The G.M. of two positive numbers is 35 and the A.M. of the same number is 43 3/4, then the greater of these numbers is :

Options

A. 28
B. 30
C. 70
D. 35

Answer

To solve the problem, we need to find two positive numbers \( a \) and \( b \) given that their geometric mean (G.M.) is 35 and their arithmetic mean (A.M.) is \( 43 \frac{3}{4} \).

### Step-by-step Solution:

1. **Understanding the Means**:
- The geometric mean of two numbers \( a \) and \( b \) is given by:
\[ G.M. = \sqrt{ab} \]
- The arithmetic mean of the same two numbers is given by:
\[ A.M. = \frac{a + b}{2} \]

2. **Setting Up the Equations**:
- From the problem, we know:
\[ \sqrt{ab} = 35 \quad \Rightarrow \quad ab = 35^2 = 1225 \]
- And,
\[ \frac{a + b}{2} = 43 \frac{3}{4} = \frac{175}{4} \quad \Rightarrow \quad a + b = 2 \times \frac{175}{4} = \frac{350}{4} = 87.5 \]

3. **Using the Equations**:
- We now have two equations:
1. \( ab = 1225 \)
2. \( a + b = 87.5 \)

4. **Forming a Quadratic Equation**:
- Let \( a \) and \( b \) be the roots of the quadratic equation:
\[ x^2 - (a + b)x + ab = 0 \]
- Substituting the values we have:
\[ x^2 - 87.5x + 1225 = 0 \]

5. **Calculating the Discriminant**:
- The discriminant \( D \) of the quadratic equation is given by:
\[ D = (a + b)^2 - 4ab = (87.5)^2 - 4 \times 1225 \]
- Calculating \( (87.5)^2 \):
\[ (87.5)^2 = 7656.25 \]
- Now calculating \( 4 \times 1225 = 4900 \):
\[ D = 7656.25 - 4900 = 2756.25 \]

6. **Finding the Roots**:
- The roots can be found using the quadratic formula:
\[ x = \frac{-(a + b) \pm \sqrt{D}}{2} \]
- Substituting the values:
\[ x = \frac{87.5 \pm \sqrt{2756.25}}{2} \]
- Calculating \( \sqrt{2756.25} = 52.5 \):
\[ x = \frac{87.5 \pm 52.5}{2} \]
- This gives us two roots:
\[ x_1 = \frac{140}{2} = 70 \quad \text{and} \quad x_2 = \frac{35}{2} = 17.5 \]

7. **Identifying the Greater Number**:
- The greater of the two numbers \( a \) and \( b \) is:
\[ \text{Greater number} = 70 \]

### Final Answer:
The greater of the two numbers is **70**.