Find the fourth propotional to the numbers :
(i) 8, 36, 6
(ii) 5, 7, 30
(iii) 2, 8, 14, 3.5

Answer

To find the fourth proportional to the given numbers, we can use the concept of proportions. The fourth proportional to the numbers \( a, b, c \) is a number \( x \) such that:

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

This means that \( a \) is to \( b \) as \( c \) is to \( x \). We will solve each part step by step.

### Part (i): Find the fourth proportional to 8, 36, 6.

1. **Set up the proportion**:
\[ \frac{8}{36} = \frac{6}{x} \]

2. **Cross-multiply**:
\[ 8 \cdot x = 6 \cdot 36 \]

3. **Calculate \( 6 \cdot 36 \)**:
\[ 6 \cdot 36 = 216 \]

4. **Now we have**:
\[ 8x = 216 \]

5. **Solve for \( x \)**:
\[ x = \frac{216}{8} \]

6. **Simplify**:
\[ x = 27 \]

**Answer for Part (i)**: The fourth proportional is **27**.

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### Part (ii): Find the fourth proportional to 5, 7, 30.

1. **Set up the proportion**:
\[ \frac{5}{7} = \frac{30}{x} \]

2. **Cross-multiply**:
\[ 5 \cdot x = 30 \cdot 7 \]

3. **Calculate \( 30 \cdot 7 \)**:
\[ 30 \cdot 7 = 210 \]

4. **Now we have**:
\[ 5x = 210 \]

5. **Solve for \( x \)**:
\[ x = \frac{210}{5} \]

6. **Simplify**:
\[ x = 42 \]

**Answer for Part (ii)**: The fourth proportional is **42**.

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### Part (iii): Find the fourth proportional to 2, 8, 14.

1. **Set up the proportion**:
\[ \frac{2}{8} = \frac{14}{x} \]

2. **Cross-multiply**:
\[ 2 \cdot x = 8 \cdot 14 \]

3. **Calculate \( 8 \cdot 14 \)**:
\[ 8 \cdot 14 = 112 \]

4. **Now we have**:
\[ 2x = 112 \]

5. **Solve for \( x \)**:
\[ x = \frac{112}{2} \]

6. **Simplify**:
\[ x = 56 \]

**Answer for Part (iii)**: The fourth proportional is **56**.

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