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 numbes is:

Options

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

Answer

To find the greater of two positive numbers given their geometric mean (G.M.) and arithmetic mean (A.M.), we can follow these steps:

### Step 1: Define the Variables
Let the two positive numbers be \( a \) and \( b \).

### Step 2: Set Up the Equations
From the problem, we know:
- The geometric mean \( G.M. = 35 \)
- The arithmetic mean \( A.M. = 43 \frac{3}{4} = \frac{175}{4} \)

Using the definitions of G.M. and A.M., we can set up the following equations:
1. \( \sqrt{a \cdot b} = 35 \)
2. \( \frac{a + b}{2} = \frac{175}{4} \)

### Step 3: Solve for \( a \cdot b \)
To eliminate the square root in the first equation, we square both sides:
\[ a \cdot b = 35^2 = 1225 \]

### Step 4: Solve for \( a + b \)
From the second equation, we can multiply both sides by 2:
\[ a + b = \frac{175}{4} \times 2 = \frac{175}{2} = 87.5 \]

### Step 5: Express \( b \) in Terms of \( a \)
From the equation \( a + b = 87.5 \), we can express \( b \) as:
\[ b = 87.5 - a \]

### Step 6: Substitute \( b \) into the Product Equation
Now, substitute \( b \) into the product equation \( a \cdot b = 1225 \):
\[ a \cdot (87.5 - a) = 1225 \]

### Step 7: Rearrange the Equation
Expanding and rearranging gives us:
\[ 87.5a - a^2 = 1225 \]
\[ -a^2 + 87.5a - 1225 = 0 \]
Multiplying through by -1:
\[ a^2 - 87.5a + 1225 = 0 \]

### Step 8: Use the Quadratic Formula
To solve for \( a \), we can use the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -87.5 \), and \( c = 1225 \):
\[ a = \frac{87.5 \pm \sqrt{(-87.5)^2 - 4 \cdot 1 \cdot 1225}}{2 \cdot 1} \]

### Step 9: Calculate the Discriminant
Calculating the discriminant:
\[ (-87.5)^2 = 7656.25 \]
\[ 4 \cdot 1 \cdot 1225 = 4900 \]
\[ 7656.25 - 4900 = 2756.25 \]

### Step 10: Calculate \( a \)
Now substituting back into the quadratic formula:
\[ a = \frac{87.5 \pm \sqrt{2756.25}}{2} \]
Calculating \( \sqrt{2756.25} = 52.5 \):
\[ a = \frac{87.5 \pm 52.5}{2} \]

### Step 11: Find the Two Possible Values for \( a \)
Calculating the two possible values:
1. \( a = \frac{140}{2} = 70 \)
2. \( a = \frac{35}{2} = 17.5 \)

### Step 12: Determine the Greater Value
Thus, the greater of the two numbers is:
\[ \text{Greater value} = 70 \]

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

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