Mental Reasoning 3- Arithmetical Reasoning: Questions and Step-by-Step Solutions for Competitive Exams SSC, BSSC, Bank, Railways

Mental Reasoning 3- Arithmetical Reasoning: Questions and Step-by-Step Solutions for Competitive Exams SSC, BSSC, Bank, Railways

This is a core topic of reasoning sections in almost all competitive exams like SSC, RRB, BPSC, Banking, Defence, and Railways.

Let’s go step-by-step from basic to advanced, covering all likely and important types of Arithmetic Reasoning questions with concepts, examples, shortcuts, and bilingual explanations.

MENTAL REASONING – ARITHMETICAL REASONING (अंकगणितीय तर्कशक्ति)

1. What is Arithmetical Reasoning?

Definition (English):
It is the branch of reasoning where numerical problems are solved using logical thinking, relationships, and basic arithmetic operations (addition, subtraction, multiplication, division, ratio, proportion, etc.)

हिंदी में:
अंकगणितीय तर्कशक्ति में हम संख्याओं से जुड़े प्रश्नों को तर्क और गणना के मिश्रण से हल करते हैं — जैसे उम्र, औसत, अनुपात, गति, लाभ-हानि, समय-कार्य आदि।

2. Major Question Types in Competitive Exams

3. Type-wise Explanation, Formula & Examples

(a) Problems on Ages (आयु संबंधी प्रश्न)

Concept:
Age changes at the same rate → Difference of ages is constant.

Formulas:
If present ages are A and B,
Then after ‘n’ years → (A + n), (B + n)
Before ‘n’ years → (A – n), (B – n)

Example:
Father is 3 times son’s age. After 10 years, father will be twice the son’s age. Find son’s age.

Solution:
Let son’s age = x
Father = 3x
After 10 years → 3x + 10 = 2(x + 10)
⇒ 3x + 10 = 2x + 20
⇒ x = 10
✅ Son = 10 yrs, Father = 30 yrs

Hindi Explanation:
पिता की उम्र पुत्र की उम्र से 3 गुना है। 10 वर्ष बाद पिता की उम्र दो गुना होगी। पुत्र = 10 वर्ष।

 (b) Average (औसत)

Formula:
Average = (Sum of quantities) ÷ (Number of quantities)

Example:
Find average of 5, 10, 15, 20, 25
= (5+10+15+20+25) / 5 = 75/5 = ✅ 15

Concept Tricks:
If one new number added or removed, adjust total accordingly.

(c) Ratio & Proportion (अनुपात व समानुपात)

Formula:
If a:b = c:d → a/b = c/d → ad = bc

Example:
Divide ₹1200 in ratio 2:3.
= 1200 × 2/5 = 480
= 1200 × 3/5 = 720
✅ Amounts = ₹480 & ₹720

Trick:
Sum of ratio = total parts.

(d) Percentage (प्रतिशत)

Formula:
Percentage = (Value / Total) × 100

Example:
What percent of 50 is 10?
= (10/50)×100 = ✅ 20%

Increase/Decrease Formula:
New value = Old value × (1 ± Rate/100)

(e) Profit & Loss (लाभ और हानि)

Formulas:

• Profit = SP – CP
• Loss = CP – SP
• Profit% = (Profit / CP) × 100
• SP = (100 + Profit%) / 100 × CP

Example:
CP = 200, Profit = 20%
SP = (120/100)×200 = ✅ ₹240

(f) Simple & Compound Interest (साधारण व चक्रवृद्धि ब्याज)

Simple Interest (SI):
= (P × R × T)/100

Example:
Find SI on ₹1000 for 2 years at 10%.
= (1000×10×2)/100 = ✅ ₹200

Compound Interest (CI):
= P[(1 + R/100)^T – 1]

Example:
Find CI on ₹1000 for 2 years @10%
= 1000[(1.1)² – 1] = 1000(1.21 – 1) = ✅ ₹210

(g) Time, Speed, and Distance (समय, गति व दूरी)

Formulas:

• Speed = Distance / Time
• Distance = Speed × Time
• Time = Distance / Speed

Example:
A car runs 120 km in 3 hours.
Speed = 120/3 = ✅ 40 km/h

Conversion:
1 km/hr = (5/18) m/s

(h) Time & Work (समय और कार्य)

Formula:
Work = Rate × Time

If A can do a work in ‘x’ days,
then A’s 1 day work = 1/x

Example:
A = 10 days, B = 15 days
1 day work = 1/10 + 1/15 = (5/30) = 1/6
✅ Together = 6 days

Shortcut:
(x × y)/(x + y)

(i) Partnership (साझेदारी)

Formula:
Profit ratio = (Capital × Time) of partners

Example:
A invests ₹10,000 for 12 months, B invests ₹15,000 for 8 months
Profit ratio = 10,000×12 : 15,000×8 = 120,000 : 120,000 → ✅ 1:1

(j) Number or Digit Problems (संख्या आधारित प्रश्न)

Example:
Find number whose digits reversed is 45 more than the number.
Let number = 10x + y
Reversed = 10y + x
→ (10y + x) = (10x + y) + 45
→ 9y – 9x = 45 → y – x = 5
✅ Difference between digits = 5

(k) Coding or Equation-type Arithmetical Logic

These are common in reasoning tests.

Example:
If 2 + 3 = 10, 4 + 5 = 40, 6 + 7 = ?

Observation:
2×3×(2) = 12 ❌ not matching
Try pattern: (a+b)×a = (2+3)×2 = 10 ✅
So, (6+7)×6 = ✅ 78

Hindi:
तर्क यह है कि पहले अंक के योग को पहले अंक से गुणा किया गया है।

(l) Arithmetic Puzzles / Statements

Example:
The sum of two numbers is 40, and their difference is 8. Find the numbers.
x + y = 40
x – y = 8
→ 2x = 48 → x = 24, y = 16

✅ Numbers = 24, 16

(m) Miscellaneous Logical-Arithmetic Questions

Examples:

• Replace signs:
  If 8 × 2 + 3 = 19, what happens if × means + and + means ×?
  Then: 8 + 2 × 3 = 8 + 6 = ✅ 14
• Balancing Equations:
  Find value that makes both sides equal.

4. Common Formula Summary Table

5. Exam Strategy & Quick Tips

1️⃣ Read word problems carefully — translate them into equations.
2️⃣ Use ratio–proportion for multi-person questions.
3️⃣ Memorize key formulas (Work, SI, CI, Speed, Profit).
4️⃣ Use shortcut formulas to save time.
5️⃣ Check units — km/hr ↔ m/s, months ↔ years.
6️⃣ For reasoning puzzles (coding equations), look for hidden multiplication/addition logic.