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

Answer

To solve the problem step by step, we will first determine the daily wages of A and B based on the information given, and then find out how many days the same sum of money can pay for both A and B working together.

### Step 1: Determine the daily wages of A and B

Let the total sum of money be \( M \).

- A's wages for 21 days means:
\[ M = 21 \times \text{(A's daily wage)} \]
Therefore, A's daily wage can be expressed as:
\[ \text{A's daily wage} = \frac{M}{21} \]

- B's wages for 28 days means:
\[ M = 28 \times \text{(B's daily wage)} \]
Therefore, B's daily wage can be expressed as:
\[ \text{B's daily wage} = \frac{M}{28} \]

### Step 2: Calculate the total daily wages when A and B work together

To find out how many days the same amount of money can pay for both A and B, we need to add their daily wages together.

- The combined daily wage of A and B is:
\[ \text{Combined daily wage} = \text{A's daily wage} + \text{B's daily wage} \]
Substituting the values we found:
\[ \text{Combined daily wage} = \frac{M}{21} + \frac{M}{28} \]

### Step 3: Find a common denominator to add the fractions

The least common multiple (LCM) of 21 and 28 is 84. We will convert each fraction to have a denominator of 84:

- For A:
\[ \frac{M}{21} = \frac{4M}{84} \]

- For B:
\[ \frac{M}{28} = \frac{3M}{84} \]

Now, we can add the two fractions:
\[ \text{Combined daily wage} = \frac{4M}{84} + \frac{3M}{84} = \frac{(4M + 3M)}{84} = \frac{7M}{84} \]

### Step 4: Calculate the number of days the money can last for both A and B

Now, we need to find out how many days the total money \( M \) can pay for both A and B working together:
Let \( D \) be the number of days they can work together:
\[ M = D \times \text{Combined daily wage} \]
Substituting the combined daily wage:
\[ M = D \times \frac{7M}{84} \]

Dividing both sides by \( M \) (assuming \( M \neq 0 \)):
\[ 1 = D \times \frac{7}{84} \]

Now, solving for \( D \):
\[ D = \frac{84}{7} = 12 \]

### Final Answer
The same sum of money is sufficient to pay the wages of both A and B for **12 days**.

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