A pipe is cut into three equal parts. The lengths are given as $0$, $30_4$, $x_6$, and $121_5$. We need to find the value of $x$.

1. Problem Description

A pipe is cut into three equal parts. The lengths are given as $0$, $30_4$, $x_6$, and $121_5$. We need to find the value of $x$.

2. Solution Steps

First, convert the given numbers to base

1

0. $30_4 = 3 \times 4^1 + 0 \times 4^0 = 12 + 0 = 12$

$121_5 = 1 \times 5^2 + 2 \times 5^1 + 1 \times 5^0 = 25 + 10 + 1 = 36$

Since the pipe is cut into three equal parts, we can say that the total length is three times the length of one part.

The length of one part in base 10 is $30_4 - 0 = 12 - 0 = 12$.

Since there are three equal parts, the total length is $3 \times 12 = 36$. This matches the last point $121_5 = 36$.

The value of $x_6$ represents the length of two parts.

So, the length from 0 to $x_6$ is $2 \times 12 = 24$.

Now, we need to convert 24 to base

6. $24 = a \times 6^1 + b \times 6^0$

$24 = 4 \times 6 + 0$

So, $x_6 = 40_6$. Therefore, $x = 40$.

3. Final Answer

40