Two whole numbers are such that the square of first number exceeds the second by 112 and the ratio of the numbers is 4:3. What is the value of smaller number?

Options

A. 3
B. 4
C. 12
D. 36

Answer

To solve the problem step by step, we will follow the given conditions and derive the values of the two whole numbers.

### Step 1: Define the numbers
Let the two whole numbers be represented as:
- First number = 4x
- Second number = 3x

### Step 2: Set up the equation based on the problem statement
According to the problem, the square of the first number exceeds the second number by 112. This can be expressed mathematically as:
\[ (4x)^2 = 3x + 112 \]

### Step 3: Simplify the equation
Now, we will simplify the equation:
\[ 16x^2 = 3x + 112 \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ 16x^2 - 3x - 112 = 0 \]

### Step 5: Solve the quadratic equation
Now, we will use the quadratic formula to solve for \(x\):
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \(16x^2 - 3x - 112 = 0\), we have:
- \(a = 16\)
- \(b = -3\)
- \(c = -112\)

Calculating the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 16 \cdot (-112) = 9 + 7168 = 7177 \]

Now substituting into the quadratic formula:
\[ x = \frac{3 \pm \sqrt{7177}}{32} \]

### Step 6: Approximate the value of \(x\)
Calculating \(\sqrt{7177}\) gives us approximately \(84.7\). Therefore:
\[ x \approx \frac{3 + 84.7}{32} \quad \text{(only considering the positive root)} \]
\[ x \approx \frac{87.7}{32} \approx 2.74 \]

### Step 7: Find the smaller number
Now we can find the smaller number:
\[ \text{Smaller number} = 3x \approx 3 \cdot 2.74 \approx 8.22 \]

Since we need whole numbers, we round \(x\) to the nearest whole number, which is \(3\):
\[ x = 3 \]
Thus, the smaller number:
\[ 3x = 3 \cdot 3 = 9 \]

### Final Answer
The value of the smaller number is **9**.