If 10 persons can clean 10 floors by 10 mops in 10 days, in how many days can 8 persons clean 8 floors by 8 mops?

Options

A. A)`12 1/2` days
B. B)8 days
C. C)10 days
D. D)`81/3` days

Answer

To solve the problem, we will use the concept of work done, which can be expressed using the formula:

\[ \frac{M_1 \times D_1 \times E_1}{W_1} = \frac{M_2 \times D_2 \times E_2}{W_2} \]

Where:
- \(M\) = Number of men
- \(D\) = Number of days
- \(E\) = Efficiency (number of mops)
- \(W\) = Work done (in terms of floors cleaned)

### Step 1: Determine the total work done in the first scenario

In the first scenario:
- \(M_1 = 10\) (10 persons)
- \(D_1 = 10\) (10 days)
- \(E_1 = 10\) (10 mops)
- \(W_1 = 10\) (10 floors)

The total work done can be calculated as:
\[ W_1 = M_1 \times D_1 \times E_1 = 10 \times 10 \times 10 = 1000 \text{ units of work} \]

### Step 2: Determine the total work done in the second scenario

In the second scenario:
- \(M_2 = 8\) (8 persons)
- \(D_2 = ?\) (we need to find this)
- \(E_2 = 8\) (8 mops)
- \(W_2 = 8\) (8 floors)

The total work done can be calculated as:
\[ W_2 = M_2 \times D_2 \times E_2 = 8 \times D_2 \times 8 = 64D_2 \text{ units of work} \]

### Step 3: Set up the equation

Since the work done in both scenarios is the same, we can set the two expressions for work equal to each other:
\[ 1000 = 64D_2 \]

### Step 4: Solve for \(D_2\)

To find \(D_2\), we rearrange the equation:
\[ D_2 = \frac{1000}{64} = 15.625 \text{ days} \]

### Step 5: Convert to mixed number

To express \(15.625\) days as a mixed number:
\[ 15.625 = 15 \text{ days and } 0.625 \text{ of a day} \]
Since \(0.625\) of a day is equivalent to \(0.625 \times 24 = 15\) hours, we can express it as:
\[ 15 \text{ days and } 15 \text{ hours} \]

### Final Answer

Thus, the answer is:
\[ \text{It will take } 15 \text{ days and } 15 \text{ hours for 8 persons to clean 8 floors with 8 mops.} \]

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