A sum of money is sufficient to pay A' s wages for 21 days and B's wages for 28 days. The same money is sufficient to pay the wages of both for:

Options

A. `12 1/4 ` days
B. `14 ` days
C. `24 1/2` days
D. 12 days

Answer

To solve the problem step by step, we need to determine how many days the same sum of money can pay the wages of both A and B when they work together.

### Step 1: Define the total money available
Let the total sum of money be represented as \( M \). According to the problem, this amount is sufficient to pay A's wages for 21 days and B's wages for 28 days.

### Step 2: Calculate A's daily wage
Since the total money \( M \) can pay A's wages for 21 days, we can express A's daily wage as:
\[ \text{A's daily wage} = \frac{M}{21} \]

### Step 3: Calculate B's daily wage
Similarly, since the total money \( M \) can pay B's wages for 28 days, we can express B's daily wage as:
\[ \text{B's daily wage} = \frac{M}{28} \]

### Step 4: Find the total daily wage of A and B combined
Now, we can find the combined daily wage of A and B:
\[ \text{Combined daily wage} = \text{A's daily wage} + \text{B's daily wage} = \frac{M}{21} + \frac{M}{28} \]

### Step 5: Find a common denominator
To add the two fractions, we need a common denominator. The least common multiple (LCM) of 21 and 28 is 84. Thus, we can rewrite the fractions:
\[ \frac{M}{21} = \frac{4M}{84} \quad \text{and} \quad \frac{M}{28} = \frac{3M}{84} \]
So,
\[ \text{Combined daily wage} = \frac{4M}{84} + \frac{3M}{84} = \frac{(4M + 3M)}{84} = \frac{7M}{84} \]

### Step 6: Calculate the number of days the money can pay both A and B
Now, we need to find out how many days the total money \( M \) can pay for both A and B:
\[ \text{Number of days} = \frac{M}{\text{Combined daily wage}} = \frac{M}{\frac{7M}{84}} = \frac{84}{7} = 12 \]

### Conclusion
Thus, the same sum of money is sufficient to pay the wages of both A and B for **12 days**.