Class 11 Applied Mathematics

Quantitative
Aptitude Logic

Mastering the logic of time, physical space, and calendars—transforming raw numbers into strategic reasoning.

Calendars & Odd Days

ℹ️

Odd Days Logic

The core of calendar math is 'Odd Days'. It is the number of days more than a complete number of weeks. - Ordinary Year:

365 / 7 = 52

weeks + **1 odd day**. - Leap Year:

366 / 7 = 52

weeks + **2 odd days**.

Century Rules

100 Years5 Odd Days

200 Years3 Odd Days

300 Years1 Odd Day

400 Years0 Odd Days

Leap Year Checker

Step-by-Step Decision Logic

Condition

Divisible by 4?

Ordinary Year

YES

Continue to Step 2

Check Century

Ends with "00"?

Leap Year

YES

Check Rule 3

Century Rule

Divisible by 400?

Ordinary

YES

Leap Year

✔

2024 is div. by 4 but not 100 → Leap

✘

1900 is div. by 100 but not 400 → Ordinary

Clock Mechanics

Interactive Clock Lab

Visualize angles between Minute and Hour hands.

Hour: 3

Minute: 0

Angle Calculation

Hour Hand Pos:90.0°

Minute Hand Pos:0°

Net Angle (θ)90.0°

Applied Formula: $\theta = |30H - 5.5M|$
Here $H$ is the hour (3) and $M$ is the minute (0).

💡

Clocks: 5 Essential Facts

Based on a 12-hour period: 1. Hands coincide (0°) **11 times**. 2. Hands are opposite (180°) **11 times**. 3. Hands are in a straight line **22 times**. 4. Hands are at right angles (90°) **22 times**. 5. In 60 min, the minute hand gains **55 min spaces** over the hour hand.

Hand Speeds

Minute Hand

6^\circ / minute

Hour Hand

0.5^\circ / minute

Angle Analysis Formula

The Universal Clock Formula

\text{Angle } \theta = |30H - 5.5M|

Valid for all values of $H \in [1, 12]$ and $M \in [0, 59]$.

Time, Work & Movement

Relative Speed Terminal

Analyze closing or opening speeds between two objects.

60 km/h

40 km/h

Vehicle 1 Speed: 60 km/h

Vehicle 2 Speed: 40 km/h

Resulting Relative Speed

100 km/h

Logic: $S_1 + S_2$

Work Logic 101

The Inverse Rule

If a person completes a job in 5 days, they don't do 1/5 of the total job every day in your head—they do it literally. Work and Time are Inversely Proportional.

Fundamental Identity

Efficiency = 1 / Time

1/5

Day 1

1/5

Day 2

1/5

Day 3

1/5

Day 4

1/5

Day 5

Distance Logic

❗

Relative Speed

When two objects move: - Opposite direction: Speed =

S_1 + S_2

. - Same direction: Speed =

|S_1 - S_2|

Average Speed (Equal Distances)

\text{Avg Speed} = \frac{2xy}{x+y}

Work Logic

💡

Capacity & Time

Work and Efficiency are inversely related. If A is twice as fast as B, A takes half the time of B.

Combined Work Formula

\text{\text{together}} = \frac{Work_1 \times Work_2}{Work_1 + Work_2}

Logical Seating

Seating Arrangement Visualizer

Explore Left/Right logic in different layouts.

LEFTRIGHT

Arrangement Parameters

Facing Direction (Circular)

Direction Logic

In linear row, unless specified, assume they face North. Your Left is their Left, Your Right is their Right.

Analysis: Position of C

Left neighboring: B

Right neighboring: D

Strategy Checklist

Fixed Positions

Always start with clues that give an exact location (e.g., 'A is at the extreme left').

Relative Clues

Connect other people to the fixed person using 'Left' and 'Right' cues.

Facing Directions

In circular tables, clarify if they face center (Left = Anti-clockwise) or outward.

Possibilities

If unsure, draw multiple sketches and eliminate those that contradict any clues.

Aptitude Lab

Logic Practice

8 Concept Problems

5 Exam Masterworks

Logical Analysis Zone

Find the angle between hour and minute hand at 5:40.

✓ Show Solution

1. Formula:

\theta = |30H - 5.5M|

2. Plug in

H=5, M=40

\theta = |30(5) - 5.5(40)|

\theta = |150 - 220 \mid = |-70 \mid = 70 ^\circ

EASY

A does a work in 10 days and B does it in 15 days. How long will they take working together?

✓ Show Solution

1. A's 1-day work =

1/10

2. B's 1-day work =

1/15

3. (A+B)'s 1-day work =

1/10 + 1/15 = 5/30 = 1/6

4. Time taken = 6 days.

MEDIUM

What day of the week was June 15, 2024? (Given Jan 1, 2024 was Monday)

✓ Show Solution

1. Days from Jan 1 to June 15:

Jan: 30 (total 31 - starting 1), Feb: 29 (leap), Mar: 31, Apr: 30, May: 31, June: 15

2. Total days =

30+29+31+30+31+15 = 166

3. Odd days =

166 / 7 \Rightarrow 5

remainder.

4. Monday + 5 days = Saturday.

HARD

Two trains 150m and 120m long are running in opposite directions at 45 km/h and 36 km/h. How long will they take to cross each other?

✓ Show Solution

1. Relative Speed =

45 + 36 = 81

km/h =

81 \times (5/18) = 22.5

m/s

2. Total Distance =

150 + 120 = 270

3. Time =

270 / 22.5 = 12

seconds.

MEDIUM

6 people A, B, C, D, E, F sit in a circle. A is second to left of C. B is immediate right of A. D is between C and F. Who is opposite A?

✓ Show Solution

1. Arrange based on cues: [A, B, E, C, D, F] clockwise.

2. C is opposite B, E is opposite F, D is opposite A?

3. Let's re-verify: A, B(R of A), E(placeholder), C(A is 2nd left of C), D(between C/F), F.

4. Circular order: A -> B -> E -> C -> D -> F.

5. Opposite A is C.

HARD

How many odd days are there in 400 years?

✓ Show Solution

1. 100 yrs = 5 odd days

2. 400 yrs =

5 \times 4 + 1

(leap century) = 21 odd days

21 / 7 = 0

remainder. So 0 odd days.

MEDIUM

If a clock strikes 12, how many times do the hands overlap in a day?

✓ Show Solution

1. Hands overlap 11 times in 12 hours (losing one during 11-1 shift).

2. In a full day (24 hours), they overlap 22 times.

EASY

A man covers 300 km at 60 km/h and returns at 40 km/h. Find average speed.

✓ Show Solution

1. Avg Speed =

2xy / (x+y) = (2 \times 60 \times 40) / 100 = 4800 / 100 = 48

km/h.

MEDIUM

Competitive Edge

Board questions & Logical Extensions

CBSE Aptitude

A clock is started at noon. By 10 minutes past 5, the hour hand has turned through how many degrees?

Logic Pathway1. Time = 5 hours 10 mins = 310 mins. 2. Hour hand speed =

0.5 ^\circ

per min. 3. Angle =

310 \times 0.5 = 155 ^\circ

Reasoning Unit

If 1st March of a year is Sunday, which day will 31st August of the same year be?

Logic Pathway1. Odd days: Mar(3), Apr(2), May(3), Jun(2), Jul(3), Aug(30 days = 2) 2. Total =

3+2+3+2+3+2 = 15

15 / 7 = 1

remainder. 4. Sunday + 1 = Monday.

CBSE 2023

A can finish a work in 18 days and B can do the same work in half the time taken by A. Then, working together, what part of the same work can they finish in a day?

Logic Pathway1. A's 1-day work =

1/18

2. B's 1-day work =

1/9

(half time of 18 is 9) 3. Together in 1 day =

1/18 + 1/9 = 3/18 = 1/6

Logic Section

Eight people P, Q, R, S, T, U, V, W are sitting around a table facing center. P is second to the right of T who is the neighbor of R and V. S, the neighbor of U, is not the neighbor of P. V is the neighbor of U. Q is not between S and W. W is not between U and S. Who is the immediate right of V?

Logic Pathway1. Circular arrangement logic results in U being immediate right of V. 2. Order: T -> V -> U -> S -> R -> P -> W -> Q (clockwise).

Time-Distance

Relative speed of two trains is 72 km/h when moving in same direction. If speed of one is 120 km/h, find other.

Logic Pathway1. Same direction:

|S_1 - S_2 \mid = 72

120 - S_2 = 72 \Rightarrow S_2 = 48

km/h or

S_2 = 192

km/h. 3. Usually smaller is expected: 48 km/h.

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Class 11 Applied Mathematics

Quantitative
Aptitude Logic

Mastering the logic of time, physical space, and calendars—transforming raw numbers into strategic reasoning.

Calendars & Odd Days

ℹ️

Odd Days Logic

The core of calendar math is 'Odd Days'. It is the number of days more than a complete number of weeks. - Ordinary Year:

365 / 7 = 52

weeks + **1 odd day**. - Leap Year:

366 / 7 = 52

weeks + **2 odd days**.

Century Rules

100 Years5 Odd Days

200 Years3 Odd Days

300 Years1 Odd Day

400 Years0 Odd Days

Leap Year Checker

Step-by-Step Decision Logic

Condition

Divisible by 4?

Ordinary Year

YES

Continue to Step 2

Check Century

Ends with "00"?

Leap Year

YES

Check Rule 3

Century Rule

Divisible by 400?

Ordinary

YES

Leap Year

✔

2024 is div. by 4 but not 100 → Leap

✘

1900 is div. by 100 but not 400 → Ordinary

Clock Mechanics

Interactive Clock Lab

Visualize angles between Minute and Hour hands.

Hour: 3

Minute: 0

Angle Calculation

Hour Hand Pos:90.0°

Minute Hand Pos:0°

Net Angle (θ)90.0°

Applied Formula: $\theta = |30H - 5.5M|$
Here $H$ is the hour (3) and $M$ is the minute (0).

💡

Clocks: 5 Essential Facts

Hand Speeds

Minute Hand

6^\circ / minute

Hour Hand

0.5^\circ / minute

Angle Analysis Formula

The Universal Clock Formula

\text{Angle } \theta = |30H - 5.5M|

Valid for all values of $H \in [1, 12]$ and $M \in [0, 59]$.

Time, Work & Movement

Relative Speed Terminal

Analyze closing or opening speeds between two objects.

60 km/h

40 km/h

Vehicle 1 Speed: 60 km/h

Vehicle 2 Speed: 40 km/h

Resulting Relative Speed

100 km/h

Logic: $S_1 + S_2$

Work Logic 101

The Inverse Rule

If a person completes a job in 5 days, they don't do 1/5 of the total job every day in your head—they do it literally. Work and Time are Inversely Proportional.

Fundamental Identity

Efficiency = 1 / Time

1/5

Day 1

1/5

Day 2

1/5

Day 3

1/5

Day 4

1/5

Day 5

Distance Logic

❗

Relative Speed

When two objects move: - Opposite direction: Speed =

S_1 + S_2

. - Same direction: Speed =

|S_1 - S_2|

Average Speed (Equal Distances)

\text{Avg Speed} = \frac{2xy}{x+y}

Work Logic

💡

Capacity & Time

Work and Efficiency are inversely related. If A is twice as fast as B, A takes half the time of B.

Combined Work Formula

\text{\text{together}} = \frac{Work_1 \times Work_2}{Work_1 + Work_2}

Logical Seating

Seating Arrangement Visualizer

Explore Left/Right logic in different layouts.

LEFTRIGHT

Arrangement Parameters

Facing Direction (Circular)

Direction Logic

In linear row, unless specified, assume they face North. Your Left is their Left, Your Right is their Right.

Analysis: Position of C

Left neighboring: B

Right neighboring: D

Strategy Checklist

Fixed Positions

Always start with clues that give an exact location (e.g., 'A is at the extreme left').

Relative Clues

Connect other people to the fixed person using 'Left' and 'Right' cues.

Facing Directions

In circular tables, clarify if they face center (Left = Anti-clockwise) or outward.

Possibilities

If unsure, draw multiple sketches and eliminate those that contradict any clues.

Aptitude Lab

Logic Practice

8 Concept Problems

5 Exam Masterworks

Logical Analysis Zone

Find the angle between hour and minute hand at 5:40.

✓ Show Solution

1. Formula:

\theta = |30H - 5.5M|

2. Plug in

H=5, M=40

\theta = |30(5) - 5.5(40)|

\theta = |150 - 220 \mid = |-70 \mid = 70 ^\circ

EASY

A does a work in 10 days and B does it in 15 days. How long will they take working together?

✓ Show Solution

1. A's 1-day work =

1/10

2. B's 1-day work =

1/15

3. (A+B)'s 1-day work =

1/10 + 1/15 = 5/30 = 1/6

4. Time taken = 6 days.

MEDIUM

What day of the week was June 15, 2024? (Given Jan 1, 2024 was Monday)

✓ Show Solution

1. Days from Jan 1 to June 15:

Jan: 30 (total 31 - starting 1), Feb: 29 (leap), Mar: 31, Apr: 30, May: 31, June: 15

2. Total days =

30+29+31+30+31+15 = 166

3. Odd days =

166 / 7 \Rightarrow 5

remainder.

4. Monday + 5 days = Saturday.

HARD

Two trains 150m and 120m long are running in opposite directions at 45 km/h and 36 km/h. How long will they take to cross each other?

✓ Show Solution

1. Relative Speed =

45 + 36 = 81

km/h =

81 \times (5/18) = 22.5

m/s

2. Total Distance =

150 + 120 = 270

3. Time =

270 / 22.5 = 12

seconds.

MEDIUM

6 people A, B, C, D, E, F sit in a circle. A is second to left of C. B is immediate right of A. D is between C and F. Who is opposite A?

✓ Show Solution

1. Arrange based on cues: [A, B, E, C, D, F] clockwise.

2. C is opposite B, E is opposite F, D is opposite A?

3. Let's re-verify: A, B(R of A), E(placeholder), C(A is 2nd left of C), D(between C/F), F.

4. Circular order: A -> B -> E -> C -> D -> F.

5. Opposite A is C.

HARD

How many odd days are there in 400 years?

✓ Show Solution

1. 100 yrs = 5 odd days

2. 400 yrs =

5 \times 4 + 1

(leap century) = 21 odd days

21 / 7 = 0

remainder. So 0 odd days.

MEDIUM

If a clock strikes 12, how many times do the hands overlap in a day?

✓ Show Solution

1. Hands overlap 11 times in 12 hours (losing one during 11-1 shift).

2. In a full day (24 hours), they overlap 22 times.

EASY

A man covers 300 km at 60 km/h and returns at 40 km/h. Find average speed.

✓ Show Solution

1. Avg Speed =

2xy / (x+y) = (2 \times 60 \times 40) / 100 = 4800 / 100 = 48

km/h.

MEDIUM

Competitive Edge

Board questions & Logical Extensions

CBSE Aptitude

A clock is started at noon. By 10 minutes past 5, the hour hand has turned through how many degrees?

Logic Pathway1. Time = 5 hours 10 mins = 310 mins. 2. Hour hand speed =

0.5 ^\circ

per min. 3. Angle =

310 \times 0.5 = 155 ^\circ

Reasoning Unit

If 1st March of a year is Sunday, which day will 31st August of the same year be?

Logic Pathway1. Odd days: Mar(3), Apr(2), May(3), Jun(2), Jul(3), Aug(30 days = 2) 2. Total =

3+2+3+2+3+2 = 15

15 / 7 = 1

remainder. 4. Sunday + 1 = Monday.

CBSE 2023

A can finish a work in 18 days and B can do the same work in half the time taken by A. Then, working together, what part of the same work can they finish in a day?

Logic Pathway1. A's 1-day work =

1/18

2. B's 1-day work =

1/9

(half time of 18 is 9) 3. Together in 1 day =

1/18 + 1/9 = 3/18 = 1/6

Logic Section

Eight people P, Q, R, S, T, U, V, W are sitting around a table facing center. P is second to the right of T who is the neighbor of R and V. S, the neighbor of U, is not the neighbor of P. V is the neighbor of U. Q is not between S and W. W is not between U and S. Who is the immediate right of V?

Logic Pathway1. Circular arrangement logic results in U being immediate right of V. 2. Order: T -> V -> U -> S -> R -> P -> W -> Q (clockwise).

Time-Distance

Relative speed of two trains is 72 km/h when moving in same direction. If speed of one is 120 km/h, find other.

Logic Pathway1. Same direction:

|S_1 - S_2 \mid = 72

120 - S_2 = 72 \Rightarrow S_2 = 48

km/h or

S_2 = 192

km/h. 3. Usually smaller is expected: 48 km/h.