To find the distance the boy traveled by bus, we can use a system of equations based on the information provided.

Step 1: Define Variables

Let's define the following variables:

• Let $d_t$ be the distance traveled by train (in miles).
• Let $d_b$ be the distance traveled by bus (in miles).

Given that the entire distance is 100 miles, we have:

$d_t + d_b = 100$

Step 2: Express Time in Terms of Distance and Speed

• The speed of the train is 30 mph. Therefore, the time taken by the train is:

$\text{Time by train} = \frac{d_t}{30}$

• The speed of the bus is 40 mph. Therefore, the time taken by the bus is:

$\text{Time by bus} = \frac{d_b}{40}$

Given that the entire journey took 3 hours, we have:

$\frac{d_t}{30} + \frac{d_b}{40} = 3$

Step 3: Solve the System of Equations

We have the following two equations:

1. $d_t + d_b = 100$
2. $\frac{d_t}{30} + \frac{d_b}{40} = 3$

From the first equation, express $d_t$ in terms of $d_b$:

$d_t = 100 - d_b$

Substitute this into the second equation:

$\frac{100 - d_b}{30} + \frac{d_b}{40} = 3$

Step 4: Simplify and Solve for $d_b$

To eliminate the fractions, find a common denominator, which is 120:

$\frac{100 - d_b}{30} = \frac{4(100 - d_b)}{120} = \frac{400 - 4d_b}{120}$ $\frac{d_b}{40} = \frac{3d_b}{120}$

Now, substitute back into the equation:

$\frac{400 - 4d_b}{120} + \frac{3d_b}{120} = 3$

Combine the fractions:

$\frac{400 - 4d_b + 3d_b}{120} = 3$ $\frac{400 - d_b}{120} = 3$

Multiply both sides by 120 to clear the fraction:

$400 - d_b = 360$

$d_b = 40$

Step 5: Find $d_t$ for Verification

Since $d_t + d_b = 100$:

$d_t = 100 - 40 = 60$

Verification

1. Time by train: $\frac{60}{30} = 2$ hours
2. Time by bus: $\frac{40}{40} = 1$