A train covers a certain distance at uniform speed. If speed had been 10 kmph faster time taken eould be 2 hrs less but if it would be 10 kmph slower, 3 hrs more time would be taken. Find distance covered. Solve in one variable

Let's denote:

- \( d \) as the distance covered by the train (in kilometers).
- \( s \) as the original speed of the train (in km/h).

Given conditions:
1. If the speed had been 10 km/h faster, the time taken would be 2 hours less.
2. If the speed had been 10 km/h slower, the time taken would be 3 hours more.

Using the relationship \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

### First Condition:
- Original time taken at speed \( s \) is \( \frac{d}{s} \).
- New time taken at speed \( s + 10 \) is \( \frac{d}{s + 10} \).
- According to the condition, \( \frac{d}{s} - \frac{d}{s + 10} = 2 \).

### Second Condition:
- New time taken at speed \( s - 10 \) is \( \frac{d}{s - 10} \).
- According to the condition, \( \frac{d}{s - 10} - \frac{d}{s} = 3 \).

### Formulating Equations:
1. \( \frac{d}{s} - \frac{d}{s + 10} = 2 \)
   \[
   \Rightarrow \frac{d(s + 10) - ds}{s(s + 10)} = 2 \quad \text{(Multiply both sides by } s(s + 10) \text{)}
   \]
   \[
   \Rightarrow \frac{10d}{s(s + 10)} = 2
   \]
   \[
   \Rightarrow 5d = s(s + 10)
   \]
   \[
   \Rightarrow s^2 + 10s - 5d = 0 \quad \text{(Equation 1)}
   \]

2. \( \frac{d}{s - 10} - \frac{d}{s} = 3 \)
   \[
   \Rightarrow \frac{d(s - 10) - ds}{(s - 10)s} = 3 \quad \text{(Multiply both sides by } s(s - 10) \text{)}
   \]
   \[
   \Rightarrow \frac{-10d}{s(s - 10)} = 3
   \]
   \[
   \Rightarrow -10d = 3s(s - 10)
   \]
   \[
   \Rightarrow 3s^2 - 30s + 10d = 0 \quad \text{(Equation 2)}
   \]

### Solving the System of Equations:
From Equation 1: \( d = \frac{s^2 + 10s}{5} \)

Substitute this into Equation 2:
\[
3s^2 - 30s + 10\left(\frac{s^2 + 10s}{5}\right) = 0
\]
\[
\Rightarrow 3s^2 - 30s + 2(s^2 + 10s) = 0
\]
\[
\Rightarrow 3s^2 - 30s + 2s^2 + 20s = 0
\]
\[
\Rightarrow 5s^2 - 10s = 0
\]
\[
\Rightarrow 5s(s - 2) = 0
\]
\[
\Rightarrow s = 0 \quad \text{or} \quad s = 2
\]


Explore how to solve a problem involving a train's speed and time relationships using quadratic equations. Understand how changes in speed affect travel time, demonstrated step-by-step with algebraic methods. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation