Answer :
Certainly! Let's go through each of the questions one by one and solve them.
### Question 3
A contractor hires 5 workers who could repair Kunal's house in 8 days. If he uses 8 workers instead of 5, how long should they take to complete the job?
Solution:
1. Calculate the total amount of work in terms of worker-days. The total work can be represented as the product of workers and days.
[tex]\[ \text{Total work} = 5 \text{ workers} \times 8 \text{ days} = 40 \text{ worker-days} \][/tex]
2. To find out how many days it would take 8 workers to complete the same amount of work, divide the total work by the number of workers.
[tex]\[ \text{Days new} = \frac{\text{Total work}}{\text{Number of workers}} = \frac{40 \text{ worker-days}}{8 \text{ workers}} = 5 \text{ days} \][/tex]
So, with 8 workers, the job will be completed in 5 days.
### Question 4
In a bicycle company, it requires 36 machines to produce some number of bicycles in 54 days. How many machines would be required to produce the same number of bicycles in 81 days?
Solution:
1. Calculate the total amount of work done in terms of machine-days.
[tex]\[ \text{Total work} = 36 \text{ machines} \times 54 \text{ days} = 1944 \text{ machine-days} \][/tex]
2. To find how many machines are required to complete the same amount of work in 81 days, divide the total work by the number of days.
[tex]\[ \text{Number of machines} = \frac{\text{Total work}}{\text{Number of days}} = \frac{1944 \text{ machine-days}}{81 \text{ days}} = 24 \text{ machines} \][/tex]
Therefore, 24 machines would be needed to produce the same number of bicycles in 81 days.
### Question 5
[tex]$y$[/tex] is inversely proportional to [tex]$x$[/tex] such that [tex]$x=2$[/tex] and [tex]$y=684$[/tex]. Find the value of [tex]$y$[/tex] if [tex]$x=114$[/tex].
Solution:
1. Since [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], the relationship can be written as:
[tex]\[ y \propto \frac{1}{x} \quad \text{or} \quad y = \frac{k}{x} \][/tex]
2. Find the constant of proportionality [tex]\( k \)[/tex] using the given values [tex]\( x = 2 \)[/tex] and [tex]\( y = 684 \)[/tex].
[tex]\[ 684 = \frac{k}{2} \implies k = 684 \times 2 = 1368 \][/tex]
3. Now, use this constant to find the value of [tex]\( y \)[/tex] when [tex]\( x = 114 \)[/tex].
[tex]\[ y = \frac{1368}{114} = 12 \][/tex]
So, when [tex]\( x = 114 \)[/tex], [tex]\( y = 12 \)[/tex].
### Question 6
A man takes 17 steps to cover 5.1 meters distance. In 92 steps how much distance will be covered?
Solution:
1. Find the distance covered per step.
[tex]\[ \text{Distance per step} = \frac{5.1 \text{ meters}}{17 \text{ steps}} \approx 0.3 \text{ meters/step} \][/tex]
2. Now, calculate the distance covered in 92 steps.
[tex]\[ \text{Distance covered} = 92 \text{ steps} \times 0.3 \text{ meters/step} = 27.6 \text{ meters} \][/tex]
So, in 92 steps, the man will cover 27.6 meters.
### Question 7
12 typists type some books in 18 days. In how many days will 4 typists type the same number of books?
Solution:
1. Calculate the total amount of work in terms of typist-days.
[tex]\[ \text{Total work} = 12 \text{ typists} \times 18 \text{ days} = 216 \text{ typist-days} \][/tex]
2. To find how many days it would take 4 typists to complete the same amount of work, divide the total work by the number of typists.
[tex]\[ \text{Days new} = \frac{\text{Total work}}{4 \text{ typists}} = \frac{216 \text{ typist-days}}{4 \text{ typists}} = 54 \text{ days} \][/tex]
So, 4 typists will take 54 days to type the same number of books.
### Question 3
A contractor hires 5 workers who could repair Kunal's house in 8 days. If he uses 8 workers instead of 5, how long should they take to complete the job?
Solution:
1. Calculate the total amount of work in terms of worker-days. The total work can be represented as the product of workers and days.
[tex]\[ \text{Total work} = 5 \text{ workers} \times 8 \text{ days} = 40 \text{ worker-days} \][/tex]
2. To find out how many days it would take 8 workers to complete the same amount of work, divide the total work by the number of workers.
[tex]\[ \text{Days new} = \frac{\text{Total work}}{\text{Number of workers}} = \frac{40 \text{ worker-days}}{8 \text{ workers}} = 5 \text{ days} \][/tex]
So, with 8 workers, the job will be completed in 5 days.
### Question 4
In a bicycle company, it requires 36 machines to produce some number of bicycles in 54 days. How many machines would be required to produce the same number of bicycles in 81 days?
Solution:
1. Calculate the total amount of work done in terms of machine-days.
[tex]\[ \text{Total work} = 36 \text{ machines} \times 54 \text{ days} = 1944 \text{ machine-days} \][/tex]
2. To find how many machines are required to complete the same amount of work in 81 days, divide the total work by the number of days.
[tex]\[ \text{Number of machines} = \frac{\text{Total work}}{\text{Number of days}} = \frac{1944 \text{ machine-days}}{81 \text{ days}} = 24 \text{ machines} \][/tex]
Therefore, 24 machines would be needed to produce the same number of bicycles in 81 days.
### Question 5
[tex]$y$[/tex] is inversely proportional to [tex]$x$[/tex] such that [tex]$x=2$[/tex] and [tex]$y=684$[/tex]. Find the value of [tex]$y$[/tex] if [tex]$x=114$[/tex].
Solution:
1. Since [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex], the relationship can be written as:
[tex]\[ y \propto \frac{1}{x} \quad \text{or} \quad y = \frac{k}{x} \][/tex]
2. Find the constant of proportionality [tex]\( k \)[/tex] using the given values [tex]\( x = 2 \)[/tex] and [tex]\( y = 684 \)[/tex].
[tex]\[ 684 = \frac{k}{2} \implies k = 684 \times 2 = 1368 \][/tex]
3. Now, use this constant to find the value of [tex]\( y \)[/tex] when [tex]\( x = 114 \)[/tex].
[tex]\[ y = \frac{1368}{114} = 12 \][/tex]
So, when [tex]\( x = 114 \)[/tex], [tex]\( y = 12 \)[/tex].
### Question 6
A man takes 17 steps to cover 5.1 meters distance. In 92 steps how much distance will be covered?
Solution:
1. Find the distance covered per step.
[tex]\[ \text{Distance per step} = \frac{5.1 \text{ meters}}{17 \text{ steps}} \approx 0.3 \text{ meters/step} \][/tex]
2. Now, calculate the distance covered in 92 steps.
[tex]\[ \text{Distance covered} = 92 \text{ steps} \times 0.3 \text{ meters/step} = 27.6 \text{ meters} \][/tex]
So, in 92 steps, the man will cover 27.6 meters.
### Question 7
12 typists type some books in 18 days. In how many days will 4 typists type the same number of books?
Solution:
1. Calculate the total amount of work in terms of typist-days.
[tex]\[ \text{Total work} = 12 \text{ typists} \times 18 \text{ days} = 216 \text{ typist-days} \][/tex]
2. To find how many days it would take 4 typists to complete the same amount of work, divide the total work by the number of typists.
[tex]\[ \text{Days new} = \frac{\text{Total work}}{4 \text{ typists}} = \frac{216 \text{ typist-days}}{4 \text{ typists}} = 54 \text{ days} \][/tex]
So, 4 typists will take 54 days to type the same number of books.
