6 men or 10 boys can complete a piece of work in 15 days. If 7 men and x boys complete the same piece of work in 9 days, then x is equal to

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem step by step, let's break it down clearly.

### Step 1: Understand the Work Done by Men and Boys
We know that:
- 6 men can complete the work in 15 days.
- 10 boys can also complete the same work in 15 days.

### Step 2: Calculate the Work Done by One Man and One Boy
First, we calculate the total work done in terms of 'man-days' and 'boy-days'.

**For Men:**
- Total work = Number of men × Number of days = 6 men × 15 days = 90 man-days.

So, 1 man can complete the work in:
- Work done by 1 man in 1 day = 1/90 of the work.

**For Boys:**
- Total work = Number of boys × Number of days = 10 boys × 15 days = 150 boy-days.

So, 1 boy can complete the work in:
- Work done by 1 boy in 1 day = 1/150 of the work.

### Step 3: Set Up the Equation for 7 Men and x Boys
Now, we need to find how many boys (x) are required if 7 men and x boys can complete the work in 9 days.

**Work done by 7 men in 1 day:**
- Work done by 7 men in 1 day = 7 × (1/90) = 7/90 of the work.

**Work done by x boys in 1 day:**
- Work done by x boys in 1 day = x × (1/150) = x/150 of the work.

**Total work done by 7 men and x boys in 1 day:**
- Total work done in 1 day = (7/90) + (x/150).

### Step 4: Calculate the Total Work Done in 9 Days
Since they complete the work in 9 days, the equation becomes:
- Total work done in 9 days = 1 (the whole work).

So, we have:
\[ 9 \left( \frac{7}{90} + \frac{x}{150} \right) = 1 \]

### Step 5: Solve the Equation
Now, let's simplify the equation:
\[ \frac{7}{90} + \frac{x}{150} = \frac{1}{9} \]

To solve this, we need a common denominator. The least common multiple of 90 and 150 is 450.

Convert each term:
- \(\frac{7}{90} = \frac{7 \times 5}{90 \times 5} = \frac{35}{450}\)
- \(\frac{x}{150} = \frac{x \times 3}{150 \times 3} = \frac{3x}{450}\)

Now, substitute back into the equation:
\[ \frac{35 + 3x}{450} = \frac{1}{9} \]

Cross-multiply:
\[ 35 + 3x = \frac{450}{9} \]
\[ 35 + 3x = 50 \]

### Step 6: Isolate x
Now, isolate x:
\[ 3x = 50 - 35 \]
\[ 3x = 15 \]
\[ x = \frac{15}{3} = 5 \]

### Final Answer
Thus, the value of x is 5.