NCERT Solutions for Class 8 Maths Ganita Prakash Chapter 7 — Figure it Out (Pages 176–177)#
Unit conversions used here: 1 acre = 43,560 sq ft, 1 litre = 1000 mL, 1 tonne = 1000 kg, 1 kg = 1000 g.
1. Anagh mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form. #
Solution
The HCF of 600 and 900 is 300:
Orange : apple $= 600 : 900$
Orange : apple $= 2 : 3$
∴ The ratio is 2 : 3.
2. Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full? #
Solution
All 3 buses were full with 162 people, so each bus holds:
Seats in one bus $= \dfrac{162}{3}$
Seats in one bus $= 54$
For 204 people:
Buses needed $= \dfrac{204}{54}$
Buses needed $= 3.78$ (about)
Part of a bus cannot be hired, so we need 4 buses.
Seats in 4 buses $= 4 \times 54 = 216$
Empty seats $= 216 - 204 = 12$
∴ We need 4 buses, and they will not all be full: 12 seats will be empty.
3. The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so? #
Solution
To compare, we find the number of people per square kilometre in each city:
Delhi $= \dfrac{30000000}{1484}$
Delhi $= 20216$ people per sq. km (about)
Mumbai $= \dfrac{20000000}{550}$
Mumbai $= 36364$ people per sq. km (about)
∴ Mumbai is more crowded. Delhi has more people in all, but Mumbai has far more people living on each square kilometre.
4. A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be? #
Solution
The neck is 4 parts out of $4 + 6 = 10$:
Neck $= \dfrac{4}{10} \times$ height
For the crane: $\dfrac{4}{10} \times 155 = 62$ cm.
For you, take your own height. For example, if you are 150 cm tall:
Neck $= \dfrac{4}{10} \times 150$
Neck $= 60$ cm
∴ Your neck would be $\dfrac{4}{10}$ (two-fifths) of your height, which is 60 cm for a height of 150 cm.
5. Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. “If 2½ palas of saffron costs 3/7 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?” #
Solution
The quantity of saffron is proportional to the money paid:
$$\dfrac{3}{7} : 2\dfrac{1}{2} :: 9 : x$$The money goes up by the factor:
$9 \div \dfrac{3}{7} = 9 \times \dfrac{7}{3}$
$9 \div \dfrac{3}{7} = 21$
So the saffron goes up by the same factor:
$x = 2\dfrac{1}{2} \times 21$
$x = \dfrac{5}{2} \times 21$
$x = 52\dfrac{1}{2}$
∴ $52\dfrac{1}{2}$ palas of saffron can be bought for 9 niskas.
6. Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain’s age when the ratio of her age to her brother’s age is 1 : 2? #
Solution
Let this happen after $x$ years. Then Harmain is $1 + x$ and her brother is $5 + x$:
$$\dfrac{1 + x}{5 + x} = \dfrac{1}{2}$$$$\Rightarrow 2(1 + x) = 5 + x$$
$$\Rightarrow 2 + 2x = 5 + x$$
$$\Rightarrow x = 3$$
So Harmain will be $1 + 3 = 4$ and her brother $5 + 3 = 8$, and $4 : 8 = 1 : 2$.
∴ Harmain will be 4 years old.
7. The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold? #
Solution
$x = 37 \times \dfrac{1}{2}$
$x = 18.5$
∴ 1 litre of gold has a mass of 18.5 kg.
8. It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy? #
Solution
Area of the plot $= 200 \times 500$
Area of the plot $= 100000$ sq ft
1 acre $= 43560$ sq ft, so manure is proportional to area:
$$43560 : 10 :: 100000 : x$$$x = 10 \times \dfrac{100000}{43560}$
$x = 22.96$ tonnes (about)
∴ The farmer should buy about 22.96 tonnes (about 22,957 kg) of manure, say 23 tonnes.
9. A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity? #
Solution
10 litres $= 10000$ mL, so the bucket holds:
Number of mugs $= \dfrac{10000}{500}$
Number of mugs $= 20$
Time $= 20 \times 15$
Time $= 300$ seconds
∴ The tap takes 300 seconds, that is 5 minutes.
10. One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land? #
Solution
1 acre $= 43560$ sq ft, and cost is proportional to area:
$$43560 : 1500000 :: 2400 : x$$$x = 1500000 \times \dfrac{2400}{43560}$
$x = 82645$ (about)
∴ 2,400 sq ft costs about ₹82,645.
11. A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take him if he decides to use a tractor instead? #
Solution
With oxen:
Time $= 20 \times 6$
Time $= 120$ hours
With a tractor: 4 times faster means a quarter of the time:
Time for 1 acre $= \dfrac{6}{4} = 1.5$ hours
Time $= 20 \times 1.5$
Time $= 30$ hours
∴ The oxen take 120 hours, and the tractor takes 30 hours.
12. The ₹10 coin is an alloy of copper and nickel called ‘cupro-nickel’. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin? #
Solution
Mass of each metal. There are $3 + 1 = 4$ parts:
Copper $= \dfrac{3}{4} \times 7.74$
Copper $= 5.805$ g
Nickel $= \dfrac{1}{4} \times 7.74$
Nickel $= 1.935$ g
Cost. 1 kg $= 1000$ g, so copper costs $₹0.906$ per gram and nickel $₹1.341$ per gram:
Cost of copper $= 5.805 \times 0.906$
Cost of copper $= ₹5.26$ (about)
Cost of nickel $= 1.935 \times 1.341$
Cost of nickel $= ₹2.59$ (about)
Total cost $= 5.26 + 2.59$
Total cost $= ₹7.85$
∴ The metals in a ₹10 coin cost about ₹7.85: ₹5.26 of copper and ₹2.59 of nickel.