📐 Round 2 — Aptitude & Logical Reasoning
Complete Guide From Scratch for Freshers
What to expect: A 30–60 minute written/online test. This round tests your speed + accuracy on maths, logic, and data interpretation. Companies like DecisionTree use this to filter candidates before technical rounds.
SECTION A: QUANTITATIVE APTITUDE
1. Percentages
📖 Theory (Start Here)
A percentage means "per hundred." It's a way to express a number as a fraction of 100.
Core Formula:
Percentage = (Part / Whole) × 100
Key Conversions to Memorize:
| Fraction | Percentage | Decimal |
|---|---|---|
| 1/2 | 50% | 0.5 |
| 1/3 | 33.33% | 0.333 |
| 1/4 | 25% | 0.25 |
| 1/5 | 20% | 0.2 |
| 1/6 | 16.67% | 0.167 |
| 1/8 | 12.5% | 0.125 |
| 1/10 | 10% | 0.1 |
| 2/3 | 66.67% | 0.667 |
| 3/4 | 75% | 0.75 |
🔑 Key Concepts
1. Percentage Increase:
% Increase = [(New - Old) / Old] × 100
2. Percentage Decrease:
% Decrease = [(Old - New) / Old] × 100
3. Successive Percentage Changes: If a value changes by a% and then by b%:
Net % change = a + b + (a × b) / 100
Example: Price increases by 20%, then decreases by 10%. Net change = 20 + (-10) + (20 × -10)/100 = 20 - 10 - 2 = +8%
4. Reverse Percentage (Finding Original from Final):
If final price after X% discount = ₹Y, then:
Original = Y / (1 - X/100)
📝 Worked Examples
Q1: A shirt costs ₹800 after a 20% discount. What was the original price?
Original × (1 - 20/100) = 800
Original × 0.80 = 800
Original = 800 / 0.80 = ₹1,000 ✅
Q2: A city's population increased from 2,00,000 to 2,50,000 in a year. What's the % increase?
% Increase = [(2,50,000 - 2,00,000) / 2,00,000] × 100
= [50,000 / 2,00,000] × 100
= 25% ✅
Q3: If A's salary is 30% more than B's, by what % is B's salary less than A's?
Let B = 100, then A = 130.
B is less than A by: [(130-100)/130] × 100 = 23.08% ✅
⚠️ Common trap: "30% more" does NOT mean "30% less" in reverse!
Practice Problems
| # | Problem | Answer |
|---|---|---|
| 1 | 40% of what number is 160? | 400 |
| 2 | A product's price goes up 10%, then 10%. Net increase? | 21% |
| 3 | If 65% students passed and 420 failed, total students? | 1,200 |
| 4 | An item costs ₹540 after successive discounts of 10% and 10%. Original? | ₹666.67 |
2. Profit, Loss & Discount
📖 Theory
Key Formulas:
| Formula | Expression |
|---|---|
| Profit | SP - CP |
| Loss | CP - SP |
| Profit% | (Profit / CP) × 100 |
| Loss% | (Loss / CP) × 100 |
| SP from Profit% | SP = CP × (1 + Profit%/100) |
| SP from Loss% | SP = CP × (1 - Loss%/100) |
| Marked Price & Discount | SP = MP × (1 - Discount%/100) |
📝 Worked Examples
Q1: A shopkeeper buys at ₹500 and sells at ₹600. Profit%?
Profit = 600 - 500 = ₹100
Profit% = (100/500) × 100 = 20% ✅
Q2: An item has a marked price of ₹1,000. After 20% discount, the shopkeeper still makes 25% profit. Find CP.
SP = 1000 × (1 - 20/100) = ₹800
SP = CP × (1 + 25/100)
800 = CP × 1.25
CP = 800/1.25 = ₹640 ✅
Q3 (Tricky): A trader cheats by using a weight of 900g instead of 1kg. Profit%?
He sells 900g at the price of 1000g.
Profit% = [(1000-900)/900] × 100 = 11.11% ✅
3. Ratios & Proportions
📖 Theory
A ratio compares two quantities. If A:B = 3:4, it means for every 3 units of A, there are 4 units of B.
A proportion states that two ratios are equal: A/B = C/D → A × D = B × C (cross-multiply).
🔑 Key Operations
Combining Ratios: If A:B = 2:3 and B:C = 4:5, find A:B:C.
Step 1: Make B equal in both ratios.
B is 3 in first, 4 in second.
LCM of 3 and 4 = 12
Step 2: Scale up:
A:B = 2:3 → multiply by 4 → 8:12
B:C = 4:5 → multiply by 3 → 12:15
Step 3: A:B:C = 8:12:15 ✅
Dividing in a Ratio: Divide ₹1,200 in the ratio 3:4:5.
Total parts = 3+4+5 = 12
Share₁ = (3/12) × 1200 = ₹300
Share₂ = (4/12) × 1200 = ₹400
Share₃ = (5/12) × 1200 = ₹500 ✅
4. Averages
📖 Theory
Average = Sum of all values / Number of values
🔑 Key Shortcuts
1. Weighted Average:
Weighted Avg = (w₁×x₁ + w₂×x₂ + ...) / (w₁ + w₂ + ...)
2. New member joins a group: If the average of N numbers is A, and a new number X is added:
New Average = (N × A + X) / (N + 1)
3. One number removed: If average of N numbers is A, and number X is removed:
New Average = (N × A - X) / (N - 1)
📝 Worked Examples
Q1: Average of 5 numbers is 40. If one number (60) is removed, find the average of remaining 4.
Total sum = 5 × 40 = 200
Remaining sum = 200 - 60 = 140
New average = 140 / 4 = 35 ✅
Q2: A batsman's average after 20 innings is 45. After the 21st inning, his average increases by 2. Runs scored in 21st inning?
Total after 20 innings = 20 × 45 = 900
New average = 47
Total after 21 innings = 21 × 47 = 987
Runs in 21st inning = 987 - 900 = 87 ✅
5. Time & Work
📖 Theory
The fundamental concept: Think of work as rates.
If A completes a job in 10 days → A's rate = 1/10 of the job per day.
If B completes a job in 15 days → B's rate = 1/15 of the job per day.
Together → Combined rate = 1/10 + 1/15 = 5/30 = 1/6 per day → 6 days.
🔑 Shortcut Formula
When two people work together:
Days together = (a × b) / (a + b)
📝 Worked Examples
Q1: A does a job in 12 days. B does it in 18 days. Together?
Days = (12 × 18) / (12 + 18) = 216 / 30 = 7.2 days ✅
Q2: A and B together complete a job in 8 days. A alone takes 12 days. How long does B take alone?
A+B rate = 1/8, A rate = 1/12
B rate = 1/8 - 1/12 = (3-2)/24 = 1/24
B alone = 24 days ✅
Q3: A is twice as efficient as B. Together they finish in 12 days. How long does A alone take?
If B's rate = x, then A's rate = 2x.
Together: 2x + x = 3x = 1/12 → x = 1/36
A's rate = 2/36 = 1/18 → A alone = 18 days ✅
6. Time, Speed & Distance
📖 Theory
Speed = Distance / Time
Distance = Speed × Time
Time = Distance / Speed
Unit Conversion:
km/hr to m/s → multiply by 5/18
m/s to km/hr → multiply by 18/5
🔑 Key Concepts
| Scenario | Formula |
|---|---|
| Average Speed (same distance, different speeds) | 2 × S₁ × S₂ / (S₁ + S₂) |
| Relative Speed (same direction) | S₁ - S₂ |
| Relative Speed (opposite direction) | S₁ + S₂ |
| Train crossing a pole | Time = Length of train / Speed |
| Train crossing a platform | Time = (Train length + Platform length) / Speed |
📝 Worked Examples
Q1: A car goes from A to B at 60 km/hr and returns at 40 km/hr. Average speed?
Average Speed = 2 × 60 × 40 / (60 + 40) = 4800/100 = 48 km/hr ✅
⚠️ Common mistake: It's NOT simply (60+40)/2 = 50!
Q2: A train 200m long crosses a platform 300m long in 25 seconds. Speed?
Total distance = 200 + 300 = 500m
Speed = 500/25 = 20 m/s = 20 × 18/5 = 72 km/hr ✅
7. Probability
📖 Theory
Probability measures the chance of an event happening.
P(Event) = Number of Favorable Outcomes / Total Possible Outcomes
Rules:
- P(Event) is always between 0 and 1 (or 0% to 100%)
- P(NOT happening) = 1 - P(happening)
- AND (both events): P(A AND B) = P(A) × P(B) — if independent
- OR (either event): P(A OR B) = P(A) + P(B) - P(A AND B)
📝 Worked Examples
Q1: A bag has 5 red, 3 blue, 2 green balls. Probability of drawing a red ball?
P(Red) = 5 / (5+3+2) = 5/10 = 1/2 ✅
Q2: Two dice are thrown. Probability that sum = 7?
Total outcomes = 6 × 6 = 36
Favorable: (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) = 6 outcomes
P(sum=7) = 6/36 = 1/6 ✅
Q3: A coin is tossed 3 times. Probability of getting at least 1 head?
P(at least 1 head) = 1 - P(no heads) = 1 - P(all tails)
P(all tails) = (1/2)³ = 1/8
P(at least 1 head) = 1 - 1/8 = 7/8 ✅
Q4: From a deck of 52 cards, probability of drawing a King OR a Heart?
P(King) = 4/52
P(Heart) = 13/52
P(King AND Heart) = 1/52 (King of Hearts)
P(King OR Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13 ✅
8. Permutations & Combinations
📖 Theory
Think of it this way: Does the order matter?
| Concept | Order Matters? | Formula | Example |
|---|---|---|---|
| Permutation | ✅ Yes | nPr = n! / (n-r)! | Arranging 3 people in a row |
| Combination | ❌ No | nCr = n! / [r! × (n-r)!] | Choosing 3 people for a team |
Factorial: n! = n × (n-1) × (n-2) × ... × 1. Example: 5! = 120. Special: 0! = 1.
📝 Worked Examples
Q1: How many 3-letter arrangements from {A, B, C, D, E}?
Order matters → Permutation
5P3 = 5! / (5-3)! = 120 / 2 = 60 ✅
Q2: How many ways to select 3 members from a team of 8?
Order doesn't matter → Combination
8C3 = 8! / (3! × 5!) = (8×7×6) / (3×2×1) = 56 ✅
Q3: How many ways to arrange the letters in "ANALYTICS"?
ANALYTICS has 9 letters. A appears 2 times.
Arrangements = 9! / 2! = 362880 / 2 = 181440 ✅
SECTION B: LOGICAL REASONING
9. Number Series
📖 Theory
Look for these common patterns in number series:
📝 Worked Examples
Q1: 2, 6, 12, 20, 30, ?
Differences: 4, 6, 8, 10 → increasing by 2
Next difference = 12
Answer = 30 + 12 = 42 ✅
Pattern: n(n+1) → 1×2, 2×3, 3×4, 4×5, 5×6, 6×7 = 42
Q2: 3, 5, 9, 17, 33, ?
Differences: 2, 4, 8, 16 → doubling
Next difference = 32
Answer = 33 + 32 = 65 ✅
Q3: 1, 1, 2, 3, 5, 8, 13, ?
Fibonacci series! Each number = sum of previous two.
Next = 8 + 13 = 21 ✅
Q4: 2, 3, 5, 7, 11, 13, ?
Prime numbers! Next prime = 17 ✅
Strategy for Solving
- First, check the differences between consecutive terms
- If differences aren't constant, check if differences form their own pattern
- Try multiplication ratios (each term ÷ previous)
- Look for squares (1,4,9,16...) or cubes (1,8,27,64...)
- Check for alternating patterns (separate odd and even positioned terms)
10. Coding-Decoding
📖 Theory
In coding-decoding, letters/words are replaced according to a rule. Your job is to find the rule and apply it.
Common Types:
| Type | Example |
|---|---|
| Letter shift | A→C, B→D (shift +2) → CAT = ECV |
| Reverse alphabet | A→Z, B→Y, C→X → CAT = XZG |
| Position numbers | A=1, B=2, C=3... → CAT = 3+1+20 = 24 |
| Mirror coding | The word is reversed → CAT = TAC |
Reverse Alphabet Mapping (Memorize this):
A↔Z B↔Y C↔X D↔W E↔V
F↔U G↔T H↔S I↔R J↔Q
K↔P L↔O M↔N
Trick: A+Z=27, B+Y=27... If a letter's position is X, its reverse is (27-X).
📝 Worked Examples
Q1: If COMPUTER = DNPQVUFS, what is DATA?
C→D (+1), O→N (-1), M→P (+3), P→Q (+1)...
Let me check pairs: C(3)→D(4), O(15)→N(14), M(13)→P(16), P(16)→Q(17)...
Pattern: +1, -1, +3, +1, -1, +3, +1, -1 (repeating +1,-1,+3)
D(4)→E, A(1)→Z(-1), T(20)→W(+3), A(1)→B(+1)
DATA = EZWB ✅
Q2: If MANGO = 51 (M=13+A=1+N=14+G=7+O=15=50... let me recheck), using position sum: M(13)+A(1)+N(14)+G(7)+O(15) = 50. If they say MANGO=51, then the rule might be position sum + 1. Apply to APPLE:
A(1)+P(16)+P(16)+L(12)+E(5) = 50 → 50 + 1 = 51 ✅
11. Syllogisms
📖 Theory
A syllogism gives you statements and asks which conclusions logically follow. Use Venn Diagrams — this is the fastest method.
Key Rules:
| Statement | Venn Diagram |
|---|---|
| "All A are B" | Circle A is completely inside circle B |
| "Some A are B" | Circles A and B overlap partially |
| "No A are B" | Circles A and B don't overlap at all |
| "Some A are not B" | Part of circle A is outside circle B |
📝 Worked Examples
Q1:
- Statement 1: All dogs are animals.
- Statement 2: Some animals are cats.
- Conclusion: Some dogs are cats. TRUE or FALSE?
Draw it: Dogs circle is inside Animals circle.
Cats circle overlaps Animals, but it could overlap
the "Animals but not Dogs" area.
→ We can't be certain some dogs are cats.
→ Conclusion: DOES NOT FOLLOW ✅
Q2:
- Statement 1: All roses are flowers.
- Statement 2: All flowers are plants.
- Conclusion: All roses are plants. TRUE or FALSE?
Roses ⊂ Flowers ⊂ Plants → Roses ⊂ Plants
→ Conclusion: FOLLOWS ✅
Q3:
- Statement 1: No fish is a bird.
- Statement 2: Some birds are animals.
- Conclusion I: Some animals are not fish.
- Conclusion II: No fish is an animal.
Fish and Bird circles don't overlap.
Some Birds are Animals (overlap).
→ Those animals that are birds are definitely not fish → Conclusion I FOLLOWS ✅
→ But some animals could still be fish (from a different part) → Conclusion II DOES NOT FOLLOW ✅
12. Blood Relations
📖 Theory
Build a family tree diagram as you read the problem. Use these conventions:
Male: □ Female: ○ Unknown gender: △
Marriage: ——— (horizontal line)
Parent-Child: | (vertical line)
Key Relationships to Know:
| Relationship | Meaning |
|---|---|
| Father's/Mother's father | Grandfather |
| Father's/Mother's mother | Grandmother |
| Father's brother | Uncle (Chacha/Tau) |
| Father's sister | Aunt (Bua) |
| Mother's brother | Uncle (Mama) |
| Mother's sister | Aunt (Mausi) |
| Brother's/Sister's son | Nephew |
| Brother's/Sister's daughter | Niece |
📝 Worked Examples
Q1: "A is B's mother. C is A's father. D is C's mother. What is A to D?"
D → C → A → B
(D is mother of C, C is father of A)
A is the granddaughter of D ✅
(D is great-grandmother of B)
Q2: "Pointing to a photo, A said: 'He is the son of my father's only daughter.' Who is in the photo?"
Father's only daughter = A herself (if female) or A's sister
If A is female: father's only daughter = A → the son is A's son ✅
If A is male: father's only daughter = A's sister → the son is A's nephew
[!TIP] Strategy: Always draw the family tree on paper. Don't try to solve blood relations in your head.
13. Seating Arrangements
📖 Theory
Two types:
- Linear — People sitting in a row (facing same or opposite direction)
- Circular — People sitting around a table
Steps:
- Read ALL clues first before drawing
- Start with the most concrete clue (e.g., "A sits at the left end")
- Use "definite" clues before "relative" clues
- Draw and test — there may be multiple valid arrangements; check which one satisfies ALL conditions
📝 Worked Example
Q: 6 people (A-F) sit in a row facing north.
- A sits at the left end.
- B is not adjacent to A.
- C sits in the middle (position 3 or 4).
- D is immediately right of C.
- E sits between A and F.
Position: 1 2 3 4 5 6 (1=left end)
From clue 1: A is at position 1.
From clue 5: E sits between A and F → A _ F with E in middle → A E F at 1,2,3
From clue 3: C is at 3 or 4. Since F is at 3, C must be at 4.
From clue 4: D is immediately right of C → D at 5.
Remaining: B at 6.
From clue 2: B (at 6) is not adjacent to A (at 1) → ✅ Satisfied.
Answer: A E F C D B ✅