The sum of third, fourth and fifth part of a number exceeds half of the number by 34. Find the number.

Options

A. 60
B. 120
C. 30
D. None of these

Answer

To solve the problem, we need to find a number based on the information given about its parts. Let's denote the unknown number as \( x \).

### Step 1: Define the parts of the number
The third part of the number is \( \frac{x}{3} \), the fourth part is \( \frac{x}{4} \), and the fifth part is \( \frac{x}{5} \).

### Step 2: Write the equation based on the problem statement
According to the problem, the sum of the third, fourth, and fifth parts exceeds half of the number by 34. This can be expressed mathematically as:
\[ \frac{x}{3} + \frac{x}{4} + \frac{x}{5} = \frac{x}{2} + 34 \]

### Step 3: Find a common denominator
To solve the equation, we need to find a common denominator for the fractions on the left side. The least common multiple of 3, 4, and 5 is 60. Therefore, we can rewrite the equation as:
\[ \frac{20x}{60} + \frac{15x}{60} + \frac{12x}{60} = \frac{30x}{60} + 34 \]

### Step 4: Combine the fractions
Now, combine the fractions on the left side:
\[ \frac{20x + 15x + 12x}{60} = \frac{30x}{60} + 34 \]
This simplifies to:
\[ \frac{47x}{60} = \frac{30x}{60} + 34 \]

### Step 5: Eliminate the fractions
Multiplying the entire equation by 60 to eliminate the denominators gives:
\[ 47x = 30x + 2040 \]

### Step 6: Solve for \( x \)
Now, isolate \( x \) by subtracting \( 30x \) from both sides:
\[ 47x - 30x = 2040 \]
\[ 17x = 2040 \]
Now, divide both sides by 17:
\[ x = \frac{2040}{17} = 120 \]

### Conclusion
The number is \( 120 \).

---