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. `1:2sqrt5`
B. `2:(3+sqrt5)`
C. `2:sqrt5`
D. can't be determined

Answer

Let's break down the problem step by step.

### Step 1: Define Variables
Let the original length of the pencil be \( L \). When the pencil is broken, let the larger part be \( x \) and the smaller part be \( y \).

### Step 2: Set Up the Equation
According to the problem, the ratio of the two broken parts (larger part \( x \) and smaller part \( y \)) is the same as the ratio of the original length of the pencil \( L \) to the larger part \( x \). This can be expressed as:

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

### Step 3: Cross Multiply
To eliminate the fractions, we can cross-multiply:

\[ y \cdot x = L \cdot x \]

### Step 4: Simplify the Equation
Since \( x \) is not zero, we can divide both sides by \( x \):

\[ y = L \]

### Step 5: Find the Ratio of the Other Part to the Original Length
Now, we need to find the ratio of the smaller part \( y \) to the original length \( L \):

\[ \text{Ratio} = \frac{y}{L} = \frac{L}{L} = 1 \]

### Conclusion
The ratio of the other part (the smaller part) to the original length of the pencil is \( 1 \).

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