The product of two positive numbers is 1344 and their ratio is 7:12. The smaller of these numbers is:

Options

A. 16
B. 28
C. 112
D. 48

Answer

To solve the problem, we need to find two positive numbers whose product is 1344 and whose ratio is 7:12. Let's denote the two numbers as \( x \) and \( y \).

### Step 1: Set up the equations based on the given information.
Given that the ratio of the two numbers is 7:12, we can express the numbers in terms of a variable \( k \):
- \( x = 7k \)
- \( y = 12k \)

### Step 2: Use the product of the numbers.
We know that the product of the two numbers is 1344:
\[ x \cdot y = 1344 \]
Substituting the expressions for \( x \) and \( y \):
\[ (7k) \cdot (12k) = 1344 \]

### Step 3: Simplify the equation.
This simplifies to:
\[ 84k^2 = 1344 \]

### Step 4: Solve for \( k^2 \).
To isolate \( k^2 \), divide both sides by 84:
\[ k^2 = \frac{1344}{84} \]
Calculating the right side:
\[ k^2 = 16 \]

### Step 5: Solve for \( k \).
Taking the square root of both sides gives us:
\[ k = 4 \]

### Step 6: Find the values of \( x \) and \( y \).
Now we can find the two numbers:
- \( x = 7k = 7 \times 4 = 28 \)
- \( y = 12k = 12 \times 4 = 48 \)

### Step 7: Identify the smaller number.
The smaller of the two numbers is:
\[ \text{Smaller number} = x = 28 \]

Thus, the smaller of the two numbers is **28**.