Henry and Ford are two different persons, but when they work together, they complete it in 10 days. Had Henry worked at half of his efficient and Ford at five times of his efficiency it would have taken them to finish the job in `50%` of the scheduled time. In how many days can Ford complete the job working alone?

Options

A. 12
B. 24
C. 15
D. 30

Answer

To solve the problem step by step, let's denote the efficiencies of Henry and Ford as follows:

- Let Henry's efficiency be \( H \) (units of work per day).
- Let Ford's efficiency be \( F \) (units of work per day).

### Step 1: Determine the total work done when both work together.
When Henry and Ford work together, they complete the work in 10 days. Therefore, the total work \( W \) can be expressed as:
\[ W = (H + F) \times 10 \]

### Step 2: Analyze the scenario when Henry works at half efficiency and Ford at five times efficiency.
If Henry works at half of his efficiency, his efficiency becomes \( \frac{H}{2} \). If Ford works at five times his efficiency, his efficiency becomes \( 5F \). Together, they complete the work in 50% of the scheduled time, which is 5 days. Thus, the total work can also be expressed as:
\[ W = \left(\frac{H}{2} + 5F\right) \times 5 \]

### Step 3: Set the two expressions for total work equal to each other.
From the two expressions for total work, we have:
\[ (H + F) \times 10 = \left(\frac{H}{2} + 5F\right) \times 5 \]

### Step 4: Simplify the equation.
Expanding both sides:
\[ 10H + 10F = \frac{5H}{2} + 25F \]

To eliminate the fraction, multiply the entire equation by 2:
\[ 20H + 20F = 5H + 50F \]

Rearranging gives:
\[ 20H - 5H = 50F - 20F \]
\[ 15H = 30F \]

### Step 5: Find the ratio of efficiencies.
Dividing both sides by 15 gives:
\[ H = 2F \]
This means the ratio of Henry's efficiency to Ford's efficiency is:
\[ H : F = 2 : 1 \]

### Step 6: Calculate the total work.
Substituting \( H = 2F \) back into the total work equation:
\[ W = (H + F) \times 10 = (2F + F) \times 10 = 3F \times 10 = 30F \]

### Step 7: Determine the time taken by Ford to complete the work alone.
Now, we know the total work \( W = 30F \). The time \( T \) taken by Ford to complete the work alone is given by:
\[ T = \frac{W}{F} = \frac{30F}{F} = 30 \text{ days} \]

Thus, Ford can complete the job working alone in **30 days**.