Alligation Rule — find ratio of mixing two quantities
Mean Price — weighted average price of mixture
Replacement — repeated removal & replacement formula
Types: Simple Mixture, Compound Mixture, Serial Dilution
1. Basic Concepts
Mixture: Combining two or more ingredients to form a new product. Alligation: A rule to find the ratio in which two or more ingredients at different prices must be mixed to produce a mixture at a given price.
ALLIGATION RULE:
Cheaper ............... Dearer
c ..................... d
Mean (m)
/ \
(d - m) (m - c)
Ratio = (d - m) : (m - c)
Cheaper quantity / Dearer quantity = (d - m) / (m - c)
TRICK: Think of it as a "criss-cross" method — subtract mean from each price diagonally! The differences give you the ratio directly.
2. Types of Problems
Type 1: Finding the Ratio
Given prices of two ingredients & mean price → find mixing ratio
Type 2: Finding the Mean Price
Given prices & quantities → find weighted average price
Mean Price = (c × q1 + d × q2) / (q1 + q2)
Type 3: Finding Quantity
Given ratio & one quantity → find the other
Type 4: Replacement
Repeated removal & replacement of a portion from a vessel
3. Solved Examples
Example 1: Rice at Rs 30/kg is mixed with rice at Rs 45/kg. Find the ratio to get mixture worth Rs 36/kg.
Using Alligation:
Cheaper = 30, Dearer = 45, Mean = 36
Ratio = (45 - 36) : (36 - 30) = 9 : 6 = 3 : 2
Answer: 3 : 2 ✓
Example 2: A shopkeeper mixes 20 kg of sugar at Rs 18/kg with 25 kg of sugar at Rs 22/kg. Find the price of the mixture.
COMMON MISTAKES TO AVOID:
1. Don't confuse which is cheaper & which is dearer in alligation
2. Water is ALWAYS Rs 0 — don't forget this!
3. In replacement: formula gives what's LEFT, not what's removed
4. Ratio from alligation = (d-m):(m-c) — the order matters!
Blood Relation refers to the relation among family members connected by birth (not marriage).
Tip: ALWAYS draw a family tree diagram. Use:
♂ for Male, ♀ for Female, — for same generation, | for parent-child
Relation
Male
Female
Parent's parent
Grandfather
Grandmother
Parent's child
Brother
Sister
Parent's brother
Uncle (Paternal: Chacha/Tau)
—
Parent's sister
—
Aunt (Bua/Mausi)
Uncle/Aunt's child
Cousin (Brother)
Cousin (Sister)
Child's child
Grandson
Granddaughter
Sibling's child
Nephew
Niece
Spouse's parent
Father-in-law
Mother-in-law
Child's spouse
Son-in-law
Daughter-in-law
Spouse's sibling
Brother-in-law
Sister-in-law
2. Paternal vs Maternal
Paternal Side (Father's Side):
Father's Father = Dada (Paternal Grandfather)
Father's Mother = Dadi (Paternal Grandmother)
Father's Brother = Chacha / Tau (Uncle)
Father's Sister = Bua (Aunt)
Maternal Side (Mother's Side):
Mother's Father = Nana (Maternal Grandfather)
Mother's Mother = Nani (Maternal Grandmother)
Mother's Brother = Mama (Uncle)
Mother's Sister = Mausi (Aunt)
TRICK — Generation Counting:
Same Generation = +0 (Siblings, Cousins, Spouse)
One Above = +1 (Parents, Uncle, Aunt)
Two Above = +2 (Grandparents)
One Below = -1 (Children, Nephew, Niece)
Two Below = -2 (Grandchildren)
3. Coded Blood Relations
In coded blood relations, relationships are expressed using symbols (+, -, ×, ÷, @, #, $, etc.). First decode the symbols, then draw the family tree.
Example 1: If A + B means "A is the father of B", A - B means "A is the mother of B", A × B means "A is the brother of B", then what is the relation of P to S in: P + Q - R × S?
Decode step by step:
P + Q → P is father of Q
Q - R → Q is mother of R
R × S → R is brother of S
Family Tree:
P (♂) — father of → Q (♀) — mother of → R (♂) & S
So P is grandfather of R and S P is the Grandfather of S ✓
4. Solved Examples
Example 2: Pointing to a photograph, Arun said "She is the daughter of my grandfather's only son." How is the girl related to Arun?
Grandfather's only son = Arun's Father
Daughter of Arun's Father = Arun's Sister
Answer: Sister ✓
Example 3: A is B's sister. C is B's mother. D is C's father. E is D's mother. How is A related to D?
Family Tree (bottom to top):
E (Great-grandmother) → D (Grandfather) → C (Mother) → B & A
A is the Granddaughter of D ✓
Example 4: If "M $ N" means M is the brother of N, "M # N" means M is the mother of N, "M @ N" means M is the father of N. What does P @ Q $ R # S mean?
P @ Q → P is father of Q
Q $ R → Q is brother of R
R # S → R is mother of S
P → Q & R (children) → S (grandson/granddaughter) P is Grandfather of S ✓
5. Practice Problems
Q1: Pointing to a man, a woman says "His mother is the only daughter of my mother." How is the woman related to the man?
Solution:
"Only daughter of my mother" = the woman herself
So the woman is the mother of the man. Answer: Mother ✓
Q2: B is the husband of P. Q is the only grandson of E, who is the wife of D. D has only one son and one daughter. B is the only brother of A. How is P related to A?
Solution:
D & E are grandparents. D has one son (B) and one daughter (A).
B is husband of P. Q is grandson of E (through B & P).
So P is married to B (A's brother). Answer: Sister-in-law (Bhabhi) ✓
COMMON MISTAKES:
1. Don't assume gender from name — always check given info
2. "Only son/daughter" is a crucial clue — narrows down possibilities
3. "Pointing to a photograph" — the photo person is different from the speaker
4. Always draw the family tree, don't solve in your head!
💺Chapter 2.3 — Seating Arrangements
SEATING ARRANGEMENTS — Overview
Linear Arrangement — people sitting in a row (one side / both sides)
Circular Arrangement — people sitting around a table
Key: Fix one person first, then place others relative to them
Directions: Left/Right in linear, Clockwise/Anti-clockwise in circular
1. Linear Arrangement
Linear Seating: People sit in a straight line, all facing the SAME direction (usually North).
Key Rule: When facing North —
• LEFT = West side
• RIGHT = East side
If facing South, LEFT & RIGHT get reversed!
TRICK — Facing Direction:
• "Facing North" → Left = your actual left
• "Facing South" → Left & Right SWAP!
• Always draw an arrow showing the facing direction first!
Approach for Linear Problems:
1. Draw a horizontal line with positions (1, 2, 3...n)
2. Note the facing direction
3. Place people with DEFINITE info first
4. Use clues like "X sits 3rd to the left of Y" to place others
5. Verify all conditions at the end
Example 1: 7 people (A-G) sit in a line facing North. C sits at one end. B sits 3rd to the right of C. A sits exactly in the middle. D sits to the immediate left of A. Find the arrangement.
Positions: 1 _ 2 _ 3 _ 4 _ 5 _ 6 _ 7
• C is at one end → C at position 1 (or 7)
• B is 3rd to right of C → C at 1, B at 4 (works) OR C at 7, B at 10 (impossible)
• So C = 1, B = 4
• A is in middle = position 4... but B is at 4!
• Wait, middle of 7 = position 4. B = 4. Contradiction.
• Re-check: If A is exactly middle, A = 4. Then B ≠ 4. Let's say C = 7.
• B is 3rd to right of C: from position 7, 3 right = doesn't exist (facing north, right = towards 1). So B at position 7-3 = 4.
• A = 4 conflicts again. So the puzzle's "3rd to left" may apply.
Key Learning: Always be careful about direction interpretation & try both ends!
2. Circular Arrangement
Circular Seating: People sit around a circular table, usually facing the center.
Key Rules:
• When facing center: LEFT = Clockwise, RIGHT = Anti-clockwise
• When facing outward: LEFT = Anti-clockwise, RIGHT = Clockwise
• "Opposite" = diametrically opposite (n/2 positions away)
TRICK — Fix & Place Method:
1. Fix one person's position (reduces complexity)
2. Place others relative to the fixed person
3. "Immediate left" means next person clockwise (when facing center)
4. "3rd to the right" = count 3 positions anti-clockwise
Example 2: 6 people (P, Q, R, S, T, U) sit around a circular table facing the center. P sits opposite Q. R is to the immediate left of P. S is between T and U. T is not adjacent to P.
Fix P at top. Q is opposite (bottom).
R is immediate left of P = clockwise next.
P
/ \
U R
| |
T S
\ /
Q
Check: S is between T and U ✓ T is not adjacent to P ✓ Final: P, R, S, Q, T, U (clockwise) ✓
Example 3: 8 people sit in a circle facing center. A is 3rd to the left of B. C is opposite A. D is between A and C (going clockwise from A). Where is D relative to B?
Fix B at 12 o'clock position.
A is 3rd to left of B = 3 clockwise from B = position 3.
C is opposite A = position 3+4 = position 7.
D is between A and C clockwise = positions 4, 5, or 6.
D is 4th/5th to the left of B ✓
3. Practice Problems
Q1: 5 friends (A, B, C, D, E) sit in a row facing North. C sits in the middle. A is to the immediate right of C. B is at one of the ends. D is not adjacent to C. Find the arrangement.
Solution:
Positions: 1_2_3_4_5. C = 3 (middle). A = 4 (right of C).
D not adjacent to C → D ≠ 2 or 4. D ≠ 4 (A). So D = 1 or 5.
B at one end. If D = 1, B = 5. E = 2. Arrangement: D, E, C, A, B ✓
Q2: 6 people sit in a circle facing center. A is opposite D. B is to the immediate right of A. C is not adjacent to A. E is to the left of D. Find F's position.
Solution:
Fix A at top. D opposite (bottom). B immediate right of A = anti-clockwise next.
C not adjacent to A. E left of D = clockwise from D.
Arrangement (clockwise): A, F, C, D, E, B F is 2nd to the right of A ✓
Syllogism: A form of logical reasoning where a conclusion is drawn from two given statements (premises).
Uses Venn Diagrams to test validity of conclusions.
BASIC RULES:
1. "All A are B" → A is completely inside B
2. "Some A are B" → A and B overlap
3. "No A are B" → A and B don't overlap at all
4. "Some A are not B" → Part of A is outside B
TRICK — Quick Conclusions:
• All A are B + All B are C → All A are C ✓
• All A are B + Some B are C → Some A are C (NOT definite)
• Some A are B + Some B are C → No definite conclusion
• No A are B + All B are C → Some C are not A ✓
• All A are B → Some B are A (always true — converse)
Example 1:
Statements: All dogs are animals. All animals are living things.
Conclusions: I. All dogs are living things. II. Some living things are dogs.
All dogs ⊂ Animals ⊂ Living things
I. All dogs are living things → ✓ (chain rule)
II. Some living things are dogs → ✓ (converse of I)
Answer: Both I and II follow ✓
Example 2:
Statements: Some cats are dogs. No dogs are birds.
Conclusions: I. Some cats are not birds. II. No cats are birds.
Some cats = dogs, and NO dogs = birds
I. Those cats that are dogs → they're not birds → Some cats are not birds ✓
II. No cats are birds → We can't say this for ALL cats (only those overlapping with dogs) ✗
Answer: Only I follows ✓
2. Statement & Assumptions
Assumption: Something taken for granted or accepted as true without proof. It is an unstated part of the statement that MUST be true for the statement to hold.
How to check if assumption is implicit:
1. Read the statement carefully
2. Ask: "Is this assumption NECESSARY for the statement to make sense?"
3. If YES → assumption is implicit
4. If it's just POSSIBLE but not necessary → NOT implicit
5. If it CONTRADICTS the statement → NOT implicit
Example 3:
Statement: "Join our coaching for guaranteed success in exams."
Assumptions: I. Coaching can help pass exams. II. People want to pass exams.
I. The ad assumes coaching helps → Implicit ✓
II. The ad targets people wanting to pass → Implicit ✓
Answer: Both are implicit ✓
3. Statement & Conclusions
Conclusion: A judgement or decision that logically follows from the given statement. It must be directly derivable — not just possible, but certain.
Example 4:
Statement: "India's population is growing faster than its resources."
Conclusions: I. Resources distribution may become a problem. II. India should control population.
I. If population grows > resources → distribution problem ✓ (logically follows)
II. "Should control" is a recommendation, not a direct conclusion ✗
Answer: Only I follows ✓
4. Cause & Effect
Types of Cause-Effect Relationships:
A. Statement I is the cause, II is the effect B. Statement II is the cause, I is the effect C. Both are effects of a common cause D. Both are causes of a common effect E. No cause-effect relationship
Example 5:
I. Government increased fuel prices by 10%.
II. Prices of essential commodities increased significantly.
Fuel prices ↑ → Transport costs ↑ → Commodity prices ↑ Answer: I is the cause, II is the effect (Type A) ✓
🔬Chapter 2.5 — Non-Verbal Reasoning
NON-VERBAL REASONING — Overview
Series Completion — find the next figure pattern
Analogy — figure A : B :: C : ?
Classification — find the odd figure out
Mirror & Water Images — reflection rules
Paper Folding & Cutting — predict the unfolded pattern
Dice & Cubes — opposite faces, numbering
1. Figure Series Completion
Figure Series: A sequence of figures following a pattern. You need to find the next figure.
Common Patterns: Rotation, Addition/Removal of elements, Size change, Shading pattern, Movement of elements
How to Approach:
1. Count elements in each figure — are they increasing/decreasing?
2. Check rotation — is something rotating 45°/90°/180°?
3. Check shading — is it alternating? quarter by quarter?
4. Check position — is an element moving clockwise/anti-clockwise?
5. Check addition/deletion — new elements added each step?
COMMON PATTERNS:
• 90° clockwise rotation each step
• One element added per step
• Shading rotates: top-left → top-right → bottom-right → bottom-left
• Alternate figures follow different rules (odd & even pattern)
• Two independent patterns superimposed
2. Mirror & Water Images
MIRROR IMAGE (vertical mirror on right):
Left-Right gets reversed. Up-Down stays same.
• Letters: b → d, p → q, E → ?, etc.
• Numbers: reversed horizontally
WATER IMAGE (horizontal reflection below):
Up-Down gets reversed. Left-Right stays same.
• Letters: b → p, d → q, M → W, etc.
TRICK — Mirror Image of Time:
Mirror image of clock time = 11:60 - given time
Example: Mirror of 3:25 = 11:60 - 3:25 = 8:35
Example: Mirror of 7:45 = 11:60 - 7:45 = 4:15
Example 1: What is the mirror image of 5:40?
Mirror time = 11:60 - 5:40 = 6:20
Answer: 6:20 ✓
3. Dice & Cubes
Standard Dice: Opposite faces always sum to 7.
1 ↔ 6, 2 ↔ 5, 3 ↔ 4
Non-Standard Dice: No fixed rule. Use the given views to determine opposite faces.
Finding Opposite Faces (from 2 views):
1. Find the COMMON face between two views
2. The remaining faces in each view are ADJACENT to the common face
3. Adjacent faces in view 1 are opposite to adjacent faces in view 2
CUBE CUTTING FORMULA:
If a cube of side n is cut into unit cubes:
Approach:
1. Track the folds — note how the paper is folded (in half, quarter, etc.)
2. Note the cut position — where and what shape is cut
3. Unfold mentally — the cut pattern will be symmetric about each fold line
4. One fold = 1 axis of symmetry, Two folds = 2 axes of symmetry
TRICK: Number of holes = 2^(number of folds) × holes in folded piece
If folded 2 times and 1 hole punched → 2² × 1 = 4 holes when unfolded!
🔄Chapter 2.6 — Input Output
INPUT OUTPUT — Overview
Machine processes input through multiple steps
Each step applies a FIXED operation (sort, swap, shift, etc.)
Find the pattern from given steps, then predict any step
1. Understanding Input-Output
Input-Output: A machine takes an input (numbers and/or words) and rearranges them through a series of steps following a fixed rule. You need to identify the rule and predict intermediate or final steps.
Common Operations:
1. Sorting — ascending/descending, one element per step
2. Shifting — smallest/largest moves to left/right end
3. Swapping — two elements exchange positions
4. Interleaving — words & numbers arranged alternately
5. Mixed — words sorted alphabetically + numbers sorted numerically simultaneously
2. Solved Examples
Example 1:
Input: 45 sand 17 big 93 red 28 fun 61 joy
Step 1: 93 45 sand 17 big red 28 fun 61 joy
Step 2: 93 big 45 sand 17 red 28 fun 61 joy
Step 3: 93 big 61 45 sand 17 red 28 fun joy
Step 4: 93 big 61 fun 45 sand 17 red 28 joy
Pattern: Each step alternately places the largest remaining number at the left end, then the first alphabetical word next to it.
Step 1: Largest (93) moves to front
Step 2: First word alphabetically (big) placed after 93
Step 3: Next largest (61) placed after big
Step 4: Next word (fun) placed after 61
Step 5: 93 big 61 fun 45 joy sand 17 red 28 Step 6: 93 big 61 fun 45 joy 28 sand 17 red Step 7: 93 big 61 fun 45 joy 28 red sand 17 Step 8 (Final): 93 big 61 fun 45 joy 28 red 17 sand ✓
SOLVING STRATEGY:
1. Compare Input → Step 1 → Step 2... and find what CHANGED
2. Check: Is one number/word moving per step? Or two?
3. Check: Is it moving to LEFT end or RIGHT end?
4. Check: Is the order ascending, descending, or alphabetical?
5. Once pattern found → apply it to find any step!
📊Chapter 2.7 — Data Interpretation
DATA INTERPRETATION — Overview
Tables — rows & columns of data
Bar Graphs — vertical/horizontal bars comparing quantities
Pie Charts — circle divided into sectors (percentages)
Line Graphs — data points connected by lines (trends)
Caselets — data given in paragraph form
1. Table-Based DI
Data is presented in rows and columns. Questions ask you to calculate totals, averages, percentages, ratios, etc.
Example 1: Sales (in units) of 5 products in 3 months:
Product
Jan
Feb
Mar
Total
A
120
150
130
400
B
200
180
220
600
C
90
110
100
300
D
160
140
170
470
E
80
95
85
260
Q: What % of total sales came from Product B?
Total all products = 400+600+300+470+260 = 2030
B's share = (600/2030) × 100 = 29.56% ✓
Q: What's the average monthly sale of Product D?
Avg = 470/3 = 156.67 units ✓
2. Bar Graph
Bar Graph: Uses rectangular bars to represent data. Height/length of bar = value. Can be vertical or horizontal, single or grouped.
QUICK CALCULATIONS for Bar Graphs:
• % Change = ((New - Old) / Old) × 100
• Ratio = directly compare bar heights
• Average = Sum of all bars / Number of bars
• Difference = taller bar - shorter bar
3. Pie Chart
PIE CHART FORMULAS:
Total = 360° = 100%
1% = 3.6°
Sector angle = (value / total) × 360°
Value from angle = (angle / 360°) × total
Value from % = (percentage / 100) × total
Q: How much is spent on Food?
Food = 25% of 50000 = 0.25 × 50000 = Rs 12,500 ✓
Q: What is the angle for Education sector?
Angle = 20% × 360° = 0.20 × 360 = 72° ✓
Q: Rent is how much more than Transport?
Difference = (30% - 15%) of 50000 = 15% × 50000 = Rs 7,500 ✓
4. Line Graph
Line Graph: Data points connected by lines showing trends over time. Multiple lines can show comparisons.
What to look for:
1. Upward trend — values increasing
2. Downward trend — values decreasing
3. Steepest rise — maximum % increase between two periods
4. Intersection — where two lines cross (equal values)
5. Highest/Lowest points — peak and valley
5. Caselets (Paragraph-based DI)
Caselet: Data is given in paragraph/story form instead of a graph/table. You must extract numbers and organize them yourself.
Approach:
1. Read the entire paragraph first
2. Underline all numbers and keywords
3. Create your own table from the data
4. Fill in given values, calculate missing ones
5. Now answer questions from your table
Example 3: "In a class of 60 students, 40% are girls. 75% of boys and 50% of girls passed the exam."
Girls = 40% of 60 = 24. Boys = 60 - 24 = 36.
Boys passed = 75% of 36 = 27. Girls passed = 50% of 24 = 12.
Total passed = 27 + 12 = 39. Failed = 60 - 39 = 21.
Q: What % students failed?
= (21/60) × 100 = 35% ✓
DI SPEED TRICKS:
1. Learn percentage equivalents: 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%
2. Approximate! In DI, options are usually far apart — exact calculation rarely needed
3. For % change: use "base value" method — 10% of base is easy to compute
4. Pie chart: if two sectors differ by x°, their difference = (x/360) × total
Handle exceptions (referred to manager, additional review, etc.)
1. Understanding the Format
Eligibility Test: A set of conditions is given for selection. A candidate's profile is provided and you must check if they meet all, some, or none of the criteria.
Typical Criteria: Age range, Minimum marks, Work experience, Entrance test score, Interview performance, Nationality, etc.
Approach:
1. Read ALL conditions carefully and list them
2. For each candidate, check each condition one by one
3. Mark ✓ or ✗ for each criterion
4. If ALL ✓ → Selected
5. If specific condition fails → check if any exception applies
6. If exception applies → Referred / Provisional selection
2. Solved Example
Example: A company selects candidates based on:
(i) Age: 21-30 years as on 01/01/2026
(ii) 10th marks: ≥ 60%
(iii) 12th marks: ≥ 55%
(iv) Graduation marks: ≥ 50%
(v) Selection test score: ≥ 40 out of 80
Exception A: If all criteria met except (v) but score ≥ 35, refer to GM. Exception B: If all criteria met except (ii) but 10th ≥ 55% and has 2 years experience, refer to Director.
Checking:
(i) Age on 01/01/2026: 27 years → 21-30 ✓
(ii) 10th: 72% ≥ 60% ✓
(iii) 12th: 61% ≥ 55% ✓
(iv) Graduation: 58% ≥ 50% ✓
(v) Test: 37/80 — NOT ≥ 40 ✗, but ≥ 35 ✓
All met except (v), but score ≥ 35 → Exception A applies Answer: Refer to GM ✓
TIPS FOR ELIGIBILITY:
1. Make a checklist table — conditions as rows, candidates as columns
2. Calculate age CAREFULLY using the reference date given
3. Read exceptions ONLY after checking main conditions
4. "At least" means ≥ (equal or more), "Not more than" means ≤
5. Don't mix up "as on date" — it's NOT the current date!
🤝Chapter 2.9 — Partnership
PARTNERSHIP — Overview
Simple Partnership — all invest for same time
Compound Partnership — different investment periods
Working vs Sleeping Partner
Profit split based on Capital × Time ratio
1. Basic Concepts
Partnership: When two or more people invest money in a business and share profits/losses.
Working Partner: Invests money AND manages the business (gets salary + profit share) Sleeping Partner: Only invests money, doesn't manage (gets only profit share)
PARTNERSHIP FORMULAS:
Simple Partnership (same time period):
Profit Ratio = Investment Ratio
A's share : B's share = A's capital : B's capital
Compound Partnership (different time periods):
Profit Ratio = (Capital × Time) Ratio
A's share : B's share = (A's capital × A's time) : (B's capital × B's time)
2. Solved Examples
Example 1: Simple Partnership
A invests Rs 40,000 and B invests Rs 60,000. If total profit is Rs 25,000, find each person's share.
Example 2: Compound Partnership
A starts a business with Rs 50,000. After 4 months, B joins with Rs 30,000. At end of year, profit = Rs 18,400. Find each share.
A's capital × time = 50000 × 12 = 600000
B's capital × time = 30000 × 8 = 240000
Ratio = 600000 : 240000 = 5 : 2
Example 3: Working Partner
A and B invest Rs 20,000 and Rs 30,000 respectively. A works as manager and gets 10% of total profit as salary. Rest is divided in capital ratio. If total profit = Rs 50,000, find each share.
A's total = 5000 + 18000 = Rs 23,000 B's total = Rs 27,000 ✓
Example 4: Change in Investment
A invests Rs 10,000 for first 6 months, then withdraws Rs 4,000 for remaining 6 months. B invests Rs 8,000 for the full year. Find profit ratio.
Q1: A, B, and C invest Rs 20,000, Rs 30,000, and Rs 50,000. Profit after 1 year = Rs 40,000. Find C's share.
Solution:
Ratio = 2 : 3 : 5. Total parts = 10.
C's share = (5/10) × 40000 = Rs 20,000 ✓
Q2: A starts with Rs 15,000. B joins after 3 months with Rs 20,000. C joins after 6 months with Rs 25,000. Year-end profit = Rs 46,000. Find A's share.
Solution:
A = 15000 × 12 = 180000
B = 20000 × 9 = 180000
C = 25000 × 6 = 150000
Ratio = 180 : 180 : 150 = 6 : 6 : 5. Total = 17.
A's share = (6/17) × 46000 = Rs 16,235.29 ✓
COMMON MISTAKES:
1. Forgetting to calculate TIME for each partner (B joins late = fewer months)
2. Working partner salary is deducted BEFORE profit split — not after!
3. When capital changes mid-year, calculate each period separately
4. Don't confuse "invests for" vs "joins after" — "joins after 3 months" = works for 9 months
🎯
Ready for Exam?
Quick tricks & formulas for last minute revision
⚡Quick Tricks — Last Minute Revision
How to Use: Quick revision of all formulas and tricks before exam! Read once, revise twice!
Mixture & Alligation
Alligation Ratio = (d - m) : (m - c)
c = cheaper, d = dearer, m = mean price
Mean Price = (c×q1 + d×q2) / (q1 + q2)
Replacement: Final = Initial × (1 - R/V)^n
V = vessel volume, R = replaced qty, n = operations
Remember: Water = Rs 0 always! Alligation is just weighted average in reverse.
Linear (facing North): Left = West, Right = East Linear (facing South): Left & Right SWAP!
Circular (facing center): Left = Clockwise, Right = Anti-CW Circular (facing out): Left = Anti-CW, Right = Clockwise
Strategy: Fix one person → place others relative to them
Syllogisms — Quick Rules
All A are B + All B are C → All A are C ✓
All A are B → Some B are A (always true)
No A are B → No B are A (always true)
Some A are B + No B are C → Some A are not C ✓
Some + Some → No definite conclusion ✗
Non-Verbal Reasoning
Mirror Image of Time: 11:60 - given time Cube Cutting (side n):
3 faces painted = 8 | 2 faces = 12(n-2)
1 face = 6(n-2)² | 0 faces = (n-2)³
Paper Folding: Holes = 2^(folds) × holes punched Standard Dice: Opposite faces sum = 7 (1↔6, 2↔5, 3↔4)
Simple: Profit Ratio = Capital Ratio Compound: Profit Ratio = (Capital × Time) Ratio
"Joins after x months" = works for (12-x) months
Working partner salary → deduct BEFORE profit split
Input-Output
1. Compare consecutive steps — what moved?
2. One element per step? Two? Words & numbers separate?
3. Moving to LEFT end or RIGHT end?
4. Sorting order: ascending / descending / alphabetical?
5. Pattern found → apply to any step!
Eligibility Test
1. List ALL conditions in a table
2. Check each candidate against EACH condition
3. All pass → Selected
4. One fails → Check exceptions
5. Exception matches → Referred to specific authority
6. No exception → Rejected
EXAM STRATEGY:
• Attempt DI, Partnership, Mixture first — formula-based = fast marks!
• Seating & Blood Relation — draw diagrams, don't solve in head
• Syllogisms — draw Venn diagrams for 100% accuracy
• Input-Output — invest 1 min finding pattern, then all Qs become easy
• Non-Verbal — practice is the only way, no shortcuts
• Time management: 1 min per question max!
All the Best! 🏆
You've got this. Formulas yaad hain, tricks pata hain, bas confidence rakhna aur diagrams banana mat bhoolna!
📝Important Questions — Exam-Focused
About This Section: Contains 15 important questions covering all 9 chapters. Click any question to see the complete solution.
Mixture & Alligation
1
Q1. Tea at Rs 126/kg is mixed with tea at Rs 135/kg in 1:2 ratio. Find the mixture price per kg.
Q2. A vessel has 60L of milk. 15L is taken out and replaced with water. This is done 3 times. Find the quantity of milk left and the ratio of milk to water.
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Answer:
Milk left = 60 × (1 - 15/60)³ = 60 × (3/4)³
= 60 × 27/64 = 25.3125 L
Water = 60 - 25.3125 = 34.6875 L
Ratio (Milk : Water) = 25.3125 : 34.6875 = 27 : 37
Blood Relation
3
Q3. Pointing to a woman, Ram said "She is the daughter of the only child of my grandmother." How is the woman related to Ram?
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Answer:
Ram's grandmother's only child = Ram's father or mother
Daughter of Ram's parent = Ram's sister
Answer: Sister
Seating Arrangement
4
Q4. 6 people (A-F) sit in a circle facing center. B is opposite E. A is to immediate left of B. C is between D and F. D is not adjacent to B. Find the arrangement.
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Answer:
Fix B at top. E is opposite (bottom).
A is immediate left of B = clockwise next from B.
D is NOT adjacent to B → D can't be next to B.
C is between D and F.
Clockwise arrangement: B, A, D, E, F, C
Verify: B opposite E ✓ | A immediate left of B ✓ | C between F and D ✓ | D not adjacent to B ✓
Syllogisms
5
Q5. Statements: All books are pens. Some pens are erasers. Conclusions: I. Some books are erasers. II. Some erasers are pens.
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Answer:
All books ⊂ pens, Some pens ∩ erasers
I. Some books are erasers → NOT definite (the overlapping pens may not include books) ✗
II. Some erasers are pens → Converse of "Some pens are erasers" = always true ✓
Answer: Only conclusion II follows
6
Q6. Statements: No fish is a bird. All birds are animals. Conclusions: I. No fish is an animal. II. Some animals are not fish.
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Answer:
No fish = bird, All birds ⊂ animals
I. No fish is animal → Can't say! Fish may still be animals through other paths ✗
II. Some animals are not fish → Birds are animals and birds are NOT fish, so YES ✓
Answer: Only conclusion II follows
Non-Verbal Reasoning
7
Q7. What is the mirror image of the time 4:50? Also, find the water image of the letter 'E'.
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Answer:
Mirror time = 11:60 - 4:50 = 7:10
Water image of 'E': flip vertically → looks like a '∃' (reverse E with horizontal flip). The top bar goes to bottom, bottom bar goes to top.
8
Q8. A cube of side 5 is painted on all faces and cut into unit cubes. How many small cubes have exactly 1 face painted?
Q9. Pie chart: Total budget = Rs 2,40,000. Education: 20%, Food: 30%, Rent: 25%, Transport: 15%, Misc: 10%. Find: (a) Amount on Food (b) Angle of Rent sector (c) Education is how much more than Misc?
Q10. Input: 72 cut 15 big 38 joy 91 red 46 ant. Step 1: 91 72 cut 15 big 38 joy red 46 ant. Step 2: 91 ant 72 cut 15 big 38 joy red 46. Find the pattern and give Step 5.
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Answer:
Pattern: Alternately place largest remaining number, then first alphabetical word, at the left end.
Step 1: 91 (largest number) → front
Step 2: ant (first word alphabetically) → after 91
Step 3: 72 (next largest) → after ant
Step 4: big (next word) → after 72 Step 5: 91 ant 72 big 46 cut 15 38 joy red
(46 = next largest → placed after big)
Partnership
11
Q11. A and B invest Rs 25,000 and Rs 35,000 respectively. If total profit = Rs 48,000, find each share.
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Answer:
Ratio = 25000 : 35000 = 5 : 7. Total = 12.
A = (5/12) × 48000 = Rs 20,000
B = (7/12) × 48000 = Rs 28,000
12
Q12. A starts business with Rs 40,000. After 3 months B joins with Rs 60,000. After 6 months from start, C joins with Rs 80,000. Year-end profit = Rs 37,500. Find each share.
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Answer:
A = 40000 × 12 = 4,80,000
B = 60000 × 9 = 5,40,000
C = 80000 × 6 = 4,80,000
Ratio = 480 : 540 : 480 = 8 : 9 : 8. Total = 25.
A = (8/25) × 37500 = Rs 12,000
B = (9/25) × 37500 = Rs 13,500
C = (8/25) × 37500 = Rs 12,000
Eligibility Test
13
Q13. Selection criteria: (i) Age 21-28 on 01/01/2026 (ii) 12th ≥ 65% (iii) Graduation ≥ 55% (iv) Test ≥ 45/100. Exception: If (iv) fails but ≥ 40, refer to Director. Candidate: Priya, DOB: 10/05/2000, 12th: 78%, Grad: 62%, Test: 42/100.
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Answer:
(i) Age on 01/01/2026: 25 years → 21-28 ✓
(ii) 12th: 78% ≥ 65% ✓
(iii) Grad: 62% ≥ 55% ✓
(iv) Test: 42 < 45 ✗ BUT ≥ 40 ✓
All pass except (iv), but score ≥ 40 → Exception applies Answer: Refer to Director
Statement & Assumptions
14
Q14. Statement: "The best way to travel is by flight — book now!" Assumptions: I. People prefer faster travel. II. Flights are cheaper than trains.
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Answer:
I. "Best way" implies speed/convenience preference → Implicit ✓
II. Nothing about cost comparison → NOT implicit ✗
Answer: Only assumption I is implicit
15
Q15. Cause: Heavy rains caused floods in the city. Effect: Schools declared a holiday. Identify the relationship.
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Answer:
Heavy rains → Floods → Schools can't function → Holiday declared
Answer: Statement I is the cause, Statement II is the effect