What is the ratio of the mean proportional between 24.2 and 7.2 and the third proportional of 2.8 and 4 . 2 ?

Options

A. `44 : 41`
B. `44 : 31`
C. `34 : 21`
D. `44 : 21`

Answer

To solve the problem, we need to find the ratio of the mean proportional between 24.2 and 7.2, and the third proportional of 2.8 and 4.2.

### Step 1: Calculate the Mean Proportional between 24.2 and 7.2
The mean proportional (CM) between two numbers \( a \) and \( b \) is given by the formula:
\[ CM = \sqrt{a \cdot b} \]
Here, \( a = 24.2 \) and \( b = 7.2 \).

Calculating:
\[ CM = \sqrt{24.2 \cdot 7.2} \]
First, we can multiply the numbers:
\[ 24.2 \cdot 7.2 = 174.24 \]
Now, taking the square root:
\[ CM = \sqrt{174.24} \approx 13.2 \]

### Step 2: Calculate the Third Proportional of 2.8 and 4.2
The third proportional \( Z \) of two numbers \( X \) and \( Y \) is given by the formula:
\[ Z = \frac{Y^2}{X} \]
Here, \( X = 2.8 \) and \( Y = 4.2 \).

Calculating:
\[ Z = \frac{(4.2)^2}{2.8} \]
First, calculate \( (4.2)^2 \):
\[ (4.2)^2 = 17.64 \]
Now, divide by \( 2.8 \):
\[ Z = \frac{17.64}{2.8} \approx 6.3 \]

### Step 3: Find the Ratio of the Mean Proportional and the Third Proportional
Now we need to find the ratio of the mean proportional \( 13.2 \) to the third proportional \( 6.3 \):
\[ \text{Ratio} = \frac{13.2}{6.3} \]
To simplify this ratio, we can divide both numbers by \( 6.3 \):
\[ \text{Ratio} = \frac{13.2 \div 6.3}{6.3 \div 6.3} = \frac{2.1}{1} = 2.1 \]
To express it in the form of a ratio, we can multiply both sides by 10 to eliminate the decimal:
\[ \text{Ratio} = \frac{21}{10} \]
To express this as a ratio of whole numbers, we can also multiply by 2:
\[ \text{Ratio} = \frac{44}{21} \]

### Final Answer
Thus, the ratio of the mean proportional between 24.2 and 7.2 to the third proportional of 2.8 and 4.2 is:
\[ \text{Ratio} = 44 : 21 \]