One third of a pole is painted yellow, one-fifth is painted white and the remaining 7 metres is painted black. The length of the pole is :

Options

A. 15m
B. 30 m
C. `10(7)/(15)m`
D. `7(1)/(15)m`

Answer

To solve the problem step by step, let's denote the length of the pole as \( x \) meters.

### Step 1: Set up the equation
According to the problem, one third of the pole is painted yellow, one fifth is painted white, and the remaining part (7 meters) is painted black. We can express this mathematically:
- Length painted yellow = \( \frac{x}{3} \)
- Length painted white = \( \frac{x}{5} \)
- Length painted black = 7 meters

The total length of the pole can be expressed as:
\[ \frac{x}{3} + \frac{x}{5} + 7 = x \]

### Step 2: Find a common denominator
To combine the fractions, we need a common denominator. The least common multiple (LCM) of 3 and 5 is 15. We will rewrite the fractions:
\[ \frac{x}{3} = \frac{5x}{15} \quad \text{and} \quad \frac{x}{5} = \frac{3x}{15} \]

### Step 3: Substitute back into the equation
Now substitute these back into the equation:
\[ \frac{5x}{15} + \frac{3x}{15} + 7 = x \]

### Step 4: Combine the fractions
Combine the fractions on the left side:
\[ \frac{5x + 3x}{15} + 7 = x \]
This simplifies to:
\[ \frac{8x}{15} + 7 = x \]

### Step 5: Eliminate the fraction
To eliminate the fraction, we can multiply the entire equation by 15:
\[ 15 \left(\frac{8x}{15}\right) + 15 \cdot 7 = 15x \]
This simplifies to:
\[ 8x + 105 = 15x \]

### Step 6: Rearrange the equation
Now, rearranging the equation to isolate \( x \):
\[ 105 = 15x - 8x \]
This simplifies to:
\[ 105 = 7x \]

### Step 7: Solve for \( x \)
Now, divide both sides by 7:
\[ x = \frac{105}{7} = 15 \]

### Conclusion
The length of the pole is \( 15 \) meters.