X and Y can finish a piece of work in 30 days. Six days they work together and then X quits the work. Y takes 32 days to finish the work. Calculate how many days Y take to complete the piece of work: (1) 32 days (2) 10 days (3) 30 days (4) 40 days   
    

Solution:

By using the unitary method, we can calculate the work done by X and Y in a day.

To calculate the total work done, we will be taking the LCM of 30 and 32.

Let's write down the prime factorization of 30 and 32.

30 = 2 × 3 × 5

32 = 2 × 2 × 2 × 2 × 2

LCM(30,  32) = 2 × 2 × 2 × 2 × 2 × 3 × 5 = 480

Thus, total work done is 480 units

Given: X and Y together complete the work in 30 days

Units of work done per day by X and Y together is (X + Y ) = 480/30  

(X + Y) = 16 units/day

According to the question,

efficiency of (X + Y) working for 6 days + efficiency of Y working for 32 days completes 480 units of work

(16 × 6) + (Y × 32) = 480

96 + 32Y = 480

32Y = 480 - 96

Y = 384 / 32

Y = 12

Thus, Y can complete 12 units/day

Time taken by Y to complete the whole work = Total work / 12

= 480/ 12 = 40

Thus, Option (4) 40 days, is the total number of days taken by Y to complete the piece of work alone.

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X and Y can finish a piece of work in 30 days. Six days they work together and then X quits the work. Y takes 32 days to finish the work. Calculate how many days Y take to complete the piece of work: (1) 32 days (2) 10 days (3) 30 days (4) 40 days

Summary:

Total number of days taken by Y to complete the piece of work alone is 40 days.