A naughty student breaks the pencil in such a way that the ratio of two broken parts is same as that of the original length of the pencil to one of the larger part of the pencil. The ratio of the other part to the original length of pencil is

Options

A. `1:2sqrt(5)`
B. `2:(3+sqrt(5))`
C. `2:sqrt(5)`
D. can't be determined

Answer

To solve the problem, let's denote the original length of the pencil as \( L \). We will break the pencil into two parts: the larger part as \( x \) and the smaller part as \( y \).

### Step 1: Set up the relationship
According to the problem, the ratio of the two broken parts \( x \) and \( y \) is the same as the ratio of the original length \( L \) to the larger part \( x \). This can be expressed mathematically as:

\[ \frac{x}{y} = \frac{L}{x} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ x^2 = Ly \]

### Step 3: Express \( y \) in terms of \( x \) and \( L \)
From the equation \( x^2 = Ly \), we can express \( y \) as:

\[ y = \frac{x^2}{L} \]

### Step 4: Find the ratio of the smaller part \( y \) to the original length \( L \)
Now, we need to find the ratio of the smaller part \( y \) to the original length \( L \):

\[ \frac{y}{L} = \frac{\frac{x^2}{L}}{L} = \frac{x^2}{L^2} \]

### Step 5: Conclusion
Thus, the ratio of the other part (the smaller part \( y \)) to the original length of the pencil \( L \) is:

\[ \frac{y}{L} = \frac{x^2}{L^2} \]