Work Calculation: A and B Working Together

Step 1: Work done by A and B in 1 day

• Work done by A in 1 day: 1/20
• Work done by B in 1 day: 1/12

Step 2: Work done by A in the first 4 days

Work done by A in 4 days: 4 × 1/20 = 1/5

Remaining work: 1 – 1/5 = 4/5

Step 3: Work done by A and B together in 1 day

The total work done by A and B together in 1 day is:

1/20 + 1/12 = 3/60 + 5/60 = 8/60 = 2/15

Step 4: Days required to finish the remaining work

Remaining work is 4/5, and A and B together can complete 2/15 of the work in 1 day. So, the number of days to complete the remaining work is:

4/5 ÷ 2/15 = 6 days

Step 5: Total time to complete the work

A worked alone for 4 days, and A and B worked together for 6 days. Therefore, the total time taken is:

4 days + 6 days = 10 days

Work Calculation: A and B Working on Alternate Days

Step 1: Work done by A and B in 1 day

• Work done by A in 1 day: 1/10
• Work done by B in 1 day: 1/15

Step 2: Work done in 2 days (A on 1st day, B on 2nd day)

The total work done in 2 days is:

1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6

Thus, in 2 days, A and B together complete 1/6 of the total work.

Step 3: Complete cycles

In 6 cycles of 2 days (12 days), they complete:

6 × 1/6 = 1 (the whole work)

Step 4: Conclusion

The work will be completed in 12 days.

Work Calculation: Vijay, Sahil, and Ranjit

Step 1: Equations for individual combinations of workers

Let the work done by Vijay in one day be V, the work done by Sahil in one day be S, and the work done by Ranjit in one day be R.

• Vijay and Sahil: V + S = 1/40
• Sahil and Ranjit: S + R = 1/48
• Ranjit and Vijay: R + V = 1/60

Step 2: Add all three equations

Adding the three equations:

2(V + S + R) = 1/40 + 1/48 + 1/60

Step 3: Calculate the sum of the right-hand side

The least common denominator (LCM) of 40, 48, and 60 is 240. Converting the fractions:

• 1/40 = 6/240
• 1/48 = 5/240
• 1/60 = 4/240

Adding them together:

6/240 + 5/240 + 4/240 = 15/240 = 1/16

Step 4: Solve for V + S + R

Now we have:

2(V + S + R) = 1/16

Dividing both sides by 2:

V + S + R = 1/32

Step 5: Conclusion

Vijay, Sahil, and Ranjit working together can complete the work in 32 days.

Work Calculation: Railway Line Laying

Step 1: Calculate the total work

Initially, 75 men were employed to complete the job in 3 months (90 days). The total work is:

Total Work = 75 × 90 = 6750 man-days

Step 2: Calculate the required number of men for 18 days

Let the total number of men required to complete the work in 18 days be x. Since the total work remains the same, we have:

x × 18 = 6750

Step 3: Solve for x

Solving for x:

x = 6750 ÷ 18 = 375

Step 4: Calculate the number of additional men needed

The original number of men employed was 75, so the additional number of men required is:

375 – 75 = 300

Thus, 300 more men should be employed to complete the work in 18 days.

Carpenters Work Calculation

Let the original number of carpenters be x.

Step 1: Calculate the total amount of work

The total work is given by the original number of carpenters and the number of days:

Total Work = x × 9 man-days

Step 2: Calculate the total work done with reduced carpenters

When 5 carpenters were absent, the remaining x – 5 carpenters took 12 days to finish the work. The total work can also be expressed as:

Total Work = (x – 5) × 12 man-days

Step 3: Equating the two expressions for total work

Since both expressions represent the same total work, we can equate them:

x × 9 = (x – 5) × 12

Step 4: Solve for x

Expanding both sides:

9x = 12(x – 5)

9x = 12x – 60

Bringing all terms involving x to one side:

9x – 12x = -60

-3x = -60

x = 60 ÷ 3 = 20

The original number of carpenters was 20.

Work Calculation

We can solve this problem using the concept of work, which is calculated as:

Total Work = Number of people × Number of days × Hours per day

Step 1: Calculate the total work with the original 5 persons

• Number of persons = 5
• Number of days = 8
• Hours per day = 7

Total Work = 5 × 8 × 7 = 280 man-hours

Step 2: Calculate the daily work requirement with 7 persons to complete the work in 4 days

• Total number of people now = 5 + 2 = 7
• Days to complete the work = 4

Let the required hours per day be x.

7 × 4 × x = 280 man-hours

Step 3: Solve for x

28x = 280

x = 280 ÷ 28 = 10 hours per day

Thus, they need to work 10 hours per day to complete the work in 4 days.