Lesson 6 Unitary Method

Study Note - Unit 1 Example 1:

If the price of 10 kg apples is Rs. 950, then what will be the price of 3 kg apples at the same rate?

Solution:

Workers	Days	Length of Wall (m)
30	20	600
15	18	x (Let)

Now, using direct variation:

x/600 = (15 × 18) / (30 × 20)

or, x = (15/30) × (18/20) × 600

∴ x = 270

Therefore, in 18 days, 15 workers can construct a 270 m long wall.

Now, using direct variation:

x/600 = (15 × 18) / (30 × 20)

or, x = (15/30) × (18/20) × 600

∴ x = 270

Therefore, in 18 days, 15 workers can construct a 270 m long wall.

Now, using direct variation:

x/600 = (15 × 18) / (30 × 20)

or, x = (15/30) × (18/20) × 600

∴ x = 270

Therefore, in 18 days, 15 workers can construct a 270 m long wall.

Now, using direct variation:

x/600 = (15 × 18) / (30 × 20)

or, x = (15/30) × (18/20) × 600

or, x = 270

Therefore, in 18 days, 15 workers can construct a 270 m long wall.

Example 2:

30 men can construct a 600 m long wall in 20 days. Calculate how long the wall will be constructed by 15 workers in 18 days.

Solution:

Here,

In 20 days, 30 workers can construct a 600 m long wall.

In 1 day, 30 workers can construct 600/20 m of the wall (worker and day: direct variation).

In 1 day, 1 worker can construct 600/(20 × 30) m of the wall (worker and day: direct variation).

In 18 days, 1 worker can construct 600 × 18/(20 × 30) m of the wall (worker and day: direct variation).

In 18 days, 15 workers can construct 600 × 18 × 15/(20 × 30) m of the wall (work and worker: direct variation).

= 270 m of the wall.

Alternative Method:

Here,

Worker	Day	Length of Wall (m)
30	20	600
15	18	x (Let)

Now, using direct variation:

x/600 = (15 × 18) / (30 × 20)

or, x = (15/30) × (18/20) × 600

or, x = 270

Therefore, in 18 days, 15 workers can construct a 270 m long wall.

Example 2

30 men can construct a 600 m long wall in 20 days. Calculate how long the wall will be constructed by 15 workers in 18 days.

Solution:

Here,

In 20 days, 30 workers can construct a 600 m long wall.

In 1 day, 30 workers can construct 600/20 m of the wall (worker and day: direct variation).

In 1 day, 1 worker can construct 600/(20 × 30) m of the wall (worker and day: direct variation).

In 18 days, 1 worker can construct (600 × 18)/(20 × 30) m of the wall (worker and day: direct variation).

In 18 days, 15 workers can construct (600 × 18 × 15)/(20 × 30) m of the wall (work and worker: direct variation).

= 270 m of the wall.

Alternative Method

Here,

Worker: 30, Day: 20, Length of Wall: 600 m

Worker: 15, Day: 18, Length of Wall: x (Let)

Using direct variation:

x/600 = (15/30) × (18/20)

or, x = (15/30) × (18/20) × 600

∴ x = 270

Therefore, in 18 days, 15 workers can construct a 270 m long wall.

Example 3

20 people require 24 days to do a work. How long will it take for 15 men to do the same work?

Solution:

Here,

No. of Workers: 20, Days: 24

No. of Workers: 15, Days: x (Let)

Number of workers and the number of days required to complete are in indirect variation.

20/15 = x/24

or, x = (15 × 24) / 20

or, x = 360 / 20

∴ x = 32

Hence, 15 people require 32 days to do the work.

Example 4

If 32 people take 24 days to paint 6 houses, then:

(a) How long will it take to paint 6 houses by 1 man?

(b) If the work is to be completed in 8 days, how many people should be added?

Solution:

(a) 32 men take 24 days to paint 6 houses.

1 man takes 24 × 32 days to paint 6 houses.

= 768 days

Alternative Method:

Now,

People: 32, Days: 24, Houses: 6

People: 1, Days: x (Let), Houses: 6

x = 32 × 24

or, x = 768

Therefore, 1 man requires 768 days to paint 6 houses.

(b) Here, 32 workers are required to paint 6 houses in 24 days.

In 1 day, (32/24) workers are required to paint 6 houses.

In 8 days, (32/24) × 8 workers are required to paint 6 houses.

= 96 workers are required.

Therefore, the number of additional workers = 96 - 32 = 64 people.

Next Method:

Here,

Working Days: 24, Houses: 6, Workers: 32

Working Days: 8, Houses: 6, Workers: x (Let)

Now,

x/32 = (24/8) × (6/6)

or, x = 32 × 3

or, x = 96

Therefore, the number of additional workers = 96 - 32 = 64 people.

Exercise 6

Write D for the direct variation and I for the indirect variation against the following statements:

a) The relationship between the length of the diameter of a circle and the measurement of its circumference. D

b) The relationship between the number of students in the hostel and the number of days the provision lasts. I

c) The relationship between the principal and interest earned on the fixed time and the fixed rate of interest. D

d) The relationship between the speed of a vehicle and the distance travelled by the vehicle. D

e) The relationship between the speed of a vehicle and time to cover a fixed distance. I

2. If the price of 4 dozen pens is Rs.576, then how many pens can be bought with Rs. 228?

Solution:

Here,

4 dozen pens = 4 × 12 = 48 pens

Pens = 48, Price = Rs. 576

Pens = x (let), Price = Rs. 228

Price is directly proportional to the number of pens.

Price for x pens = x × 12

or, x × 12 = 228

Therefore, x = 228 / 12 = 19 pens

Hence, 19 pens can be bought with Rs. 228.

3. If an athlete can complete 18 km distance in 45 minutes, then estimate the time to cover a distance of 30 km by the athlete.

Solution:

Here,

Distance covered in 45 minutes = 18 km

Time for 1 km = 45 minutes / 18 km = 2.5 minutes per km

For 30 km, time required = 2.5 × 30 = 75 minutes

Hence, the athlete will take 75 minutes to cover 30 km.

4. The loaded truck can cover a distance in 6 hours at the speed of 48 km per hour. If the speed of the truck decreases to 36 km per hour, then how long will it take to cover that distance?

Solution:

Here,

Time taken at 48 km/h = 6 hours

Distance covered = speed × time = 48 × 6 = 288 km

Now, at the new speed of 36 km/h, time = distance / speed

Time = 288 / 36 = 8 hours

Hence, the truck will take 8 hours to cover the same distance at the speed of 36 km/h.

5. If 20 workers can complete one task in 15 days. How many workers are to be added to complete the same work in 12 days?

Solution:

Here,

Number of workers = 20, Number of days = 15

Let the new number of workers be x, and the number of days = 12

Number of workers and number of days are inversely proportional.

20 × 15 = x × 12

or, x = (20 × 15) / 12 = 25 workers

Therefore, the new number of workers required is 25. Hence, 5 more workers should be added.

6. It takes 14 days to complete the work by 12 workers. If 21 workers are added, then how many days will it take to complete the work?

Solution:

Here,

Number of workers = 12, Number of days = 14

Let the new number of days be x, and the number of workers = 12 + 21 = 33

Number of workers and number of days are inversely proportional.

12 × 14 = 33 × x

or, x = (12 × 14) / 33 = 5.09 days (approx.)

Hence, it will take approximately 5.09 days to complete the work with 33 workers.

7. In a barrack, there is a provision of food for 200 soldiers for 30 days. How many soldiers should be transferred to another place in order to last the food for 40 days?

Solution:

Here,

Number of soldiers= 200, Days= 30

Number of soldiers= x (let), Days= 40

The remaining number of soldiers = 200 - x

Number of soldiers and number of days are inversely proportional.

200 × 30 = (200 - x) × 40

or, 6000 = (200 - x) × 40

or, 200 - x = 6000 / 40 = 150

or, x = 200 - 150 = 50 soldiers

Hence, 50 soldiers should be transferred to another place.

8. A motorcycle takes 6 hours to cover a certain distance at the speed of 30 km per hour. What should be its speed to cover the same distance in 5 hours?

Solution:

Here,

Time = 6 hours, Speed = 30 km/h

Distance covered = speed × time = 30 × 6 = 180 km

Now, time = 5 hours, and the distance remains the same.

Speed = distance / time = 180 / 5 = 36 km/h

Hence, the speed required to cover the same distance in 5 hours is 36 km/h.

9. The total price of 3 chairs and 4 tables is Rs.7,540. If the price of one chair is Rs. 220, then find the price of one table.

Solution:

Here,

Price of 1 chair = Rs. 220

Price of 3 chairs = 3 × 220 = Rs. 660

Total price for 3 chairs and 4 tables = Rs. 7540

Price of 4 tables = Rs. 7540 - Rs. 660 = Rs. 6880

Price of 1 table = Rs. 6880 / 4 = Rs. 1720

Hence, the price of 1 table is Rs. 1720.

10. The total price of 5 cows and 2 oxen is Rs.1,35,000. If the price of one ox is Rs.17500, then, find the price of one cow.

Solution:

Here,

Price of 1 ox = Rs. 17,500

Price of 2 oxen = 2 × 17,500 = Rs. 35,000

Total price for 5 cows and 2 oxen = Rs. 1,35,000

Price of 5 cows = Rs. 1,35,000 - Rs. 35,000 = Rs. 1,00,000

Price of 1 cow = Rs. 1,00,000 / 5 = Rs. 20,000

Hence, the price of 1 cow is Rs. 20,000.

11. If 4 men can earn Rs.28,000 in 10 days, then

a) How much will a man earn in a day?

b) How much will 3 men earn in 15 days?

c) By what percent will it be less or more between the earning of 4 men in 10 days and earning of 3 men in 15 days?

Solution:

Here,

4 men earn Rs. 28,000 in 10 days.

1 man earns in 10 days = Rs. 28,000 / 4 = Rs. 7,000

1 man earns in 1 day = Rs. 7,000 / 10 = Rs. 700

Earning of 1 man in 1 day = Rs. 700

Earning of 3 men in 15 days = 3 × 15 × 700 = Rs. 31,500

Earning of 4 men in 10 days = Rs. 28,000

Earning of 3 men in 15 days = Rs. 31,500

Difference = Rs. 31,500 - Rs. 28,000 = Rs. 3,500

Percentage difference = (3,500 / 28,000) × 100 = 12.5%

Therefore, the earning of 3 men in 15 days is 12.5% more than the earning of 4 men in 10 days.

12. If 48 men require 700 kg of rice for 30 days, then

a) How much rice is needed for 40 men for 36 days?

b) What is the total price of rice for 40 men for 36 days at the rate of Rs.60 per kg?

Solution:

Rice needed for 1 man for 1 day = 700 / (48 × 30) = 0.486 kg

Rice needed for 40 men for 36 days = 40 × 36 × 0.486 = 699.84 kg

Total rice = 699.84 kg

Price per kg = Rs. 60

Total price = 699.84 × 60 = Rs. 41,990.4

Hence, the total price of rice is Rs. 41,990.4.

13. If 15 men can construct a 50 m long wall in 8 days, then find the length of the wall that can be constructed by 16 men in 12 days working at the same pace.

Solution:

Wall constructed by 1 man in 1 day = 50 / (15 × 8) = 0.4167 m

Wall constructed by 16 men in 12 days = 16 × 12 × 0.4167 = 80 m

Hence, the wall constructed is 80 m long.

14. If 20 men can construct an 80 m long wall in 8 days, then how many men are required to construct a 60 m long wall in 24 days?

Solution:

Work done by 1 man in 1 day = 80 / (20 × 8) = 0.5 m

Men required = 60 / (24 × 0.5) = 5 men

Hence, 5 men are required to construct the wall.

15. Bhargav deposited Rs.6,000 for 3 years in Nepal Bank Limited. If he receives Rs.1,800 as interest at the end of 3 years, calculate the interest for Rs.10,000 for 4 years at the same rate of interest.

Solution:

Interest rate = (1800 / (6000 × 3)) × 100 = 10%

Interest for Rs. 10,000 for 4 years = (10,000 × 10 × 4) / 100 = Rs. 4,000

Hence, the interest is Rs. 4,000.

Simplified Steps:

Calculating the Rate of Interest,

Interest earned for Rs.6,000 in 3 years = Rs.1,800

Interest for 1 year = Rs.1,800 ÷ 3 = Rs.600

Interest for 1 year on Rs.1 = Rs.600 ÷ 6,000 = 0.1

Interest rate per year (for Rs.100) = 0.1 × 100 = 10%

Calculating the Interest for Rs.10,000 for 4 Years,

Interest for Rs.1 in 1 year = 10 ÷ 100 = 0.1

Interest for Rs.10,000 in 1 year = Rs.10,000 × 0.1 = Rs.1,000

Interest for Rs.10,000 in 4 years = Rs.1,000 × 4 = Rs.4,000

The interest for Rs.10,000 for 4 years is Rs.4,000

16. 6 men and 8 boys can harvest the crops of 30 ropanis in 4 days. If the capacity to harvest of 4 boys is equal to 2 men, then, at the same rate, how much ropanis can 14 men and 8 boys harvest in 8 days?

Solution:

4 boys = 2 men, so 8 boys = 4 men.

Total capacity of 6 men and 8 boys = 6 + 4 = 10 men.

10 men can harvest 30 ropanis in 4 days.

1 man can harvest = 30 / (10 × 4) = 0.75 ropani/day.

14 men + 8 boys = 14 + 4 = 18 men.

Ropanis harvested by 18 men in 8 days = 18 × 8 × 0.75 = 108 ropanis.

Hence, 108 ropanis can be harvested.

17. If 10 men can construct a road in 2 days working 7 hours a day, then, working at the same pace, calculate the number of hours per day 5 men should work to construct a road in 14 days?

Solution:

Total work = 10 × 2 × 7 = 140 man-hours.

Work required per day for 5 men = 140 / 14 = 10 hours.

Hence, 5 men should work 10 hours per day.

18. If 18 men can construct a 54 m long wall in 10 days, then how many men of the same capacity are required to construct a 66 m long wall in 22 days?

Solution:

Work done by 18 men in 10 days = 54 m

Work done by 1 man in 1 day = 54 / (18 × 10) = 0.3 m.

Men required = 66 / (22 × 0.3) = 10 men.

Hence, 10 men are required.

19. If 20 men construct a park having 40 m length and 20 m width in 25 days, then how long will it take for 50 men to construct a park of 50 m length and 40 m width?

Solution:

Work = length × width = 40 × 20 = 800 m².

New work = 50 × 40 = 2000 m².

Work done by 1 man in 1 day = 800 / (20 × 25) = 1.6 m².

Time required by 50 men = 2000 / (50 × 1.6) = 25 days.

Hence, it will take 25 days.

20. If 25 men earn Rs.5,00,000 in 30 days, then

a) How much does a man earn in a day?

b) How much will a man earn in 10 days?

c) How many men will earn Rs.5,00,000 in 10 days?

d) What is the earning of 5 men in 40 days?

Solution:

Earning of 25 in 30 days = Rs.5,00,000

Earning of 1 man in 1 day = Rs. 5,00,000 / (25 × 30) = Rs. 666.67.

Earning in 10 days = 666.67 × 10 = Rs. 6,666.67.

Men required = Rs. 5,00,000 / (666.67 × 10) = 75 men.

Earning = 5 × 40 × 666.67 = Rs. 1,33,333.33.

Hence, the earning is Rs. 1,33,333.33.

Solution of selected questions from Excel in Mathematics - Book 8

1. 4 machines of a factory can finish the required amount of production in 6 days. If one machine had machinery defect and stop functioning, in how many days would the remaining machines finish the same quantity of production at the same rate of production?

Solution:

Here, number of machines = 4 - 1 = 3

4 machines can finish the required amount of production in 4 × 6 days = 24 days.

∴ 3 machines can finish the same amount of production in 24 / 3 = 8 days.

2. 12 workers were employed to complete the construction of a building in 60 days. How many additional numbers of workers should be employed to complete the construction in 45 days?

Solution:

In 60 days, 12 workers can construct a building.

In 1 day, 60 × 12 = 720 workers can construct the building.

In 45 days, 720 / 45 = 16 workers can construct the building.

Hence, the additional number of workers = 16 - 12 = 4.

3. 120 students of hostel have food enough for 60 days. If 30 more students join the hostel after 15 days, how long will the remaining food last?

Solution:

Here, remaining number of days = 60 - 15 = 45.

Total number of students with 30 more students = 120 + 30 = 150.

Now,

120 students have food for 45 days.

1 student has food enough for 120 × 45.

150 students have the food enough for 5400 / 150 = 36 days.

Hence, the remaining provisions will last in 36 days.

4. A garrison of 110 people had provisions for 30 days. If 22 people leave the garrison after 10 days, how long does the remaining provision last?

Solution:

Here,

After 10 days, remaining number of people = 110 - 22 = 88.

Remaining number of days = 30 - 10 = 20.

Now,

110 people have provision for 20 days.

1 person has provision for 110 × 20 = 2200 days.

88 people have provision for 2200 / 88 = 25 days.

Hence, the remaining provisions will last in 25 days.

5. A computer can finish downloading an application file in 4 minutes at the rate of 600 MB per minute.

(i) Find the size of the application file?

(ii) How long does it take to download the file when the download rate increases to 800 MB per minute?

Solution:

Here,

(i) Size of the application file = 4 minutes × 600 MB/minute = 2400 MB.

(ii) Time taken to download the file at 800 MB/minute = 2400 MB / 800 MB/minute = 3 minutes.

6. The size of movie file in YouTube is 1.8 gigabyte (GB) and the rate of download of the file is 900 megabyte (MB) per minute.

Saturday, January 4, 2025

Class 8 Maths Solution | Lesson 6 Unitary Method | Curriculum Development Centre (CDC)

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