Find the third proportional to:
8 and 12
12 and 18
4.5 and 6

Answer

To find the third proportional to the given pairs of numbers, we can use the property of continued proportion. If \( A \), \( B \), and \( C \) are in continued proportion, then:

\[ \frac{A}{B} = \frac{B}{C} \]

From this, we can derive that:

\[ C = \frac{B^2}{A} \]

Now, let's solve the three pairs step by step.

### 1. Finding the third proportional to 8 and 12

Let \( A = 8 \) and \( B = 12 \). We need to find \( C \).

Using the formula:

\[ C = \frac{B^2}{A} = \frac{12^2}{8} \]

Calculating \( 12^2 \):

\[ 12^2 = 144 \]

Now substituting back:

\[ C = \frac{144}{8} = 18 \]

**Answer for the first pair:** The third proportional to 8 and 12 is **18**.

### 2. Finding the third proportional to 12 and 18

Let \( A = 12 \) and \( B = 18 \). We need to find \( C \).

Using the formula:

\[ C = \frac{B^2}{A} = \frac{18^2}{12} \]

Calculating \( 18^2 \):

\[ 18^2 = 324 \]

Now substituting back:

\[ C = \frac{324}{12} = 27 \]

**Answer for the second pair:** The third proportional to 12 and 18 is **27**.

### 3. Finding the third proportional to 4.5 and 6

Let \( A = 4.5 \) and \( B = 6 \). We need to find \( C \).

Using the formula:

\[ C = \frac{B^2}{A} = \frac{6^2}{4.5} \]

Calculating \( 6^2 \):

\[ 6^2 = 36 \]

Now substituting back:

\[ C = \frac{36}{4.5} \]

To simplify \( \frac{36}{4.5} \), we can multiply the numerator and denominator by 10 to eliminate the decimal:

\[ C = \frac{360}{45} \]

Now simplifying \( \frac{360}{45} \):

\[ C = 8 \]

**Answer for the third pair:** The third proportional to 4.5 and 6 is **8**.

### Summary of Answers:
1. The third proportional to 8 and 12 is **18**.
2. The third proportional to 12 and 18 is **27**.
3. The third proportional to 4.5 and 6 is **8**.