The ratio of the mean proportional between 0.7 and 2.8 and the third proportional to 2.5 and 3.5 is:

Options

A. `7:2`
B. `2:7`
C. `7:3`
D. `3:7`

Answer

To solve the problem, we need to find the ratio of the mean proportional between 0.7 and 2.8 to the third proportional to 2.5 and 3.5. Let's break this down step by step.

### Step 1: Find the Mean Proportional
The mean proportional \( x \) between two numbers \( a \) and \( b \) can be found using the formula:
\[ x^2 = a \cdot b \]
In our case, \( a = 0.7 \) and \( b = 2.8 \).

Calculating:
\[ x^2 = 0.7 \cdot 2.8 \]
\[ x^2 = 1.96 \]
Taking the square root of both sides:
\[ x = \sqrt{1.96} = 1.4 \]

### Step 2: Find the Third Proportional
The third proportional \( y \) to two numbers \( a \) and \( b \) can be found using the formula:
\[ \frac{a}{b} = \frac{b}{y} \]
Rearranging gives:
\[ y = \frac{b^2}{a} \]
In our case, \( a = 2.5 \) and \( b = 3.5 \).

Calculating:
\[ y = \frac{(3.5)^2}{2.5} \]
\[ y = \frac{12.25}{2.5} \]
\[ y = 4.9 \]

### Step 3: Find the Ratio of Mean Proportional to Third Proportional
Now we need to find the ratio of \( x \) to \( y \):
\[ \text{Ratio} = \frac{x}{y} = \frac{1.4}{4.9} \]
To simplify this ratio, we can convert it into a fraction:
\[ \frac{1.4}{4.9} = \frac{14}{49} \]
Now, simplifying \( \frac{14}{49} \):
\[ \frac{14 \div 7}{49 \div 7} = \frac{2}{7} \]

### Final Answer
Thus, the ratio of the mean proportional between 0.7 and 2.8 to the third proportional to 2.5 and 3.5 is:
\[ \text{Ratio} = 2 : 7 \]