The ratio of third proportional to 21 and 42 and mean proportinal to 16 and 49 is

Options

A. `3:1`
B. `2:3`
C. `4:3`
D. `1:3`

Answer

To solve the problem, we need to find the ratio of the third proportional to 21 and 42, and the mean proportional to 16 and 49. Let's break this down step by step.

### Step 1: Find the Third Proportional to 21 and 42

The third proportional \( x \) to two numbers \( a \) and \( b \) can be found using the property of continued proportion. If \( a, b, x \) are in continued proportion, then:

\[ \frac{a}{b} = \frac{b}{x} \]

Substituting \( a = 21 \) and \( b = 42 \):

\[ \frac{21}{42} = \frac{42}{x} \]

### Step 2: Cross Multiply to Solve for \( x \)

Cross multiplying gives us:

\[ 21 \cdot x = 42 \cdot 42 \]

Calculating \( 42 \cdot 42 \):

\[ 42 \cdot 42 = 1764 \]

So we have:

\[ 21x = 1764 \]

### Step 3: Solve for \( x \)

Now, divide both sides by 21:

\[ x = \frac{1764}{21} = 84 \]

### Step 4: Find the Mean Proportional to 16 and 49

The mean proportional \( y \) between two numbers \( a \) and \( b \) can be found using the formula:

\[ y = \sqrt{a \cdot b} \]

Substituting \( a = 16 \) and \( b = 49 \):

\[ y = \sqrt{16 \cdot 49} \]

### Step 5: Calculate \( y \)

Calculating \( 16 \cdot 49 \):

\[ 16 \cdot 49 = 784 \]

Now, take the square root:

\[ y = \sqrt{784} = 28 \]

### Step 6: Find the Ratio of \( x \) and \( y \)

Now we need to find the ratio of \( x \) to \( y \):

\[ \text{Ratio} = \frac{x}{y} = \frac{84}{28} \]

### Step 7: Simplify the Ratio

To simplify \( \frac{84}{28} \):

\[ \frac{84}{28} = 3 \]

Thus, the ratio of the third proportional to 21 and 42, and the mean proportional to 16 and 49 is:

\[ \text{Ratio} = 3:1 \]