Figure It Out: Unit Conversions

1. Orange : Apple Juice

• $600 : 900 \rightarrow 6 : 9 \rightarrow 2 : 3$.

2. Bus Trip

• Ratio: 162 people : 3 buses $\rightarrow$ 54 people/bus.
• New people: 204.
• Buses needed: $204 \div 54 \approx 3.77$.
• You need 4 buses. They will not be full.

3. Population Density

• Delhi: $30,000,000 / 1484 \approx 20,215$ people/sq km.
• Mumbai: $20,000,000 / 550 \approx 36,363$ people/sq km.
• Mumbai is more crowded (higher ratio).

4. Crane Neck

• Ratio 4:6 $\rightarrow$ 2:3. (Neck is 2/5 of total height).
• If your height is 150 cm: Neck = $\frac{2}{5} \times 150 = 60$ cm. (That’s a very long neck! Cranes are different).

5. Lilavati Problem

• $2.5$ palas = $3/7$ niskas.
• Cost of 1 pala = $(3/7) \div 2.5 = (3/7) \times (2/5) = 6/35$ niskas.
• How much for 9 niskas?
  • Quantity = Total Money / Cost per unit.
  • $9 \div (6/35) = 9 \times (35/6) = 1.5 \times 35 = 52.5$ palas.

6. Harmain’s Age

• Current: Harmain 1, Brother 5. (Gap 4 years).
• Target Ratio 1:2.
• Let Harmain’s age be $x$. Brother is $x+4$.
• $x : x+4 :: 1 : 2 \implies \frac{x}{x+4} = \frac{1}{2}$.
• $2x = x + 4 \implies x = 4$.
• Answer: Harmain will be 4 years old.

9. Tap Rate

• 500 mL in 15 sec.
• 10 Litres = 10,000 mL.
• Ratio: $500 : 15 :: 10000 : x$.
• Factor: $10000 / 500 = 20$.
• Time: $15 \times 20 = 300$ seconds = 5 minutes.

12. Coin Cost

• Mass 7.74g. Ratio 3:1 (Cu:Ni).
• Cu mass = $0.75 \times 7.74 = 5.805$ g.
• Ni mass = $0.25 \times 7.74 = 1.935$ g.
• Prices (per gram): Cu = Rs 0.906, Ni = Rs 1.341.
• Cost Cu: $5.805 \times 0.906 \approx \text{Rs }5.26$.
• Cost Ni: $1.935 \times 1.341 \approx \text{Rs }2.59$.
• Total Cost: $\approx \text{Rs }7.85$.