Question

A and B are employed to complete a work. When A worked at twice his normal efficiency and B worked at half his normal efficiency, the work was completed in 17 days. If A had worked at five times his normal efficiency and B worked at one-fourth of his normal efficiency, the work would have been completed in 10 days. If B works alone, how many days does he need to complete the work?

• A. 21.25
• B. 23.75
• C. 25.25
• D. 20.75
• E. None of these

Hint:

In problems with efficiency and time, use the relationship between time, work, and efficiency to set up equations and solve for the unknowns.

Answer 1

The Correct Option is A

Step 1: Define variables for efficiencies.
Let the normal efficiency of A be \( a \), and the normal efficiency of B be \( b \). The total work is considered as 1 unit.
Step 2: Work equations for the first scenario.
In the first scenario, A works at twice his efficiency and B works at half his efficiency. The work is completed in 17 days. Thus, the work equation is: \[ \frac{2a}{1} \times 17 + \frac{b}{2} \times 17 = 1 \] \[ 34a + 8.5b = 1 \quad \text{(Equation 1)} \] Step 3: Work equations for the second scenario.
In the second scenario, A works at five times his normal efficiency and B works at one-fourth his normal efficiency. The work is completed in 10 days. Thus, the work equation is: \[ \frac{5a}{1} \times 10 + \frac{b}{4} \times 10 = 1 \] \[ 50a + 2.5b = 1 \quad \text{(Equation 2)} \] Step 4: Solve the system of equations.
Now solve the system of equations:
From Equation 1: \[ 34a + 8.5b = 1 \] From Equation 2: \[ 50a + 2.5b = 1 \] Multiply Equation 2 by 10 to eliminate the decimal: \[ 500a + 25b = 10 \] Multiply Equation 1 by 10 as well: \[ 340a + 85b = 10 \] Now subtract the first equation from the second equation: \[ (500a + 25b) - (340a + 85b) = 10 - 10 \] \[ 160a - 60b = 0 \] Solving for \( a \) in terms of \( b \): \[ 160a = 60b \quad \Rightarrow \quad a = \frac{3}{8}b \] Step 5: Substitute into one of the original equations.
Substitute \( a = \frac{3}{8}b \) into Equation 1: \[ 34 \times \frac{3}{8}b + 8.5b = 1 \] Simplifying: \[ \frac{102}{8}b + 8.5b = 1 \quad \Rightarrow \quad \frac{102}{8}b + \frac{17}{2}b = 1 \] Multiply through by 8 to eliminate the denominator: \[ 102b + 68b = 8 \quad \Rightarrow \quad 170b = 8 \quad \Rightarrow \quad b = \frac{8}{170} = \frac{4}{85} \] Step 6: Find the number of days B works alone.
Now we know that \( b = \frac{4}{85} \). Since B works alone, the time taken is the inverse of B's efficiency: \[ \text{Time} = \frac{1}{b} = \frac{85}{4} = 21.25 \text{ days} \] Step 7: Conclusion.
Thus, B would take 21.25 days to complete the work. The correct answer is (1) 21.25.