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The respective ratio of the time taken b...

The respective ratio of the time taken by A and B individually to finish a piece of work is 4 : 5. The respective ratio of the time taken by B and C individually to finish the same piece of work is 3 :4. If A and C together can finish `1/3`rd of the same piece of work in 5 days,, in how many days can B alone finish the same piece of work?

A

27

B

40

C

24

D

30

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, let's break it down: ### Step 1: Understand the Ratios The time taken by A and B is in the ratio of 4:5. This means: - Let the time taken by A be 4x days. - Let the time taken by B be 5x days. The time taken by B and C is in the ratio of 3:4. This means: - Let the time taken by B be 3y days. - Let the time taken by C be 4y days. ### Step 2: Equate the Time Taken by B From the above, we have: - 5x = 3y (since both represent the time taken by B) ### Step 3: Express A and C in Terms of y From the equation 5x = 3y, we can express x in terms of y: - x = (3/5)y Now substituting x into the expressions for A and C: - Time taken by A = 4x = 4 * (3/5)y = (12/5)y - Time taken by C = 4y ### Step 4: Find the Efficiency of A, B, and C Efficiency is inversely proportional to time. Therefore, the efficiencies can be expressed as: - Efficiency of A = 1/(12/5)y = 5/12y - Efficiency of B = 1/(3y) = 1/3y - Efficiency of C = 1/(4y) = 1/4y ### Step 5: Combine Efficiencies of A and C A and C together can finish 1/3 of the work in 5 days. First, we need to find their combined efficiency: - Combined efficiency of A and C = (5/12y) + (1/4y) To add these, we need a common denominator: - (5/12y) + (3/12y) = (8/12y) = (2/3y) ### Step 6: Calculate Work Done in 5 Days In 5 days, A and C together can complete: - Work done = Efficiency * Time = (2/3y) * 5 = (10/3y) This amount of work equals 1/3 of the total work, so: - Total work = 3 * (10/3y) = 10y ### Step 7: Find Efficiency of B We know the efficiency of B is: - Efficiency of B = 1/3y ### Step 8: Calculate the Time Taken by B to Complete the Work The time taken by B to finish the total work can be calculated as: - Time = Total Work / Efficiency of B = 10y / (1/3y) = 10y * (3/y) = 30 days ### Final Answer Thus, B alone can finish the work in **30 days**. ---
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