Unitary Method - Exercise 8(A)

12 bags of rice having equal weights weigh 174 kg. How much will 20 such bags weigh?

Answer

Weight of 12 bags = 174 kg.

Weight of 1 bag = $\Big(\dfrac{174}{12}\Big) \text{ kg}$

[Less bags, Less weight]

Weight of 20 bags = $\Big(\dfrac{174}{12} \times 20\Big) \text{ kg}$

[More bags, More weight]

= $\Big(\dfrac{174}{3} \times 5\Big)$ kg

= 58 x 5 kg $\quad$ (Dividing 174 and 3 by 3)

= 290 kg.

Hence, the weight of 20 bags is 290 kg.

8 kg of tomatoes cost ₹ 84. What is the cost of 14 kg of tomatoes?

Answer

Cost of 8 kg tomatoes = ₹ 84.

Cost of 1 kg tomato = ₹ $\Big(\dfrac{84}{8}\Big)$

[Less tomatoes, Less cost]

Cost of 14 kg tomatoes = ₹ $\Big(\dfrac{84}{8} \times 14\Big)$

[More tomatoes, More cost]

= ₹ $\Big(\dfrac{21}{2} \times 14\Big)$

= ₹ $\Big(\dfrac{294}{2}\Big)$

= ₹ 147

Hence, the cost of 14 kg of tomatoes is ₹ 147.

16 m of cloth costs ₹ 2200. What is the cost of 6 m of cloth?

Answer

Cost of 16 m of cloth = ₹ 2200.

Cost of 1 m of cloth = ₹ $\Big(\dfrac{2200}{16}\Big)$

[Less cloth, Less cost]

Cost of 6 m of cloth = ₹ $\Big(\dfrac{2200}{16} \times 6\Big)$

[More cloth, More cost]

$= ₹ \Big(\dfrac{2200}{16} \times 6\Big) \\[1em] = ₹ \Big(\dfrac{275}{2} \times 6\Big) \quad \text{(Dividing 2200 and 16 by 8)} \\[1em] = ₹ \Big(\dfrac{1650}{2}\Big) \\[1em] = ₹ 825$

Hence, the cost of 6 m of cloth is ₹ 825.

If 8.5 m of a uniform rod weighs 30.6 kg, what will be the weight of 5 m of the same rod?

Answer

Weight of 8.5 m rod = 30.6 kg.

Weight of 1 m rod = $\Big(\dfrac{30.6}{8.5}\Big) \text{ kg}$

[Less length, Less weight]

Multiply by 10 to remove decimals.

Weight of 1 m rod = $\Big(\dfrac{30.6 \times 10}{8.5 \times 10}\Big) \text{ kg} = \Big(\dfrac{306}{85}\Big) \text{ kg}$

Weight of 5 m rod = $\Big(\dfrac{306}{85} \times 5\Big) \text{ kg}$

[More length, More weight]

= $\Big(\dfrac{306}{17}\Big) \text{ kg} \quad \text{(Dividing 5 and 85 by 5)}$

= 18 kg.

Hence, the weight of 5 m of the rod is 18 kg.

The cost of 6 purses is ₹ 453. How many such purses can be bought for ₹ 604?

Answer

For ₹ 453, number of purses bought = 6.

For ₹ 1, number of purses bought = $\Big(\dfrac{6}{453}\Big) \text{ purses}$

[Less money, Less purses]

For ₹ 604, number of purses bought = $\Big(\dfrac{6}{453} \times 604\Big) \text{ purses}$

[More money, More purses]

= $\Big(\dfrac{2}{151} \times 604\Big) \text{ purses} \quad \text{(Dividing 6 and 453 by 3)}$

= $\Big(\dfrac{1208}{151}\Big)$ purses

= 8 purses.

Hence, 8 purses can be bought for ₹ 604.

12 apples weigh 2 kg 500 g. How many apples will weigh 15 kg?

Answer

First, convert weight to a single unit:

1000 g = 1 kg

500 g = 0.5 kg

∴ 2 kg 500 g = (2 + 0.5) kg = 2.5 kg.

For 2.5 kg, number of apples = 12.

For 1 kg, number of apples = $\Big(\dfrac{12}{2.5}\Big)$

[Less weight, Less apples]

Multiply by 10 to remove decimals.

For 1 kg, number of apples = $\Big(\dfrac{12 \times 10}{2.5 \times 10}\Big) = \Big(\dfrac{120}{25}\Big)$ apples

For 15 kg, number of apples = $\Big(\dfrac{120}{25} \times 15\Big) \text{ apples}$

[More weight, More apples]

= $\Big(\dfrac{120}{5} \times 3\Big) \text{ apples} \quad \text{(Dividing 15 and 25 by 5)}$

= 24 x 3 apples $\quad$ (Dividing 120 and 5 by 5)

= 72 apples.

Hence, 72 apples will weigh 15 kg.

55 m of cloth is required to make 25 shirts. How much cloth will be required to make 35 shirts of the same size?

Answer

Cloth required for 25 shirts = 55 m.

Cloth required for 1 shirt = $\Big( \dfrac{55}{25}\Big) \text{m}$

[Less shirts, Less cloth]

Cloth required for 35 shirts = $\Big( \dfrac{55}{25} \times 35\Big) \text{m}$

[More shirts, More cloth]

= $\Big( \dfrac{55}{5} \times 7\Big)$ m

= 11 x 7 m $\quad$ (Dividing 55 and 5 by 5)

= 77 m.

Hence, 77 m of cloth will be required to make 35 shirts.

A scooter consumes 2.5 litres of petrol in covering a distance of 85 km. What distance will it cover in 8 litres of petrol?

Answer

In 2.5 litres, distance covered = 85 km.

In 1 litre, distance covered = $\Big( \dfrac{85}{2.5}\Big)$ km.

[Less petrol, Less distance]

Multiply by 10 to remove decimals.

In 1 litre, distance covered = $\Big( \dfrac{85 \times 10}{2.5 \times 10}\Big) = \Big( \dfrac{850}{25}\Big)$ km

In 8 litres, distance covered = $\Big( \dfrac{850}{25} \times 8\Big) \text{ km}$

[More petrol, More distance]

= 34 x 8 km $\quad$ (Dividing 850 and 25 by 25)

= 272 km.

Hence, the scooter will cover 272 km in 8 litres of petrol.

A journey of 124 km costs ₹ 1395. How much will a journey of 240 km cost?

Answer

Cost of 124 km journey = ₹ 1395.

Cost of 1 km journey = ₹ $\Big(\dfrac{1395}{124}\Big)$

[Less distance, Less cost]

Cost of 240 km journey = ₹ $\Big(\dfrac{1395}{124} \times 240\Big)$

[More distance, More cost]

= ₹ 11.25 x 240 $\quad$ (Dividing 1395 and 124 by 124)

= ₹ 2700

Hence, a journey of 240 km will cost ₹ 2700.

A compositor takes 1 hour 45 minutes to compose 24 pages of a book. How long will he take to compose 64 pages?

Answer

First, convert time into minutes:

1 hour = 60 minutes

∴ 1 hour 45 minutes = 60 + 45 = 105 minutes.

Time taken to compose 24 pages = 105 minutes.

Time taken to compose 1 page = $\Big( \dfrac{105}{24}\Big) \text{ minutes}$

[Less pages, Less time]

Time taken to compose 64 pages = $\Big(\dfrac{105}{24} \times 64\Big) \text{ minutes}$

[More pages, More time]

= $\Big(\dfrac{105}{3} \times 8\Big) \text{ minutes} \quad \text{(Dividing 24 and 64 by 8)}$

= 35 x 8 minutes

= 280 minutes (or 4 hours 40 minutes)

Hence, he will take 4 hours 40 minutes to compose 64 pages.

If a man walks 16 km in 5 hours, how long would he take to walk 11.2 km?

Answer

Time taken to walk 16 km = 5 hours.

Time taken to walk 1 km = $\Big(\dfrac{5}{16}\Big) \text{ hours}$

[Less distance, Less time]

Time taken to walk 11.2 km = $\Big(\dfrac{5}{16} \times 11.2\Big) \text{ hours}$

[More distance, More time]

= $\Big(\dfrac{5 \times 11.2}{16}\Big)$ hours

= $\dfrac{56}{16}$ hours

= 3.5 hours (or 3 hours 30 minutes).

Time taken to walk 11.2 km = 3 hours 30 minutes

5 men can paint a hall in 18 hours. How many men will be able to paint it in 10 hours?

Answer

Men required to paint in 18 hours = 5 men.

Men required to paint in 1 hour = (5 x 18) men.

[Less time, More men]

Men required to paint in 10 hours = $\Big(\dfrac{5 \times 18}{10}\Big) \text{ men}$

[More time, Less men]

= $\dfrac{90}{10}$ men

= 9 men.

9 men are required to paint in 10 hours.

30 workers can dig a trench in 5 days. How many workers will be required to dig it in 6 days?

Answer

Workers required to dig in 5 days = 30 workers.

Workers required to dig in 1 day = (30 x 5) workers.

[Less days, More workers]

Workers required to dig in 6 days = $\Big(\dfrac{30 \times 5}{6}\Big) \text{ workers}$

[More days, Less workers]

= (5 x 5) workers $\quad$ [Dividing 30 and 6 by 6]

= 25 workers.

25 workers are required to dig in 6 days.

In a fort, 360 men had provisions for 21 days. If 60 more men join them, how long will the provisions last?

Answer

Original men = 360.

New men = 360 + 60 = 420.

Provisions for 360 men last = 21 days.

Provisions for 1 man last = (21 x 360) days.

[Less men, More days]

Provisions for 420 men last = $\Big(\dfrac{21 \times 360}{420}\Big) \text{ days}$

[More men, Less days]

= $\Big(\dfrac{1 \times 360}{20}\Big) \text{ days} \quad \text{[Dividing 21 and 420 by 21]}$

= 18 days.

420 men had provisions for 18 days.

In an army camp, 300 soldiers had provisions for 13 days. If 40 of them are transferred to the other camp, how long will the provisions last?

Answer

Original soldiers = 300.

Remaining soldiers = 300 - 40 = 260.

Provisions for 300 soldiers last = 13 days.

Provisions for 1 soldier last = (13 x 300) days.

[Less soldiers, More days]

Provisions for 260 soldiers last = $\Big(\dfrac{13 \times 300}{260}\Big) \text{ days}$

[More soldiers, Less days]

= $\Big(\dfrac{1 \times 300}{20}\Big) \text{ days} \quad \text{[Dividing 13 and 260 by 13]}$

= 15 days.

260 soldiers had provisions for 15 days

Moving at the rate of 70 km/hr, a car completes a journey in 18 minutes. How long would it take to complete this journey, if the speed is increased to 84 km/hr?

Answer

Time taken at 70 km/hr speed = 18 minutes.

Time taken at 1 km/hr speed = (18 x 70) minutes.

[Less speed, More time]

Time taken at 84 km/hr speed = $\Big(\dfrac{18 \times 70}{84}\Big) \text{ minutes}$

[More speed, Less time]

= $\Big(\dfrac{3 \times 70}{14}\Big) \text{ minutes} \quad$ [Dividing 18 and 84 by 6]

= 3 x 5 minutes $\quad$ [Dividing 70 and 14 by 14]

= 15 minutes.

At the rate of 84 km/hr, the car completes the journey in 15 minutes.