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Three people A,B and C can do a piece of work in 45 hours,40hours and 30hours respectively. How long can B take to complete the remaining work alone,if he starts after A and  C have worked for 13hours each?

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Rate of A = 1 job / 45 hours = 1/45 job per hour

Rate of B = 1 job / 40 hours = 1/40 job per hour

Rate of C = 1 job / 30 hours = 1/30 job per hour

Combined rate = Rate of A + Rate of B + Rate of C

= 1/45 + 1/40 + 1/30

= (8/360 + 9/360 + 12/360)

= 29/360 job per hour

Work completed by A = Rate of A * Time

= (1/45) * 13

= 13/45 job

Work completed by C = Rate of C * Time

= (1/30) * 13

= 13/30 job

To solve this problem, we need to determine the combined rate at which A, B, and C complete the work. Once we have that, we can calculate the remaining work and find out how long it would take for B to complete it alone.

Let's first find the rates at which A, B, and C complete the work:

Rate of A = 1 job / 45 hours = 1/45 job per hour

Rate of B = 1 job / 40 hours = 1/40 job per hour

Rate of C = 1 job / 30 hours = 1/30 job per hour

Since they work together, their combined rate is the sum of their individual rates:

Combined rate = Rate of A + Rate of B + Rate of C

= 1/45 + 1/40 + 1/30

= (8/360 + 9/360 + 12/360)

= 29/360 job per hour

Now, let's calculate the work completed by A and C after 13 hours:

Work completed by A = Rate of A * Time

= (1/45) * 13

= 13/45 job

Work completed by C = Rate of C * Time

= (1/30) * 13

= 13/30 job

Since A and C have worked for 13 hours each, the remaining work is:

Remaining work = 1 job - (Work completed by A + Work completed by C)

= 1 - (13/45 + 13/30)

= 1 - (26/90 + 39/90)

= 1 - (65/90)

= 25/90 job

Time taken by B = Remaining work / Rate of B

= (25/90) / (1/40)

= (25/90) * (40/1)

= (25 * 40) / (90 * 1)

= 1000 / 90

≈ 11.11 hours

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