A boy traveled by train which moved at the speed of 30 mph. He then boarded a bus that moved at the speed of 40 mph and reached his destination. The entire distance covered was 100 miles and the entire duration of the journey was 3 hours. Find the distance he traveled by bus.

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: \( \frac{40}{40} = 1 \)

Learn how to calculate the distance traveled by bus in a speed, distance, time problem. This problem involves using linear equations to find distances based on speeds and total journey time. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation