mathematical Equations
Topics Covered
This cheat sheet covers all important Arithmetic Reasoning topics for quick revision:
| Topic | Description | Key Formulas / Tips |
|---|---|---|
| Age Problems | Finding present, past, and future ages using given conditions and ratios | Use linear equations; represent ages as variables |
| Number Problems | Finding numbers based on given conditions, digit problems, consecutive numbers | Let digits be x, y; form equations |
| Ratio & Proportion | Simple ratio, compound ratio, direct and inverse proportion | a:b :: c:d → ad = bc |
| Partnership | Capital ratio, time ratio, profit distribution among partners | Profit ∝ Capital × Time |
| Speed, Time & Distance | Relative speed, average speed, train problems, boat and stream | Speed = Distance / Time; Relative speed = sum/difference |
| Work & Time | Individual work, combined work, pipes and cisterns, efficiency | Work = Rate × Time; 1/A + 1/B = 1/T |
| Percentages | Increase/decrease, successive percentage, population problems | Successive: a + b + ab/100 |
| Averages | Basic average, weighted average, missing numbers | Average = Sum / Number of items |
| Profit & Loss | Cost price, selling price, discount, marked price, profit/loss percentage | Profit% = (Profit/CP) × 100; SP = CP × (100 ± P%)/100 |
| Simple & Compound Interest | SI, CI, principal, rate, time, amount | SI = (P×R×T)/100; CI = P(1+R/100)^T - P |
| Mixture & Alligation | Alligation rule, replacement problems, mixing ratios, concentration | Alligation: (Cheaper - Mean) / (Mean - Dearer) |
| Mensuration | Perimeter, area, volume of 2D and 3D shapes | Formulas for circle, triangle, rectangle, cylinder, sphere |
| Mathematical Puzzles | BODMAS rules, equations, logical arithmetic problems | Follow BODMAS; form equations from conditions |
Why This Cheat Sheet is Important
Master Core Concepts
Understand fundamental arithmetic concepts that form the backbone of quantitative aptitude.
Solve with Speed
Apply shortcut techniques and formula-based approaches to solve problems quickly.
Crack Varied Questions
Handle diverse question types from age problems to mixture and alligation.
Boost Exam Scores
Arithmetic Reasoning carries significant weight in all competitive exams.
Competitive Exams Covered
- SSC CGL
- SSC CHSL
- SSC MTS
- Bank PO & Clerk
- IBPS Exams
- Railway Recruitment Exams
- UPSC CSAT
- Defence Exams
- Insurance Exams
- CAT
- State Government Exams
Quick Exam-Oriented Examples
Question 1:
The sum of the ages of a father and his son is 60 years. Five years ago, the father was 4 times as old as his son. Find their present ages.
Step 1: Let father's age = x, son's age = y.
Step 2: x + y = 60 ...(i)
Step 3: Five years ago: (x - 5) = 4(y - 5) → x - 5 = 4y - 20 → x = 4y - 15 ...(ii)
Step 4: Substitute (ii) in (i): 4y - 15 + y = 60 → 5y = 75 → y = 15
Step 5: x = 60 - 15 = 45
Answer: Father = 45 years, Son = 15 years
Question 2:
The sum of three consecutive even numbers is 72. Find the numbers.
Step 1: Let the numbers be x, x+2, x+4.
Step 2: x + (x+2) + (x+4) = 72
Step 3: 3x + 6 = 72 → 3x = 66 → x = 22
Answer: 22, 24, 26
Question 3:
Two numbers are in the ratio 5:7. If their sum is 144, find the numbers.
Step 1: Let the numbers be 5x and 7x.
Step 2: 5x + 7x = 144 → 12x = 144 → x = 12
Step 3: Numbers = 5×12 = 60, 7×12 = 84
Answer: 60 and 84
Question 4:
A and B invest ₹40,000 and ₹60,000 in a business. If the total profit is ₹25,000, find A's share.
Step 1: Ratio of capitals = 40000 : 60000 = 2 : 3
Step 2: Profit is distributed in the same ratio.
Step 3: A's share = (2/5) × 25000 = 10000
Answer: A's share = ₹10,000
Question 5:
A train travels 240 km in 4 hours. What is its speed in m/s?
Step 1: Speed = Distance / Time = 240 / 4 = 60 km/hr
Step 2: Convert to m/s: 60 × (5/18) = 300/18 = 16.67 m/s
Answer: 16.67 m/s
Question 6:
A can do a work in 10 days, B can do it in 15 days. How many days will they take to complete the work together?
Step 1: A's 1 day work = 1/10, B's 1 day work = 1/15
Step 2: Combined work in 1 day = 1/10 + 1/15 = (3+2)/30 = 5/30 = 1/6
Step 3: Time = 1 / (1/6) = 6 days
Answer: 6 days
Question 7:
If 20% of a number is 80, find the number.
Step 1: Let the number be x.
Step 2: 20% of x = 80 → (20/100) × x = 80
Step 3: x = 80 × (100/20) = 80 × 5 = 400
Answer: 400
Question 8:
The average of 5 numbers is 25. If one number is removed, the average becomes 22. Find the removed number.
Step 1: Sum of 5 numbers = 5 × 25 = 125
Step 2: Sum of remaining 4 numbers = 4 × 22 = 88
Step 3: Removed number = 125 - 88 = 37
Answer: 37
Question 9:
An article is bought for ₹800 and sold for ₹1000. Find the profit percentage.
Step 1: Profit = SP - CP = 1000 - 800 = ₹200
Step 2: Profit% = (Profit/CP) × 100 = (200/800) × 100 = 25%
Answer: 25%
Question 10:
In what ratio should tea costing ₹60/kg and ₹80/kg be mixed to get a mixture worth ₹72/kg?
Step 1: Apply alligation rule:
Cheaper (60) Dearer (80)
\ /
Mean (72)
/ \
(72-60)=12 (80-72)=8
Step 2: Ratio = 12 : 8 = 3 : 2
Answer: 3 : 2
Question 11:
Find the simple interest on ₹5000 at 10% per annum for 2 years.
Step 1: SI = (P × R × T) / 100
Step 2: SI = (5000 × 10 × 2) / 100 = 1000
Answer: ₹1000
Final Takeaway
Arithmetic Reasoning tests your ability to apply mathematical concepts to real-world problems. The key is to identify the underlying relationship, form the correct equation, and solve systematically.
Regular practice of age problems, number problems, ratio and proportion, partnership, speed-distance, work-time, percentages, averages, profit-loss, interest, mixture and alligation, mensuration, and mathematical puzzles will significantly improve your speed and accuracy. This cheat sheet serves as a complete revision resource for mastering Arithmetic Reasoning questions in competitive examinations.
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