Chapter 8: Weight and Capacity

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.