If 12 boys can earn Rs.240 in 5 days. How many boys can earn Rs. 420 in 21 days ?

Options

A. 15 boys
B. 5 boys
C. 17 boys
D. None of these

Answer

To solve the problem step by step, we can use the concept of work and the relationship between the number of boys, the days they work, and the earnings they generate.

### Step 1: Understand the relationship between boys, days, and earnings
We know that work (W) is directly proportional to the number of boys (M) and the number of days (D). This can be expressed as:
\[ W = M \times D \]

### Step 2: Calculate the total work done in the first scenario
In the first scenario, we have:
- Number of boys (M1) = 12
- Number of days (D1) = 5
- Earnings (W1) = Rs. 240

Using the formula, we can calculate the work done:
\[ W1 = M1 \times D1 = 12 \times 5 = 60 \text{ (work units)} \]

### Step 3: Calculate the total work required in the second scenario
In the second scenario, we have:
- Earnings (W2) = Rs. 420
- Number of days (D2) = 21
- Number of boys (M2) = ? (this is what we need to find)

### Step 4: Set up the proportion based on work
Since work is directly proportional to earnings, we can set up the following proportion:
\[ \frac{M1 \times D1}{W1} = \frac{M2 \times D2}{W2} \]

Substituting the known values:
\[ \frac{12 \times 5}{240} = \frac{M2 \times 21}{420} \]

### Step 5: Simplify the equation
First, simplify the left side:
\[ \frac{60}{240} = \frac{M2 \times 21}{420} \]
\[ \frac{1}{4} = \frac{M2 \times 21}{420} \]

Now, simplify the right side:
\[ \frac{M2 \times 21}{420} = \frac{M2}{20} \]

### Step 6: Set the two sides equal
Now we have:
\[ \frac{1}{4} = \frac{M2}{20} \]

### Step 7: Cross-multiply to solve for M2
Cross-multiplying gives:
\[ 1 \times 20 = 4 \times M2 \]
\[ 20 = 4M2 \]

### Step 8: Solve for M2
Now, divide both sides by 4:
\[ M2 = \frac{20}{4} = 5 \]

### Conclusion
Thus, the number of boys required to earn Rs. 420 in 21 days is **5 boys**.