7.7 Unit Conversions

We have already seen (the bus and the honey) that a fair comparison needs the same units. Here is a reference table.

Handy Conversions

Length   1 \text{ metre} = 3.281 \text{ feet}

Area   1 \text{ m}^2 = 10.764 \text{ ft}^2,   1 \text{ acre} = 43{,}560 \text{ ft}^2,   1 \text{ hectare} = 10{,}000 \text{ m}^2 = 2.471 \text{ acres}

Volume   1 \text{ mL} = 1 \text{ cc},   1 \text{ litre} = 1000 \text{ mL}

Temperature   0^\circ\text{C} = 32^\circ\text{F}, and F = \tfrac{9}{5}\,C + 32, \qquad C = \tfrac{5}{9}\,(F - 32). (For example, 35^\circ\text{C} = \tfrac{9}{5}\times 35 + 32 = 95^\circ\text{F}.)

Figure it Out: Conversions & Mixed Problems

Practice

  1. Farida mixes 750 mL of mango pulp with 1000 mL of guava pulp. Write the ratio in simplest form.
  2. Last year 4 full buses carried 220 students and teachers. This year there are 285 students. How many buses are needed, and will they all be full?
  3. Delhi has area 1484 sq. km and population \approx 34 million; Bengaluru has area 741 sq. km and population \approx 14 million. Which city is more crowded?
  4. A flamingo of height 120 cm has neck-to-rest-of-body ratio 3 : 7. If your body followed the same ratio, how long would your neck be?
  5. From the Līlāvatī: if 3 palas of saffron cost 48 niskas, what quantity can be bought for 10 niskas?
  6. Today a child is 2 and her brother is 8. At what age of the child will their ages be in the ratio 1 : 2?
  7. Equal volumes of silver and water have masses in the ratio 21 : 2. If 1 litre of water is 1 kg, what is the mass of 1 litre of silver?
  8. Good practice is 8 tonnes of compost per acre. A plot is 300 ft by 600 ft. How much compost is needed?
  9. A tap fills a 400 mL mug in 12 s. How long to fill an 8-litre bucket?
  10. One acre of land costs ₹18{,}00{,}000. What is the cost of 3000 sq. ft of the same land?
  11. A tractor ploughs 5 times faster than a pair of oxen. The field is 16 acres, and oxen take 5 hours per acre. How long with oxen, and how long with the tractor?
  12. The ₹5 coin is cupro-nickel, copper and nickel in the ratio 3 : 1. The coin’s mass is 6 g. If copper costs ₹720/kg and nickel ₹1400/kg, what is the cost of the metals in one coin?
  1. 750 : 1000; HCF = 250, so 750 : 1000 = \mathbf{3 : 4}.
  2. Each bus held 220 \div 4 = 55 people. For 285 students, 285 \div 55 \approx 5.18, so 6 buses are needed, and they will not all be full (5 buses hold 275, leaving 10 in the sixth bus).
  3. Compare people per sq. km. Delhi: 34{,}000{,}000 \div 1484 \approx 22{,}911. Bengaluru: 14{,}000{,}000 \div 741 \approx 18{,}893. Delhi is more crowded (more people per sq. km).
  4. Parts = 3 + 7 = 10; each part = 120 \div 10 = 12 cm. Neck = 3 \times 12 = \mathbf{36} cm.
  5. Set up 3 : 48 :: x : 10. Cross-multiplying, 48x = 3 \times 10 = 30, so x = \dfrac{30}{48} = \dfrac{5}{8} = \mathbf{0.625\text{ palas}}.
  6. After y years the ratio is (2+y) : (8+y). We need \dfrac{2+y}{8+y} = \dfrac{1}{2}, so 2(2+y) = 8+y, giving 4 + 2y = 8 + y, so y = 4. The child will be 2 + 4 = \mathbf{6} years old (brother 12, ratio 6 : 12 = 1 : 2).
  7. 1 litre of water = 1 kg, and silver : water = 21 : 2, so 1 litre of silver = \dfrac{21}{2} \times 1 = \mathbf{10.5\text{ kg}}.
  8. Plot area = 300 \times 600 = 180{,}000 ft². In acres, 180{,}000 \div 43{,}560 \approx 4.13 acres. Compost = 8 \times 4.13 \approx \mathbf{33\text{ tonnes}}.
  9. 400 mL takes 12 s, so 8{,}000 mL takes \dfrac{12}{400} \times 8{,}000 = \mathbf{240\text{ s}} (i.e. 4 minutes).
  10. 1 acre = 43{,}560 ft² costs ₹18{,}00{,}000. So 3000 ft² costs \dfrac{3000}{43{,}560} \times 1{,}800{,}000 \approx \mathbf{₹1{,}23{,}967}.
  11. Oxen take 5 hours per acre \times 16 acres = \mathbf{80\text{ hours}}. The tractor is 5 times faster, so it takes 80 \div 5 = \mathbf{16\text{ hours}}.
  12. Parts = 3 + 1 = 4; each part = 6 \div 4 = 1.5 g. Copper = 3 \times 1.5 = 4.5 g; nickel = 1.5 g. Cost of copper = \dfrac{4.5}{1000} \times 720 \approx ₹3.24; cost of nickel = \dfrac{1.5}{1000} \times 1400 \approx ₹2.10. Total metal cost \approx \mathbf{₹5.34}.