Class 5 Maths — NCERT Solutions (CBSE 2026-27)

Fraction Identification:
• (a) Circle: Total parts = 8, Shaded = 3. Fraction = 3/8 (three-eighths).
• (b) Rectangle: Total parts = 12, Shaded = 7. Fraction = 7/12 (seven-twelfths).
• (c) Strip: Total parts = 10. Shaded parts = 10 - 4 = 6. Fraction = 6/10 = 3/5 (three-fifths).

Finding Equivalent Fractions:
• (a) 3/4:
- (3 × 2)/(4 × 2) = 6/8
- (3 × 3)/(4 × 3) = 9/12
- (3 × 4)/(4 × 4) = 12/16.
• (b) 2/7:
- (2 × 2)/(7 × 2) = 4/14
- (2 × 3)/(7 × 3) = 6/21
- (2 × 5)/(7 × 5) = 10/35.
• (c) 18/24:
- Divide by 2: (18 ÷ 2)/(24 ÷ 2) = 9/12
- Divide by 3: (18 ÷ 3)/(24 ÷ 3) = 6/8
- Divide by 6: (18 ÷ 6)/(24 ÷ 6) = 3/4 (simplest form).

Comparison by Cross Multiplication:
• (a) 3/5 vs 4/7: Cross multiply: 3 × 7 = 21, and 5 × 4 = 20. Since 21 > 20, 3/5 > 4/7.
• (b) 5/8 vs 7/12: Cross multiply: 5 × 12 = 60, and 8 × 7 = 56. Since 60 > 56, 5/8 > 7/12.
• (c) 2/3 vs 8/12: Cross multiply: 2 × 12 = 24, and 3 × 8 = 24. Since 24 = 24, 2/3 = 8/12 (they are equal equivalent fractions).

Number Line Representation:
1. Divide the segment from 0 to 1 into 4 equal segments.
2. Mark points at 1/4, 2/4, 3/4, and 4/4.
• Another name for 2/4: Simplifying gives 1/2 (One-Half).
• Another name for 4/4: 4 ÷ 4 = 1 (One Whole).

Calculations (Total = 16 pieces):
• (a) Sister's Share: 1/4 of 16 = 16 ÷ 4 = 4 pieces.
• (b) Mother's Share: 1/2 of 16 = 16 ÷ 2 = 8 pieces.
• (c) Rohan's Share:
- Pieces left = 16 - (4 + 8) = 16 - 12 = 4 pieces.
- Fraction = 4/16 = 1/4 of the chocolate bar.

Step-by-Step Addition:
• (a) 2/9 + 4/9: (2 + 4)/9 = 6/9 = 2/3.
• (b) 3/10 + 5/10: (3 + 5)/10 = 8/10 = 4/5.
• (c) 7/15 + 8/15: (7 + 8)/15 = 15/15 = 1 (Whole).

Calculations:
• (a) 7/8 - 3/8: (7 - 3)/8 = 4/8 = 1/2.
• (b) 11/12 - 5/12: (11 - 5)/12 = 6/12 = 1/2.
• (c) 1 - 3/5: Write 1 as 5/5: 5/5 - 3/5 = (5 - 3)/5 = 2/5.

Calculations:
• (a) 3/4 of 24: (24 ÷ 4) × 3 = 6 × 3 = 18 apples.
• (b) 2/5 of 45: (45 ÷ 5) × 2 = 9 × 2 = 18 students.
• (c) 5/6 of 60: (60 ÷ 6) × 5 = 10 × 5 = 50 minutes.

Conversions & Comparison:
• (a) 14/3: 14 ÷ 3 gives quotient 4 with remainder 2 → $4\frac{2}{3}$.
• (b) $3\frac{2}{5}$: (3 × 5 + 2)/5 = (15 + 2)/5 = 17/5.
• (c) Comparison: Convert $2\frac{1}{2}$ to improper: 5/2 = 10/4. Since 10/4 > 9/4, $2\frac{1}{2} > 9/4$.

Analysis (Total = 12 beds):
• (a) Tomatoes: 4/12 = 1/3 of the field.
• (b) Carrots: Beds left = 12 - (4 + 3 + 2) = 12 - 9 = 3 beds. Fraction = 3/12 = 1/4 of the field.
• (c) Potatoes + Spinach: 3/12 + 2/12 = 5/12. Half the field is 6/12. Since 5/12 < 6/12, No, they occupy less than half of the field.

Working:
• (a) Current Petrol: 3/8 of 40 L = (40 ÷ 8) × 3 = 5 × 3 = 15 litres.
• (b) Petrol to Fill: 40 L - 15 L = 25 litres (or 5/8 of 40 = 25 L).

Finding Common Denominator (LCM of 3, 6, 4, 12 is 12):
• 2/3 = (2 × 4)/(3 × 4) = 8/12
• 1/6 = (1 × 2)/(6 × 2) = 2/12
• 3/4 = (3 × 3)/(4 × 3) = 9/12
• 5/12 = 5/12
Comparing numerators: 2 < 5 < 8 < 9.
Ascending Order: 1/6 < 5/12 < 2/3 < 3/4.

Geometry & Fractions:
• (a) Equal Parts: 2 × 2 = 4 equal squares.
• (b) Fraction Coloured: 3 out of 4 = 3/4 (three-fourths).
• (c) Total Area: 4 × 9 cm² = 36 cm².

Unit Fractions Explanation:
• (a) Definition: A fraction with a numerator of 1 is called a unit fraction. Examples: 1/2, 1/5, 1/10.
• (b) Comparison: 1/7 > 1/9.
Rule: When two fractions have the same numerator (1), the fraction with the smaller denominator is greater, because dividing a whole into fewer parts makes each individual part larger.

Working:
• (a) Science Books: 1/3 of 60 = 60 ÷ 3 = 20 books.
• (b) Storybooks: 2/5 of 60 = (60 ÷ 5) × 2 = 12 × 2 = 24 books.
• (c) Math Puzzle Books: 60 - (20 + 24) = 60 - 44 = 16 books.

Conversions:
• (a) 3/10: 0.3.
• (b) 75/100: 0.75.
• (c) 2/5: Multiply top and bottom by 2: 4/10 = 0.4.
• (d) 1/4: Multiply top and bottom by 25: 25/100 = 0.25.

Algebraic Reasoning:
1. Let the fraction be (2 × k) / (3 × k).
2. Sum of numerator and denominator: 2k + 3k = 25.
3. 5k = 25 → k = 5.
4. Numerator = 2 × 5 = 10.
5. Denominator = 3 × 5 = 15.
The fraction is 10/15 (Check: 10 + 15 = 25 and 10/15 reduces to 2/3).

1 Hour = 60 Minutes:
• (a) 1/4 of 60: 60 ÷ 4 = 15 minutes.
• (b) 2/3 of 60: (60 ÷ 3) × 2 = 20 × 2 = 40 minutes.
• (c) 1/2 of 1/2 of 60: 1/4 of 60 = 15 minutes.

Financial Calculations:
• (a) Geometry Box: 1/3 of 240 = 240 ÷ 3 = ₹80.
• (b) Sketch Pens: 1/4 of 240 = 240 ÷ 4 = ₹60.
• (c) Money Left: 240 - (80 + 60) = 240 - 140 = ₹100.

Simplification:
• (a) 24/36: HCF(24, 36) = 12. (24 ÷ 12)/(36 ÷ 12) = 2/3.
• (b) 15/45: HCF(15, 45) = 15. (15 ÷ 15)/(45 ÷ 15) = 1/3.
• (c) 28/70: HCF(28, 70) = 14. (28 ÷ 14)/(70 ÷ 14) = 2/5.

1 kg = 1,000 g:
• (a) In grams:
- Biscuits: 3/8 of 1,000 = (1000 ÷ 8) × 3 = 125 × 3 = 375 g.
- Bread: 1/2 of 1,000 = 500 g.
• (b) Total Flour Used: 375 g + 500 g = 875 g.
• (c) Unused Flour: 1,000 g - 875 g = 125 g (or 1/8 kg).

Calculations:
• (a) Fraction of a Metre: Total length = 2 m. Length per piece = 2 ÷ 8 = 2/8 = 1/4 of a metre.
• (b) In Centimetres: 1 m = 100 cm. 1/4 of 100 cm = 100 ÷ 4 = 25 cm.

Justifications:
• (a) False: 2/3 of 12 = 8; 3/4 of 12 = 9. Since 8 < 9, the statement is False.
• (b) True: When numerator = denominator (e.g. 5/5, 8/8), dividing a quantity into $n$ parts and taking all $n$ parts equals exactly 1 whole.
• (c) True: 4/6 simplifies to 2/3 (divide by 2). 6/9 simplifies to 2/3 (divide by 3). Both equal 2/3, so they are equivalent.

Working (Total Matches = 20):
• (a) Matches Won: 3/5 of 20 = (20 ÷ 5) × 3 = 4 × 3 = 12 matches.
• (b) Matches Lost: 1/4 of 20 = 20 ÷ 4 = 5 matches.
• (c) Matches Tied: 20 - (12 + 5) = 20 - 17 = 3 matches.
- Fraction tied = 3/20 of the matches.

Clock Hand Angle Analysis: (Each hour mark = 360° ÷ 12 = 30°)
• (a) 3:00: Hour at 3, minute at 12. Separation = 3 hours = 3 × 30° = 90° (Right Angle).
• (b) 6:00: Hour at 6, minute at 12. Separation = 6 hours = 6 × 30° = 180° (Straight Angle).
• (c) 9:30: Minute hand at 6, hour hand halfway between 9 and 10 (at 9.5). Distance = 3.5 marks = 3.5 × 30° = 105° (Obtuse Angle).
• (d) 1:15: Minute hand at 3, hour hand slightly past 1. Distance ≈ 1.75 marks ≈ 52.5° (Acute Angle).

Direction Analysis:
• (a) 1/4 Turn Clockwise: 90° turn to the right from North → East.
• (b) 1/2 Turn: 180° turn (opposite direction) from North → South.
• (c) 3/4 Turn Anticlockwise: 270° counter-clockwise: North → West (90°) → South (180°) → East (270°).

Geometric Angle Count in Letters:
• (a) Letter 'H': The horizontal bar meets the two vertical uprights creating 2 interior right angles on the left and 2 on the right = 4 right angles (all 90°).
• (b) Letter 'E': Three horizontal bars meet the vertical spine: top bar (1 right angle), middle bar (2 right angles above and below), bottom bar (1 right angle) = 4 right angles (all 90°).

Cardboard Angle Tester:
1. Construction: Cut two equal strips of stiff cardboard (about 15 cm long). Fasten them together at one end using a drawing pin or split paper pin so they can pivot freely.
2. Testing a Textbook Corner:
• Open the two arms so that one arm lies along the bottom edge and the other arm lies along the side edge of the textbook.
• If the arms open to form an 'L' shape matching the corner perfectly without any gap or overlap, the corner is exactly a Right Angle (90°).
• If the corner opens less than the 'L' shape, it is an Acute Angle. If more, it is an Obtuse Angle.

Yoga Asana Angle Identification:
• (a) Dandasana: The spine is vertical (90°) to the horizontal legs lying flat on the mat. Angle = Right Angle (90°).
• (b) Bhujangasana: The chest and head lift upward and backward away from the floor, opening past 90°. Angle = Obtuse Angle (greater than 90°).
• (c) Uttanasana: The upper body folds deeply forward against the legs. The angle at the hip joint is significantly less than 90°. Angle = Acute Angle (less than 90°).

Degree Computations (Full Turn = 360°):
• (a) 1/4 Turn: 360° ÷ 4 = 90° (Right angle).
• (b) 1/2 Turn: 360° ÷ 2 = 180° (Straight angle).
• (c) 3/4 Turn: (360° ÷ 4) × 3 = 90° × 3 = 270° (Reflex angle).
• (d) 1/8 Turn: 360° ÷ 8 = 45° (Half of a right angle).

Analyzing Letters in 'RAMAN':
• R: Upright and top loop has 2 right angles.
• A: 3 acute angles (1 apex + 2 at crossbar base), 2 obtuse angles (inside crossbar junction).
• M: 3 acute angles (1 central valley + 2 outer peaks).
• A: 3 acute angles, 2 obtuse angles.
• N: 2 acute angles (top and bottom corners).
Summary Count:
• (a) Right Angles: 2.
• (b) Acute Angles: 3 + 3 + 3 + 2 = 11.
• (c) Obtuse Angles: 2 + 2 = 4.

Protractor Alignment Steps:
1. Place the centre point (cross-mark) of the protractor directly over the vertex point $B$ of the angle.
2. Align the baseline of the protractor precisely along ray $BC$ so that the $0^\circ$ mark lies directly on line $BC$.
3. Look at the second ray $BA$. Follow the scale starting from $0^\circ$ on the baseline to where ray $BA$ passes through the curved scale.
4. Read the degree value on that scale. That number gives the exact measurement of $\angle ABC$ in degrees.

Classification:
• (a) 35°: Less than 90° → Acute Angle.
• (b) 90°: Exactly 90° → Right Angle.
• (c) 128°: Between 90° and 180° → Obtuse Angle.
• (d) 89°: Less than 90° → Acute Angle.
• (e) 180°: Exactly 180° → Straight Angle.
• (f) 92°: Greater than 90° → Obtuse Angle.

Billiards / Carrom Angle Geometry:
• (a) Angle Type: 45° is less than 90°, so it is an Acute Angle (half of a right angle).
• (b) Rebound Angle: By the law of reflection, angle of reflection equals angle of incidence: Rebound angle = 45° with the cushion border.
• (c) Angle of Deviation: The straight line path (180°) minus the sum of two cushion angles (45° + 45° = 90°) gives a deflection of 180° - 90° = 90° (a right-angle turn).

Angle Variation in Hinged Tools:
• (a) Blade Opening: When scissors are opened wider, the angle between the blades increases in degree measure.
• (b) Combined Angle: Initial angle = 30°. Additional opening = 60°. Total angle = 30° + 60° = 90°. This forms an exact Right Angle.

Complementary Angle Calculations:
• (a) Complement of 35°: 90° - 35° = 55°.
• (b) Complement of 60°: 90° - 60° = 30°.
• (c) Two Obtuse Angles: No. An obtuse angle is greater than 90°. The sum of two obtuse angles will always exceed 180°, so their sum can never be 90°.

Supplementary Calculations:
• (a) Adjacent Angle: 180° - 115° = 65°.
• (b) Two Equal Angles: Let each angle be $x$. Then $x + x = 180° \rightarrow 2x = 180° \rightarrow x = 180° ÷ 2 = \mathbf{90°}$ (both are Right Angles).

Interior Angles of Polygons:
• (a) Equilateral Triangle: Sum of 3 angles = 180°. Each angle = 180° ÷ 3 = 60° (Acute).
• (b) Square: Sum of 4 angles = 360°. Each angle = 360° ÷ 4 = 90° (Right angle).
• (c) Regular Hexagon: Each interior angle = 120°. Since 120° > 90°, all angles in a regular hexagon are Obtuse Angles.

Minute Hand Speed: 60 minutes = 360°, so 1 minute = 360° ÷ 60 = 6° per minute.
• (a) 15 Minutes: 15 × 6° = 90° (1/4 turn).
• (b) 30 Minutes: 30 × 6° = 180° (1/2 turn).
• (c) 45 Minutes: 45 × 6° = 270° (3/4 turn).
• (d) 10 Minutes: 10 × 6° = 60° (1/6 turn).

Polygon Properties:
1. Triangle Closure: Three line segments meet at 3 vertices to enclose a 2D space. The sum of the 3 interior angles formed must strictly equal 180° for the triangle to close.
2. Two Sticks: No, it is impossible to create a closed polygon with only two straight sticks. Two line segments meeting at an angle leave an open mouth between their other endpoints; a minimum of 3 sides (a triangle) is mathematically required to enclose an area on a plane.

Degree Clock Geometry:
• (a) Number of Sectors: 360° ÷ 45° = 8 equal sectors.
• (b) Angle between Markings: Each step from one mark to the next is exactly 45°.

Geometric Rigidity of Triangles:
• In a quadrilateral or rectangle, the 4 sides can deform or shift into a parallelogram under heavy load without changing the lengths of the sides because angles can change freely.
• In contrast, a triangle is a rigid shape: once the lengths of the 3 sides are fixed, the 3 angles cannot change unless a side bends or breaks.
• Therefore, triangular structures with strong acute angles distribute heavy loads safely and do not deform under stress.

Working:
1. Let the angle be $A$.
2. Condition: $A + 25° = 90°$ (Right angle).
3. $A = 90° - 25° = \mathbf{65°}$.
4. Since 65° is less than 90°, it is an Acute Angle.

Hour Hand Movement:
• Time elapsed = 5:00 - 2:00 = 3 hours.
• In 12 hours, the hour hand moves 360°, so in 1 hour it moves 360° ÷ 12 = 30°.
• In 3 hours: 3 × 30° = 90° (a quarter turn / Right Angle).

Origami Geometry:
• (a) Shapes Formed: Folding along the diagonal divides the square into two identical right-angled isosceles triangles.
• (b) Angles of the Triangles:
- The original corner of the square remains a 90° right angle.
- The diagonal bisects the two other 90° corners into two equal halves: 90° ÷ 2 = 45° each.
- Therefore, the angles of each triangle are 45°, 45°, and 90°.

Reflex Angles:
• (a) Definition: An angle that is greater than 180° but less than 360° is called a reflex angle.
• (b) Exterior Reflex Angle: Complete circle = 360°. Reflex angle = 360° - 70° = 290°.
• (c) Clock Hands: 240° = 8 hour divisions (8 × 30° = 240°). At 8:00, the angle measured clockwise from the hour hand (at 8) to the minute hand (at 12) is 4 × 30° = 120°, and the major angle on the other side is 360° - 120° = 240°.

Justifications:
• (a) False: 90.5° is greater than 90°, so it is an Obtuse Angle, not acute.
• (b) True: For example, 50° + 60° = 110°. Since 110° > 90°, the sum of two acute angles can indeed be obtuse.
• (c) True: A straight angle formed by two opposite rays is defined as exactly 180°.

Net Angle Calculation:
1. Start direction = East (90° from North).
2. Turn 1 (Clockwise): +45° → Position = South-East.
3. Turn 2 (Anticlockwise): -180° → Position = North-West.
4. Turn 3 (Clockwise): +90° → Position = North-East.
Mathematical Check:
Net turn = +45° - 180° + 90° = -45° (i.e. 45° anticlockwise from East).
45° anticlockwise from East points directly to North-East.

Calculations (Duration = 3 nights, 4 persons):
• (a) Package A Total: 3 nights × ₹2,800 = ₹8,400.
• (b) Package B Total:
- Room rent: 3 × ₹2,200 = ₹6,600.
- Breakfast cost: 4 persons × ₹250 × 3 days = ₹3,000.
- Total = ₹6,600 + ₹3,000 = ₹9,600.
• (c) Comparison: Package A is cheaper. Difference = ₹9,600 - ₹8,400 = ₹1,200. Package A saves ₹1,200.

Expense Calculations (45 Students):
• Bus rental = ₹14,500
• Entry tickets = 45 × ₹90 = ₹4,050
• Packed meals = 45 × ₹150 = ₹6,750
• Guide charges = ₹1,500
• (a) Grand Total: ₹14,500 + ₹4,050 + ₹6,750 + ₹1,500 = ₹26,800.
• (b) Per Student Share: ₹26,800 ÷ 45 = ₹595.55 ≈ ₹596 per student.

Calculations:
• (a) Petrol Needed: Total distance ÷ Mileage = 1,080 ÷ 15 = 72 litres.
• (b) Fuel Cost: 72 litres × ₹96 = ₹6,912.

Working:
• (a) Train Cost: 28 × ₹385 = ₹10,780.
• (b) Comparison:
- Bus = ₹11,200
- Train = ₹10,780
- Difference = ₹11,200 - ₹10,780 = ₹420.
Traveling by train is more economical by ₹420.

Unitary Method Steps:
• (a) Cost of 1 Thali: ₹480 ÷ 4 = ₹120.
• (b) Cost of 9 Thalis: 9 × ₹120 = ₹1,080.
• (c) Thalis for ₹1,200: ₹1,200 ÷ ₹120 = 10 thalis.

Step-by-Step Schedule:
• (a) Distance: First leg = 3 h × 60 km/h = 180 km. Second leg = 140 km. Total distance = 180 + 140 = 320 km.
• (b) Driving Time: First leg = 3 hours. Second leg = 140 ÷ 70 = 2 hours. Total driving time = 3 + 2 = 5 hours.
• (c) Arrival Time:
- Total elapsed time = 5 hours driving + 45 min breakfast = 5 hours 45 minutes.
- 6:00 AM + 5 h 45 min = 11:45 AM.

Percentage Calculation:
• (a) GST Amount: 5% of ₹35,000 = (5 ÷ 100) × 35,000 = 5 × 350 = ₹1,750.
• (b) Final Total: ₹35,000 + ₹1,750 = ₹36,750.

Calculations:
• (a) Rupees Received: 150 × ₹82 = ₹12,300.
• (b) Total Spent:
- 3 handicrafts: 3 × ₹2,400 = ₹7,200.
- 2 shawls: 2 × ₹1,800 = ₹3,600.
- Total spent = ₹7,200 + ₹3,600 = ₹10,800.
• (c) Balance: ₹12,300 - ₹10,800 = ₹1,500.

Calculations:
• (a) Excess Weight: 22 kg - 15 kg = 7 kg excess.
• (b) Excess Fee: 7 kg × ₹350 = ₹2,450.

Mental Calculations:
• (a) 45 × 200: 45 × 2 = 90, append two zeros → 9,000.
• (b) 60 × 700: 6 × 7 = 42, append three zeros → 42,000.
• (c) 1,500 × 40: 15 × 4 = 60, append three zeros → 60,000.

Working:
• (a) Total Cost:
- Adults: 3 × ₹220 = ₹660.
- Children: 2 × ₹140 = ₹280.
- Total = ₹660 + ₹280 = ₹940.
• (b) Change: ₹2,000 - ₹940 = ₹1,060.

Rounding to Nearest 100:
• Petrol: ₹3,240 rounds to ₹3,200.
• Food: ₹1,890 rounds to ₹1,900.
• Toll tax: ₹460 rounds to ₹500.
• Sightseeing tickets: ₹1,120 rounds to ₹1,100.
• Estimated Total: ₹3,200 + ₹1,900 + ₹500 + ₹1,100 = ₹6,700.
(Exact sum = ₹6,710; difference is only ₹10).

Calculations:
• (a) Normal Speed: 120 km ÷ 2 h = 60 km/h.
• (b) Rainy Journey Time: 120 km ÷ 40 km/h = 3 hours.
• (c) Extra Time Lost: 3 hours - 2 hours = 1 hour (60 minutes).

Working:
• (a) Total Rent: 4 days × ₹6,500 = ₹26,000.
• (b) Equal Share (All 4 days): ₹26,000 ÷ 3 = ₹8,666.67 per family.
• (c) Fair Room-Days Division:
- Total family-days = 3 + 4 + 4 = 11 family-days.
- Cost per family-day = ₹26,000 ÷ 11 = ₹2,363.64.
- Family 1 (3 days) pays: 3 × ₹2,363.64 = ₹7,091.
- Family 2 & 3 (4 days each) pay: 4 × ₹2,363.64 = ₹9,454.50 each.

Berth Pattern in 8-berth Compartment:
• 1: Lower, 2: Middle, 3: Upper, 4: Lower, 5: Middle, 6: Upper, 7: Side Lower, 8: Side Upper.
• (a) Berth 25: 25 ÷ 8 gives remainder 1. Remainder 1 corresponds to a Lower Berth.
• (b) Berth 32: 32 ÷ 8 gives remainder 0 (exact multiple of 8). This corresponds to a Side Upper Berth (SU).

Working:
• (a) Maximum Range: 65 L × 14 km/L = 910 km.
• (b) Remaining Range: 12 L × 14 km/L = 168 km.
- Since 168 km < 180 km, No, the SUV cannot reach the station 180 km away without refueling (it will run out of fuel 12 km before reaching).

Working:
• (a) Total Bill: ₹2,850 + ₹150 = ₹3,000.
• (b) Individual Share: ₹3,000 ÷ 6 = ₹500 per person.

Rate Calculations:
• (a) Pedal Boat Rate: ₹180 ÷ 30 min = ₹6 per minute.
• (b) Motor Boat Rate: ₹450 ÷ 20 min = ₹22.50 per minute.
• (c) Difference: ₹22.50 - ₹6.00 = ₹16.50 per minute higher for the motor boat.

Relative Speed Concept:
1. Since the trains are moving toward each other, their combined speed is: 60 km/h + 80 km/h = 140 km/h.
2. Total distance to cover = 420 km.
3. Time taken to meet = Distance ÷ Relative Speed = 420 ÷ 140 = 3 hours.
4. Verification: In 3 hours, Train 1 covers 3 × 60 = 180 km; Train 2 covers 3 × 80 = 240 km. Total = 180 + 240 = 420 km. Verified!

Calculations:
• (a) Surge Amount: 40% of ₹3,500 = (40 ÷ 100) × 3,500 = 40 × 35 = ₹1,400.
• (b) New Price: ₹3,500 + ₹1,400 = ₹4,900.

Calculations:
• (a) Total Earnings: 160 × ₹45 = ₹7,200.
• (b) Target Difference: ₹8,000 - ₹7,200 = ₹800 short of target.

Calculations:
• (a) Total Distance: 345 + 412 + 288 + 355 = 1,400 km.
• (b) Average Daily Distance: Total distance ÷ 4 days = 1,400 ÷ 4 = 350 km per day.

Cash Calculations:
• (a) Cash Tendered: 4 × ₹2,000 = ₹8,000.
• (b) Change: ₹8,000 - ₹7,850 = ₹150.
• (c) Minimum Notes for ₹150: One ₹100 note + One ₹50 note = 2 notes.

Financial Review:
• (a) Total Expenses: ₹18,600 + ₹16,400 + ₹8,500 + ₹4,750 = ₹48,250.
• (b) Budget Comparison: Budget = ₹50,000. Difference = ₹50,000 - ₹48,250 = ₹1,750.
Yes, they stayed within budget, saving ₹1,750.

Scale Conversions (Scale: 1 cm = 250 m):
• (a) Map Distance = 6 cm:
- Actual distance = 6 × 250 m = 1,500 metres.
- In kilometres = 1,500 ÷ 1,000 = 1.5 km.
• (b) Actual Distance = 2,000 m:
- Map distance = 2,000 ÷ 250 = 8 cm on the map.

Calculations:
• (a) Total Map Distance: 3.5 cm + 8.5 cm = 12 cm.
• (b) Real Ground Distance: 12 cm × 100 m = 1,200 metres.
• (c) In Kilometres: 1,200 ÷ 1,000 = 1.2 km.

Calculations:
• (a) Actual Distance: 7.5 cm × 60 km = 450 km.
• (b) Average Speed: Speed = Distance ÷ Time = 450 km ÷ 5 h = 90 km/h.

Compass Directions:
• (a) Opposite to North-West: Directly opposite (180°) is South-East.
• (b) Position: Moving South then East places you in the South-East direction from the starting point.
• (c) Angle East to South: East is 90° and South is 180°, so the angle is 180° - 90° = 90° (Right angle).

Grid Calculations:
• (a) Grid Units: Both are in Row 2. Difference in columns from B to E: C, D, E = 3 grid units.
• (b) Ground Distance: 3 grid units × 50 m = 150 metres.

Perspective Comparison:
1. Top View (Aerial View): Shows objects as viewed from directly overhead. Heights of buildings cannot be seen; only flat boundaries, roofs, streets, and footprints are visible.
2. Side View: Shows vertical height and facade of structures, but objects in front hide objects behind them.
3. Why Maps Use Top View: A top view allows accurate 2D geometric scaling, uninterrupted representation of every road, plot, and landmark without foreground objects blocking background buildings.

Thread Measurement Technique:
1. Take a piece of flexible cotton thread and tie a knot at one end.
2. Place the knot at the starting point of the river on the map.
3. Carefully trace the thread along all bends, curves, and meanders of the river until reaching the endpoint.
4. Mark the endpoint on the thread with a pen or pinch it with fingers.
5. Stretch the thread straight along a calibrated centimetre ruler to measure its length.
6. Multiply the thread length in centimetres by the map scale factor to find the real ground length of the river.

Working:
• (a) Real Distances:
- Route 1: 14 × 15 km = 210 km.
- Route 2: 11 × 15 km = 165 km.
• (b) Difference: 210 km - 165 km = 45 km shorter.

Area Scaling Calculations:
• (a) Real Dimensions:
- Length = 4 cm × 50 m = 200 metres.
- Width = 4 cm × 50 m = 200 metres.
• (b) Ground Area: Side × Side = 200 m × 200 m = 40,000 sq m.

Calculations:
• (a) Dimensions: Length = 6 m, Breadth = 4.5 m.
• (b) Floor Area: 6 m × 4.5 m = 27 sq m.
• (c) Tiling Cost: 27 sq m × ₹120 = ₹3,240.

Scale Analysis:
• Real side of park = 2 cm × 100 m = 200 metres.
• (a) Side on Map 2: 200 m ÷ 50 m = 4 cm.
• (b) Area Comparison on Paper:
- Map 1 area = 2 cm × 2 cm = 4 cm².
- Map 2 area = 4 cm × 4 cm = 16 cm².
- Ratio = 16 ÷ 4 = 4 times larger.

Geographical Directions:
• (a) Kerala from Delhi: Kerala lies directly towards the South.
• (b) Assam from Delhi: Assam lies towards the East (North-East).
• (c) Sea to the West: The Arabian Sea lies to the West of the Indian peninsula.

Working:
• (a) Distance Covered: 15 km/h × 2 h = 30 km.
• (b) Map Distance: 30 km ÷ 5 km/cm = 6 cm on the map.

Geometry & Scale:
• (a) Shape: Opposite sides are equal (6 and 8 units) and meet at right angles → Rectangle.
• (b) Perimeter:
- Total grid units = 6 + 8 + 6 + 8 = 28 units.
- Real perimeter = 28 × 25 m = 700 metres.

Map Legend:
1. Definition: A map legend (or key) is an explanatory box that explains the meaning of various colours, signs, and symbols used on the map.
2. Common Symbols:
• Blue lines/areas: Rivers, canals, or water bodies.
• Black parallel lines with cross-ties: Railway tracks.
• Red cross (+): Hospital or clinic.
• Small flag or building: School.

Angle Calculations:
• (a) North to North-East: 90° ÷ 2 = 45°.
• (b) North-East to East: 90° ÷ 2 = 45°.
• (c) North-East to South-West: They are directly opposite directions along a straight line = 180° (Straight angle).

Step-by-Step Working:
• (a) Actual Length: 2.4 cm × 500 m = 1,200 metres.
• (b) In Kilometres: 1,200 ÷ 1,000 = 1.2 km.
• (c) Crossing Time:
- Speed = 60 km/h = (60 × 1,000) ÷ 3,600 = 16.67 m/s.
- Time = Distance ÷ Speed = 1,200 m ÷ 16.67 m/s = 72 seconds (1 minute 12 seconds).

Grid Street Distance:
• (a) East Units: 5 - 1 = 4 units East.
• (b) North Units: 4 - 1 = 3 units North.
• (c) Total Travel Distance:
- Total grid units = 4 + 3 = 7 units.
- Real distance = 7 × 200 m = 1,400 metres (1.4 km).

Deduction:
1. Actual Ground Distance: From Map X: 8 cm × 20 km = 160 km.
2. Scale of Map Y: On Map Y, 4 cm represents 160 km.
3. Therefore, 1 cm on Map Y represents: 160 km ÷ 4 = 40 km.
Scale of Map Y: 1 cm = 40 km.

Scaling:
• (a) Blueprint (1 cm = 25 m): 150 ÷ 25 = 6 cm.
• (b) City Map (1 cm = 50 m): 150 ÷ 50 = 3 cm.

Scale Concepts:
• (a) Definition: The scale of a map is the constant ratio between the distance measured on the map and the corresponding actual distance measured on the ground.
• (b) Ratio 1:100,000:
- 1 cm on map = 100,000 cm on ground.
- 100,000 cm ÷ 100 = 1,000 metres.
- 1,000 m = 1 kilometre.
- So 1 cm on the map represents exactly 1 km on the ground.

Contour Line Principles:
• (a) Close Contour Lines: Indicate a steep slope or cliff, where the elevation changes rapidly over a very short horizontal ground distance.
• (b) Far Apart Contour Lines: Indicate a gentle slope or flat plain, where elevation changes very gradually over a long distance.

Justifications:
• (a) True: At 1 cm = 10 km, each centimetre covers 10,000 metres of ground, so the same paper sheet covers a vastly larger geographic territory (small-scale map).
• (b) False: East is 90° clockwise from North (180° clockwise is South).
• (c) True: By international cartographic standard convention, the top edge of a map points toward true North (unless explicitly indicated otherwise by a north arrow).

Comprehensive Route Analysis:
• (a) Coordinates:
- Depot: (0, 0)
- Station 1: 4 km North → (0, 4)
- Station 2: 3 km East → (3, 4)
- Station 3: 4 km South → (3, 0).
• (b) Relative Position of Station 3: Station 3 is at (3, 0), which is exactly 3 km East of Central Depot.
• (c) Path Shape: The path goes North (4 km), East (3 km), South (4 km), and West back to depot (3 km). This forms a perfect Rectangle (4 km × 3 km) with perimeter = 4 + 3 + 4 + 3 = 14 km.

Calculations:
• (a) Daily Milk: 12 cows × 8 L/cow = 96 litres per day.
• (b) June Production (30 days): 96 L × 30 = 2,880 litres.
• (c) June Revenue: 2,880 L × ₹52 = ₹1,49,760.

Volume Calculations:
• (a) Packaged Milk:
- 40 packets × 500 mL = 20,000 mL = 20 Litres.
- 30 bottles × 1 L = 30 Litres.
- Total packaged = 20 L + 30 L = 50 Litres.
• (b) Loose Milk Remaining: 65 L - 50 L = 15 Litres.

Calculations:
• (a) Addition: 450 mL + 850 mL = 1,300 mL = 1 L 300 mL. 15 L + 8 L + 1 L = 24 L 300 mL.
• (b) Subtraction: 20 L 000 mL - 14 L 350 mL = 19 L 1,000 mL - 14 L 350 mL = 5 L 650 mL.
• (c) Conversion: 3,750 mL = 3 Litres 750 mL (or 3.75 L).

Finding Common Multiples:
1. Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40...
2. Multiples of 6: 6, 12, 18, 24, 30, 36, 42...
3. Common Multiples: Numbers appearing in both lists are 12, 24, 36.
4. LCM: The smallest common multiple is 12. Thus, LCM(4, 6) = 12.

Factor Trees for 60:
1. Tree A (Starting with 6 × 10):
• 60 = 6 × 10
• 6 splits into 2 × 3
• 10 splits into 2 × 5
• Prime factors: 2, 3, 2, 5.
2. Tree B (Starting with 4 × 15):
• 60 = 4 × 15
• 4 splits into 2 × 2
• 15 splits into 3 × 5
• Prime factors: 2, 2, 3, 5.
3. Prime Factorization: 60 = 2 × 2 × 3 × 5. Both trees yield identical prime factors.

LCM Application:
• (a) Smallest Number (LCM):
- Multiples of 6: 6, 12, 18, 24, 30...
- Multiples of 8: 8, 16, 24, 32...
- LCM(6, 8) = 24 packets.
• (b) Number of Boxes:
- Boxes of 6 = 24 ÷ 6 = 4 boxes.
- Boxes of 8 = 24 ÷ 8 = 3 boxes.

Factors Analysis:
• (a) Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
• (b) Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
• (c) Common Factors: 1, 2, 3, 4, 6, 12.
- Highest Common Factor (HCF): 12.

Working:
• (a) Societies Combined: 2,450 L + 3,180 L = 5,630 litres.
• (b) Processing Plant: 8,000 L - 5,630 L = 2,370 litres.

Proportion Calculations:
• (a) Paneer from 45 L: 45 ÷ 5 = 9 kg of paneer.
• (b) Milk for 8.5 kg Paneer: 8.5 × 5 = 42.5 litres of milk.

Measuring Jug Solutions:
• (a) For 900 mL:
- Take two 200 mL cans and one 500 mL can: 200 + 200 + 500 = 900 mL (3 pours).
• (b) For 1,000 mL (1 Litre):
- Use the 500 mL can twice: 500 + 500 = 1,000 mL, OR
- Using both cans: one 200 mL can and two 500 mL cans minus one 200 mL can, or simpler: five 200 mL cans.
- Best combination using both cans: two 200 mL + two 300 mL (or two 500 mL cans).

Calculations:
• (a) Cost of 1 kg: 1 kg = 1,000 g = two 500 g jars. Cost = 2 × ₹340 = ₹680 per kg.
• (b) 6 Jars Purchase:
- Total weight = 6 × 500 g = 3,000 g = 3 kg.
- Total bill = 6 × ₹340 = ₹2,040.

Schedule Breakdown:
• (a) Unloading Time: 8 booths × 15 min = 120 minutes = 2 hours.
• (b) Driving Time: 7 trips × 10 min = 70 minutes (1 hour 10 minutes).
• (c) Completion Time:
- Total elapsed time = 2 h + 1 h 10 min = 3 hours 10 minutes.
- 4:30 AM + 3 h 10 min = 7:40 AM.

Venn Diagram Factors:
• Factors of 20: 1, 2, 4, 5, 10, 20.
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
• (c) Overlapping Intersection (Common Factors): 1, 2, 5, 10.
- Non-shared in 20 only: 4, 20.
- Non-shared in 30 only: 3, 6, 15, 30.

Feed Calculations:
• (a) One Cow Daily:
- Per feeding = 4 kg + 2.5 kg = 6.5 kg.
- Twice a day = 6.5 kg × 2 = 13 kg per day.
• (b) 12 Cows Daily: 12 × 13 kg = 156 kg of feed.

Density Analysis:
• (a) Comparison: Milk weighs 1,032 g while water weighs 1,000 g. Milk weighs more than water by 32 grams per litre.
• (b) Weight of 5 Litres: 5 × 1,032 g = 5,160 g = 5 kg 160 g.

Prime vs Composite Classification:
• Definition: A prime number is a natural number greater than 1 that has exactly two distinct factors: 1 and itself.
• Classification:
- Prime Numbers: 11, 19, 29, 37 (only divisible by 1 and itself).
- Composite Numbers: 15 (3 × 5), 21 (3 × 7), 33 (3 × 11).

Calculations:
• (a) Total Volume: 12 L + 18 L = 30 Litres.
• (b) Number of Glasses:
- 30 L = 30,000 mL.
- Glasses = 30,000 ÷ 250 = 120 glasses.
• (c) Total Collection: 120 × ₹15 = ₹1,800.

Fraction & Volume Deduction:
1. Fraction added = 4/5 - 3/5 = 1/5 of the container.
2. Given: 1/5 of the total capacity = 14 litres.
3. Total capacity = 14 × 5 = 70 litres.
4. Verification: 3/5 of 70 = 42 L. Adding 14 L gives 42 + 14 = 56 L. 4/5 of 70 = 56 L. Solution verified!

Working:
• (a) Annual Savings: 52 × ₹4,200 = ₹2,18,400.
• (b) Remaining Savings: ₹2,18,400 - ₹1,50,000 = ₹68,400.

Co-prime Explanation:
• (a) Definition: Two numbers are co-prime if they have no common factor other than 1 (i.e. HCF = 1).
• (b) Analysis of 8 and 15:
- Factors of 8: 1, 2, 4, 8.
- Factors of 15: 1, 3, 5, 15.
- Common factor: Only 1.
- Conclusion: Yes, 8 and 15 are co-prime numbers.

Calculations:
• (a) Milk per Tub: 150 L = 150,000 mL. Milk per tub = 150,000 ÷ 600 = 250 mL per tub.
• (b) 50% Increase: 150 L + (50% of 150 L) = 150 + 75 = 225 litres daily.

Shift Decimal 3 Places Left (÷ 1,000):
• (a) 4,500 mL: 4,500 ÷ 1,000 = 4.5 L.
• (b) 850 mL: 850 ÷ 1,000 = 0.85 L.
• (c) 25 mL: 25 ÷ 1,000 = 0.025 L.
• (d) 10,250 mL: 10,250 ÷ 1,000 = 10.25 L.

Justifications:
• (a) True: Since 6 = 2 × 3, any multiple of 6 contains both 2 and 3 as factors, so it is divisible by both 2 and 3.
• (b) False: 1 has only one factor (itself), whereas prime numbers require exactly two distinct factors. The smallest prime number is 2.
• (c) False: 50 mL = 50 ÷ 1,000 = 0.05 L. Therefore 2 L 50 mL = 2.05 Litres (2.5 L is 2 L 500 mL).

Stock Balance:
• (a) Closing Stock:
- Opening stock = 4,200 L.
- Total available = 4,200 + 8,400 = 12,600 L.
- Dispatched = 7,850 L.
- Closing stock = 12,600 - 7,850 = 4,750 litres.
• (b) Unused Capacity: 15,000 L - 4,750 L = 10,250 litres unused.

Rotational Symmetry Properties:
• (a) 1/2 Turn: 360° ÷ 2 = 180°.
• (b) 1/4 Turn: 360° ÷ 4 = 90°.
• (c) Shapes that look identical after 1/2 turn (180°):
- Rectangle: Yes (180° rotation matches perfectly).
- Equilateral Triangle: No (apex points downward after 1/2 turn).
- Letter 'S': Yes (180° rotation gives 'S').
- Letter 'M': No (turns upside down into 'W').
Identical shapes: Rectangle and Letter 'S'.

Pattern Progression:
• Line 4: 1234 × 9 + 5 = 11,111.
• Line 5: 12345 × 9 + 6 = 1,11,111.
• Rule: In each successive line, the multiplicand adds the next natural digit on the right (1, 12, 123, 1234, 12345), the added number increases by 1 (2, 3, 4, 5, 6), and the resulting product consists entirely of the digit 1 repeated one more time than in the previous line.

Pattern Analysis:
• (a) Next Two Lines:
- Line 4: 4 + 5 + 6 = 15.
- Line 5: 5 + 6 + 7 = 18.
• (b) Mathematical Relationship:
- For 1, 2, 3: Middle number is 2; Sum = 6 = 3 × 2.
- For 2, 3, 4: Middle number is 3; Sum = 9 = 3 × 3.
- For 3, 4, 5: Middle number is 4; Sum = 12 = 3 × 4.
Rule: The sum of three consecutive numbers is always exactly 3 times the middle number ($3 \times M$).

Magic Square of 1 to 9 (Sum = 15):

  8   1   6  → 15
  3   5   7  → 15
  4   9   2  → 15
  ↓   ↓   ↓
 15  15  15
Diagonals: 8+5+2 = 15; 6+5+4 = 15.

• Center Number: The number 5 must be placed in the center.
• Reason: The center cell is part of 4 lines (1 row, 1 column, 2 diagonals). The average of numbers 1 to 9 is (1 + 9) ÷ 2 = 5, which is the median value required to balance all pairs adding to 10 (1+9, 2+8, 3+7, 4+6).

Rule: Square = Circle 1 × Circle 2:
• (a) Circles with 7 and 9: Square = 7 × 9 = 63.
• (b) Square is 72, Circle is 8: Other circle = 72 ÷ 8 = 9.

Square Numbers:
• (a) Next Three Numbers: 6² = 36, 7² = 49, 8² = 64.
• (b) Sum of Odd Numbers:
- 1 = 1
- 4 = 1 + 3
- 9 = 1 + 3 + 5
- 16 = 1 + 3 + 5 + 7
- 25 = 1 + 3 + 5 + 7 + 9
- 36 = 1 + 3 + 5 + 7 + 9 + 11.

Palindrome Calculations:
• (a) Process for 48:
- Step 1: Number = 48. Reverse = 84.
- Step 2: Sum = 48 + 84 = 132 (not yet a palindrome).
- Step 3: Reverse of 132 = 231.
- Step 4: Sum = 132 + 231 = 363 (which reads the same forwards and backwards → Palindrome!).
• (b) 1,001: Yes, 1,001 is a palindrome. The next 4-digit palindrome is 1,111.

Coding & Decoding (+1 Shift):
• (a) Decoding 'CPPL': Shift backward by 1 letter (-1):
- C - 1 = B
- P - 1 = O
- P - 1 = O
- L - 1 = K
Decoded Word: BOOK.
• (b) Encoding 'CLEVER': Shift forward by 1 letter (+1):
- C → D, L → M, E → F, V → W, E → F, R → S.
Encoded Message: DMFWFS.

Tessellation Geometry:
• (a) Definition: Tessellation is the covering of a plane using one or more geometric shapes with no gaps and no overlaps.
• (b) Regular Hexagons: Yes. Each interior angle of a regular hexagon is 120°. At each meeting vertex, 3 hexagons meet: 120° × 3 = 360°. Since their angles sum to exactly 360°, they tile seamlessly (honeycomb pattern).
• (c) Regular Octagons: No, octagons alone cannot tile a floor. Each interior angle is 135°. Two octagons give 135° + 135° = 270°, leaving a gap of 360° - 270° = 90° (which requires a square to fill the gap).

Rotational Angles:
• (a) 1/3 Turn: 360° ÷ 3 = 120°.
• (b) 1/6 Turn: 360° ÷ 6 = 60°.
• (c) Identical after 1/3 turn: An Equilateral Triangle (has 3-fold rotational symmetry).
• (d) Identical after 1/6 turn: A Regular Hexagon (or 6-pointed star, has 6-fold symmetry).

Triangular Numbers:
• Differences are consecutive integers: +2, +3, +4, +5...
• (a) Next Two Numbers:
- 15 + 6 = 21.
- 21 + 7 = 28.
• (b) Sum of Consecutive Triangular Numbers:
- 1 + 3 = 4 = 2² (Square number)
- 3 + 6 = 9 = 3² (Square number)
- 6 + 10 = 16 = 4² (Square number)
- 10 + 15 = 25 = 5² (Square number)
Rule: The sum of any two consecutive triangular numbers is always a Square Number!

Magic Triangle Solution (Side Sum = 9):
1. Place the smallest numbers 1, 2, 3 at the three corner vertices.
2. Corner sum = 1 + 2 + 3 = 6.
3. Sides:
• Side between 1 and 2: needs 9 - (1 + 2) = 6 in the middle.
• Side between 2 and 3: needs 9 - (2 + 3) = 4 in the middle.
• Side between 3 and 1: needs 9 - (3 + 1) = 5 in the middle.
Check: 1+6+2 = 9; 2+4+3 = 9; 3+5+1 = 9. All numbers 1 to 6 used exactly once!

Modulo Arithmetic in Repeating Motifs (Period = 4 shapes):
• Pattern sequence: 1 = UP, 2 = Square, 3 = DOWN, 4 = Circle.
• (a) 15th Shape: 15 ÷ 4 = 3 with remainder 3. Position 3 is Triangle pointing DOWN.
• (b) 28th Shape: 28 ÷ 4 = 7 with remainder 0 (exact multiple of 4). Position 4 is Circle.

Symmetric Number Products (Repunits):
• (a) Product: 11,111 × 11,111 = 123454321 (One crore twenty-three lakh forty-five thousand four hundred twenty-one).
• (b) Symmetry Explanation: The digits count up from 1 to the number of ones in the factor (5 digits: 1, 2, 3, 4, 5) and then mirror backwards symmetrically down to 1 (4, 3, 2, 1), forming a numerical palindrome.

Order of Rotational Symmetry:
• (a) Square: Rotates onto itself 4 times in a 360° turn (every 90°) → Order 4.
• (b) Regular Pentagon: Rotates onto itself 5 times (every 72°) → Order 5.
• (c) Regular Octagon: Rotates onto itself 8 times (every 45°) → Order 8.
• (d) Circle: Any rotation around its center leaves it unchanged → Infinite order.

Fibonacci Sequence:
• (a) Next Three Numbers:
- 8 + 13 = 21.
- 13 + 21 = 34.
- 21 + 34 = 55.
• (b) Occurrences in Nature: Number of petals on flowers (daisies, sunflowers), spiral arrangement of sunflower seeds, pine cone scales, and pineapple skins.

Letter Symmetry Analysis:
• (a) Both Vertical & Horizontal Lines of Symmetry: H, O, X (can be reflected across both axes).
• (b) Rotational Symmetry but No Line Symmetry: Z.
- Letter 'Z' has no axis of reflection (horizontal or vertical reflection reverses the diagonal), but rotating by 180° (half-turn) reproduces an identical 'Z'.

Calculations:
1. Total sum of numbers from 6 to 14 = (6 + 14) × 9 ÷ 2 = 20 × 4.5 = 90.
2. Row Sum (Magic Constant): 90 ÷ 3 rows = 30.
3. Center Number: Median value = (6 + 14) ÷ 2 = 10.
Check: 3 × Center number = 3 × 10 = 30 = Magic constant.

Pattern Analysis:
• Difference sequence: 6 - 2 = 4, 12 - 6 = 6, 20 - 12 = 8, 30 - 20 = 10.
• The difference increases by +2 each time (consecutive even numbers: 4, 6, 8, 10, 12, 14).
• Next difference must be +12: 30 + 12 = 42.
• Verification: 42 + 14 = 56. Correct!
Alternative Rule: $n \times (n + 1)$: 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42, 7×8=56.

Rotational Geometry of 4-Blade Windmill:
• (a) Fraction of a Turn: 4 blades = 1/4 turn.
• (b) Degrees: 360° ÷ 4 = 90°.
• (c) Symmetry: Yes, it has 4-fold rotational symmetry and looks identical after a 1/4 turn (90°), 1/2 turn (180°), and 3/4 turn (270°).

Geometric Properties:
• (a) Number of Triangles: Exactly 6 equilateral triangles meet at a central vertex to form a regular hexagon.
• (b) Sum of Angles: Each equilateral triangle has an angle of 60°. 6 × 60° = 360° (a complete circle).

Calendar Math Trick:
• (a) Rule: Because calendar numbers are symmetrically distributed horizontally (±1) and vertically (±7), the sum of all 9 numbers is always exactly 9 times the center number ($9 \times C$).
• (b) Calculation for Center = 14: Sum = 9 × 14 = 126.

Justifications:
• (a) False: In a 1/2 turn (180°), an equilateral triangle flips upside down with its base at the top and apex at the bottom.
• (b) True: By the divisibility rule for 9, the digital sum of any multiple of 9 is always 9, 18, 27... (a multiple of 9).
• (c) False: A circle has infinite lines of symmetry (any diameter passing through the center is a line of symmetry).

Fibonacci Square Spirals:
• (a) Next Sides: Sum of previous two sides: 5 + 8 = 13 cm, then 8 + 13 = 21 cm.
• (b) Series & Areas:
- Name: Fibonacci Sequence.
- Area of 1 cm square: 1 × 1 = 1 cm².
- Area of 2 cm square: 2 × 2 = 4 cm².
- Area of 3 cm square: 3 × 3 = 9 cm².
- Area of 5 cm square: 5 × 5 = 25 cm².

Weight Balancing (1 kg = 1,000 g):
• (a) For 1 kg (1,000 g): 1,000 ÷ 200 = 5 weights of 200 g.
• (b) For 2 kg (2,000 g): 2,000 ÷ 250 = 8 weights of 250 g.
• (c) Weighing 800 g: Combine one 500 g weight, one 200 g weight, and one 100 g weight: 500 + 200 + 100 = 800 g (3 weights total).

Cost Calculations:
• (a) Individual Costs:
- Onions: 2.5 × ₹36 = ₹90.
- Tomatoes: 1.5 × ₹40 = ₹60.
- Ginger: 750 g = 0.75 kg. Cost = 0.75 × ₹120 = ₹90.
• (b) Total Bill: ₹90 + ₹60 + ₹90 = ₹240.
• (c) Change: ₹500 - ₹240 = ₹260.

Pan Balance Subtraction Principle:
1. Place the 5 kg weight on the left pan.
2. Place the 2 kg weight on the right pan.
3. Pour rice into the right pan (alongside the 2 kg weight) until both pans balance perfectly.
4. Since Left Pan = Right Pan:
5 kg = 2 kg + Rice Weight → Rice Weight = 5 kg - 2 kg = 3 kg of rice.

Conversions (1 kg = 1,000 g):
• (a) 4 kg 350 g: (4 × 1,000) + 350 = 4,350 g.
• (b) 8,075 g: 8,000 g + 75 g = 8 kg 75 g.
• (c) 3/4 kg: 3/4 of 1,000 = 1,000 × 3 ÷ 4 = 750 g.
• (d) 2.25 kg: 2.25 × 1,000 = 2,250 g.

Water Consumption:
• (a) Total Used: 350 L + 180 L + 420 L = 950 litres.
• (b) Remaining Water: 1,200 L - 950 L = 250 litres.

Working:
• (a) Total Kilograms: 6 quintals × 100 kg/quintal = 600 kg.
• (b) Number of Sacks: 600 kg ÷ 25 kg/sack = 24 sacks.

Dosage Calculations:
• (a) Daily Dose: 3 times × 5 mL = 15 mL per day.
• (b) Total for 5 Days: 5 days × 15 mL = 75 mL.
• (c) Bottle Sufficiency: Yes, 75 mL < 100 mL, so a 100 mL bottle is sufficient.
- Remaining syrup = 100 mL - 75 mL = 25 mL remaining.

Calculations:
• (a) Daily Rice: 140 × 100 g = 14,000 g = 14 kg of rice.
• (b) Daily Dal: 140 × 30 g = 4,200 g = 4.2 kg (4 kg 200 g) of dal.
• (c) Monthly Rice (22 days): 22 × 14 kg = 308 kg of rice.

Working:
• (a) Weight of 1,000 Coins: 1,000 × 9 g = 9,000 g = 9 kg.
• (b) Number of Coins:
- Total weight = 4 kg 500 g = 4,500 grams.
- Number of coins = 4,500 ÷ 9 g = 500 coins.

Capacity Calculations (15 L = 15,000 mL):
• (a) Mugs to Fill: 15,000 mL ÷ 300 mL = 50 mugs.
• (b) Remaining Volume:
- Water removed = 12 × 300 mL = 3,600 mL = 3.6 L.
- Water left = 15 L - 3.6 L = 11.4 litres (11 L 400 mL).

Converting all to Grams:
• 2.8 kg = 2,800 g
• 2,750 g = 2,750 g
• 3 kg 50 g = 3,050 g
• 2,890 g = 2,890 g
• 3,000 g = 3,000 g
Ordering from largest to smallest: 3,050 g > 3,000 g > 2,890 g > 2,800 g > 2,750 g.
Descending Order: 3 kg 50 g > 3,000 g > 2,890 g > 2.8 kg > 2,750 g.

Calculations:
• (a) Doses per Vial: 50 mL ÷ 0.5 mL = 100 doses (100 patients).
• (b) Vials Needed: 400 children ÷ 100 doses/vial = 4 vials.

Comparison:
• (a) Children Equivalent: 240 kg ÷ 30 kg = 8 children.
• (b) Adult vs Baby Elephant: 3,600 kg ÷ 240 kg = 15 baby elephants.

Kitchen Measurements:
• (a) Teaspoons in a Tablespoon: 15 mL ÷ 5 mL = 3 teaspoons.
• (b) 60 mL Olive Oil: 60 ÷ 15 = 4 tablespoons.
• (c) 25 mL Vanilla: 25 ÷ 5 = 5 teaspoons.

Postal Step Tariff:
• (a) 180 g Parcel:
- First 50 g: ₹20.
- Remaining weight = 180 - 50 = 130 g.
- Additional 50 g slabs = 130 ÷ 50 = 2.6 → 3 slabs.
- Surcharge = 3 × ₹10 = ₹30.
- Total postage = ₹20 + ₹30 = ₹50.
• (b) 320 g Parcel:
- First 50 g: ₹20.
- Remaining = 320 - 50 = 270 g.
- Additional slabs = 270 ÷ 50 = 5.4 → 6 slabs.
- Surcharge = 6 × ₹10 = ₹60.
- Total postage = ₹20 + ₹60 = ₹80.

Water Wastage Calculations:
• (a) In 1 Hour (60 min): 20 mL × 60 = 1,200 mL (1.2 Litres).
• (b) In 1 Day (24 hours): 1.2 L × 24 = 28.8 litres wasted daily.
• (c) In 30 Days: 28.8 L × 30 = 864 litres.
- Financial loss = (864 ÷ 1,000) × ₹5 = ₹4.32.

Density Analysis:
• (a) Heavier Liquid: For 500 mL, honey weighs 710 g while water weighs 500 g. Therefore, honey is denser and heavier.
• (b) Difference per Litre (1,000 mL):
- Weight of 1 L water = 2 × 500 g = 1,000 g.
- Weight of 1 L honey = 2 × 710 g = 1,420 g.
- Difference = 1,420 g - 1,000 g = 420 grams heavier per litre.

Working:
• (a) Atta Produced:
- Loss = 5% of 1,500 kg = 75 kg.
- Net atta = 1,500 kg - 75 kg = 1,425 kg.
• (b) Bags Packed: 1,425 kg ÷ 10 kg/bag = 142 full bags (with 5 kg remaining).

Algebraic Derivation:
1. Let the weight of 1 whole brick be $W$ kg.
2. Given balance condition: $W = \frac{3}{4}W + \frac{3}{4}\text{ kg}$.
3. Subtract $\frac{3}{4}W$ from both sides:
$W - \frac{3}{4}W = \frac{3}{4}\text{ kg} \rightarrow \frac{1}{4}W = \frac{3}{4}\text{ kg}$.
4. Multiply both sides by 4: $W = 4 \times \frac{3}{4} = \mathbf{3\text{ kg}}$.
5. Verification: 3/4 of 3 kg = 2.25 kg. 2.25 kg + 0.75 kg = 3.0 kg. Balanced!

Calculations (25 L = 25,000 mL):
• (a) Total Cups: 25,000 mL ÷ 200 mL = 125 cups.
• (b) Consumption Analysis: Since 1 cooler holds only 125 cups, serving 160 cups requires more than 1 cooler (32 litres total). For 125 cups from 1 cooler, all 25 L are used. If 160 cups were served from a larger reservoir, 160 × 200 mL = 32 L were used.

Calculations:
• (a) Weight of 500 L: 500 L × 0.8 kg/L = 400 kg.
• (b) Litres for 4,000 kg: 4,000 kg ÷ 0.8 kg/L = 5,000 litres.

Mental Calculations:
• (a) Half of 1.5 L: 1,500 mL ÷ 2 = 750 mL.
• (b) Double of 750 mL: 750 × 2 = 1,500 mL = 1.5 Litres.
• (c) 1/4 of 1 L: 1,000 mL ÷ 4 = 250 mL.

Justifications:
• (a) True: 1.45 kg = 1,450 g. Since 1,500 g > 1,450 g, it is heavier.
• (b) True: By scientific and metric definition, 1 mL is exactly the volume of a cube with 1 cm sides (1 cm × 1 cm × 1 cm = 1 cm³).
• (c) False: 1/4 Litre = 1,000 ÷ 4 = 250 mL. Since 200 mL < 250 mL, it holds less than 1/4 litre.

Relief Camp Provision Distribution:
• (a) Food Grains Weight:
- Rice = 1.2 × 1,000 kg = 1,200 kg.
- Total grains = 1,200 kg + 450 kg = 1,650 kg.
• (b) Total Oil: 200 cans × 5 L = 1,000 Litres.
• (c) Per Family Share (500 Families):
- Rice = 1,200 kg ÷ 500 = 2.4 kg of rice per family.
- Oil = 1,000 L ÷ 500 = 2 Litres of oil per family.

Multiplication Working (24 × 18):
• (a) Box Method: Split 24 into 20 + 4, and 18 into 10 + 8:
- 20 × 10 = 200
- 4 × 10 = 40
- 20 × 8 = 160
- 4 × 8 = 32
- Sum = 200 + 40 + 160 + 32 = 432 trees.
• (b) Column Method: 24 × 8 = 192; 24 × 10 = 240; 192 + 240 = 432 trees. Both methods match!

Working:
• (a) Sacks Packed: 2,160 ÷ 45 = 48 sacks.
• (b) Total Earnings: 48 sacks × ₹1,125 = ₹54,000.

Profit & Revenue Calculations:
• (a) Profit per Coconut: ₹45 - ₹28 = ₹17 profit per coconut.
• (b) Sunday Sales (65 Coconuts):
- Total revenue = 65 × ₹45 = ₹2,925.
- Cost price = 65 × ₹28 = ₹1,820.
- Net profit = 65 × ₹17 = ₹1,105.

Division with and without Remainder:
• (a) 576 Saplings in 18 Rows: 576 ÷ 18 = 32 saplings per row.
• (b) 600 Saplings in 18 Rows:
- 600 ÷ 18 gives quotient 33 with remainder 6 (18 × 33 = 594; 600 - 594 = 6).
- 33 saplings per row with 6 saplings left over.

Proportion Calculations:
• (a) Oil Extracted: 1,800 ÷ 15 = 120 litres of oil.
• (b) Market Value: 120 L × ₹240 = ₹28,800.

Mental Arithmetic Strategies:
• Method 1 (Distributive): 48 × (20 + 5) = (48 × 20) + (48 × 5) = 960 + 240 = 1,200.
• Method 2 (Quarter of 100): 25 = 100 ÷ 4.
48 × 25 = (48 × 100) ÷ 4 = 4,800 ÷ 4 = 1,200.

Wage Calculations:
• (a) Daily Total for 8 Workers: 8 × ₹450 = ₹3,600 per day.
• (b) 6-Day Wage Bill: 6 × ₹3,600 = ₹21,600.

Payload Calculations:
• (a) Maximum Bags: 1,500 kg ÷ 30 kg/bag = 50 bags.
• (b) Total Coconuts: 50 bags × 50 coconuts/bag = 2,500 coconuts.

Water Delivery Calculations:
• (a) Water per Tree: 4 hours × 2.5 L/h = 10 litres per tree.
• (b) Total Daily Water for 160 Trees: 160 × 10 L = 1,600 litres.

Division & Proof:
1. 1,485 ÷ 12:
• 12 × 100 = 1,200 (Remainder 285)
• 12 × 20 = 240 (Remainder 45)
• 12 × 3 = 36 (Remainder 9)
Quotient = 123, Remainder = 9.
2. Verification:
(12 × 123) + 9 = 1,476 + 9 = 1,485. The division is verified!

Working:
• (a) Coir Fibre: (1,500 ÷ 100) × 8 kg = 15 × 8 = 120 kg of coir fibre.
• (b) Rope Length: 120 kg × 15 m/kg = 1,800 metres of rope.

Grid Spacing:
• (a) Trees per Row: Number of 7-metre intervals = 84 ÷ 7 = 12 intervals.
- Trees per row = Intervals + 1 (posts and spaces rule) = 12 + 1 = 13 trees.
• (b) Total Trees in Square Orchard: 13 rows × 13 trees/row = 169 trees.

Calculations:
• (a) Bearing Period: 35 years.
• (b) Lifetime Yield: 35 years × 140 coconuts/year = 4,900 coconuts.

Financial Calculations:
• (a) Gross Revenue: 4,500 × ₹22 = ₹99,000.
• (b) Total Expenses:
- Climber charges: 4,500 × ₹3 = ₹13,500.
- Cartage = ₹1,500.
- Market cess = ₹1,200.
- Total expenses = ₹13,500 + ₹1,500 + ₹1,200 = ₹16,200.
• (c) Net Profit: ₹99,000 - ₹16,200 = ₹82,800.

Shortcuts:
• (a) 63 × 11: Sum of digits = 6 + 3 = 9. Place between: 693.
• (b) 45 × 12: (45 × 10) + (45 × 2) = 450 + 90 = 540.
• (c) 250 × 16: 250 × 4 × 4 = 1,000 × 4 = 4,000.

Working:
• (a) Days Required: 150 ÷ 6 = 25 working days.
• (b) Total Earnings: 150 × ₹85 = ₹12,750.

Long Division Steps:
• Divide thousands: 4 ÷ 4 = 1 (Remainder 0).
• Bring down 2: 2 cannot be divided by 4 (2 < 4). We must write 0 in the quotient.
• Bring down 0 to make 20: 20 ÷ 4 = 5 (Remainder 0).
• Bring down 8: 8 ÷ 4 = 2 (Remainder 0).
Quotient = 1,052, Remainder = 0.
Explanation: Whenever the brought-down digit is smaller than the divisor, a 0 is placed in that place value of the quotient before bringing down the next digit.

Calculations:
• (a) Jaggery Yield: 120 L ÷ 8 L/kg = 15 kg of jaggery.
• (b) Sales Value: 15 kg × ₹180 = ₹2,700.

LCM and Remainder Principle:
1. The number leaves a remainder of 2 when divided by both 4 and 5.
2. LCM of 4 and 5 = 4 × 5 = 20.
3. The number must be of the form: $\text{LCM}(4, 5) \times k + 2 = 20k + 2$.
4. For $k = 1$: Number = 20 + 2 = 22 coconuts.
5. Verification:
- 22 ÷ 4 = 5 with remainder 2.
- 22 ÷ 5 = 4 with remainder 2.
The smallest possible number of coconuts is 22.

Calculations:
• (a) Total Pumping Hours: 48,000 ÷ 1,600 = 30 hours.
• (b) Days Required: 30 hours ÷ 6 hours/day = 5 days.