To do a certain work, the ratio of the efficiencies of A and B is 7:6. Working together, they can complete the same work in 21 days. What part of the same work will be completed by B alone in 26 days ?

Options

A. `5/8`
B. `4/7`
C. `5/6`
D. `2/3`

Answer

To solve the problem step by step, we will use the information given about the efficiencies of A and B and how long they take to complete the work together.

### Step 1: Understand the ratio of efficiencies
The ratio of the efficiencies of A and B is given as 7:6. This means if A can do 7 units of work in a day, B can do 6 units of work in a day.

### Step 2: Calculate the total efficiency when A and B work together
To find the total efficiency of A and B working together, we add their efficiencies:
- Efficiency of A = 7x (where x is a common factor)
- Efficiency of B = 6x

Total efficiency of A and B together = 7x + 6x = 13x.

### Step 3: Calculate the total work done in 21 days
We know that A and B together can complete the work in 21 days. Therefore, the total work (W) can be calculated as:
\[ W = \text{Total Efficiency} \times \text{Number of Days} \]
\[ W = 13x \times 21 = 273x \]

### Step 4: Calculate the efficiency of B
From the ratio, the efficiency of B is 6x.

### Step 5: Calculate the work done by B in 26 days
Now, we need to find out how much work B can complete in 26 days:
\[ \text{Work done by B} = \text{Efficiency of B} \times \text{Number of Days} \]
\[ \text{Work done by B} = 6x \times 26 = 156x \]

### Step 6: Find the fraction of the total work done by B
Now, we need to express the work done by B in terms of the total work:
\[ \text{Fraction of work done by B} = \frac{\text{Work done by B}}{\text{Total Work}} \]
\[ \text{Fraction of work done by B} = \frac{156x}{273x} = \frac{156}{273} \]

### Step 7: Simplify the fraction
To simplify \(\frac{156}{273}\):
- The greatest common divisor (GCD) of 156 and 273 is 39.
- Dividing both the numerator and the denominator by 39 gives:
\[ \frac{156 \div 39}{273 \div 39} = \frac{4}{7} \]

### Final Answer
Thus, the part of the work that will be completed by B alone in 26 days is \(\frac{4}{7}\).

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