If `[4- (5)/(1 + (1)/(3 + (1)/(2 + (1)/(4))))]^(th)` part of a journey takes 10 minutes, then to complete `(3)/(5)` th of that journey, it will take

Options

A. 40 minutes
B. 45 minutes
C. 48 minutes
D. 36 minutes

Answer

To solve the problem step by step, we need to simplify the expression and then calculate the time required for \( \frac{3}{5} \) of the journey.

### Step 1: Simplify the Expression
We start with the expression:
\[ 4 - \frac{5}{1 + \frac{1}{3 + \frac{1}{2 + \frac{1}{4}}}} \]

First, we will simplify the innermost fraction:
\[ 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]

Now we substitute this back into the expression:
\[ 3 + \frac{1}{\frac{9}{4}} = 3 + \frac{4}{9} = \frac{27}{9} + \frac{4}{9} = \frac{31}{9} \]

Next, we substitute this back into the expression:
\[ 1 + \frac{1}{\frac{31}{9}} = 1 + \frac{9}{31} = \frac{31}{31} + \frac{9}{31} = \frac{40}{31} \]

Now we substitute this back into the original expression:
\[ 4 - \frac{5}{\frac{40}{31}} = 4 - \frac{5 \times 31}{40} = 4 - \frac{155}{40} \]

Convert 4 into a fraction with a denominator of 40:
\[ 4 = \frac{160}{40} \]
Thus, we have:
\[ \frac{160}{40} - \frac{155}{40} = \frac{5}{40} = \frac{1}{8} \]

### Step 2: Determine the Time for \( \frac{3}{5} \) of the Journey
Since \( \frac{1}{8} \) of the journey takes 10 minutes, we need to find out how long \( \frac{3}{5} \) of the journey will take.

First, we find the total time for the whole journey:
\[ 1 \text{ part of the journey} = 10 \text{ minutes} \times 8 = 80 \text{ minutes} \]

Now, we calculate the time for \( \frac{3}{5} \) of the journey:
\[ \frac{3}{5} \text{ of the journey} = \frac{3}{5} \times 80 \text{ minutes} = \frac{240}{5} = 48 \text{ minutes} \]

### Final Answer
The time to complete \( \frac{3}{5} \) of the journey is **48 minutes**.

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