https://file.aiquickdraw.com/mathbot/file/1730649750487jg08l0q0.jpg

Let's solve this step-by-step.

### Step 1: Define the Variables
1. Let \( t \) be the time in hours after Lauren starts her journey until she overtakes Hannah.
2. Hannah starts her journey 2 hours before Lauren, so the time Hannah has been traveling when Lauren starts is 2 hours.

### Step 2: Write Equations for Distance
Since both trains are traveling towards the same destination, we can set up equations based on the distance each train covers by the time Lauren catches up.

#### Hannah's Distance
The distance Hannah has traveled when Lauren overtakes her is:
\[
\text{Distance of Hannah} = 40 \times (t + 2)
\]
where \( 40 \) mph is Hannah's speed and \( t + 2 \) is her total travel time.

#### Lauren's Distance
The distance Lauren has traveled when she overtakes Hannah is:
\[
\text{Distance of Lauren} = 60 \times t
\]
where \( 60 \) mph is Lauren's speed and \( t \) is her total travel time.

### Step 3: Set Up the Equation
Since Lauren overtakes Hannah, the distances they have traveled will be equal at that moment:
\[
40 \times (t + 2) = 60 \times t
\]

### Step 4: Solve the Equation
Expanding and solving for \( t \):
\[
40t + 80 = 60t
\]
\[
80 = 20t
\]
\[
t = 4
\]

### Step 5: Interpret the Result
Lauren will overtake Hannah 4 hours after Lauren starts her journey.

### Final Answer
**It will take 4 hours for Lauren to overtake Hannah.**

Would you like more details on any of the steps, or have any questions? 

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Here are some related questions that might deepen your understanding:
1. What would happen if both trains started at the same time?
2. How would the time change if Lauren's train speed was reduced to 50 mph?
3. If Hannah's train increased speed to 50 mph, how would that impact the time it takes Lauren to overtake her?
4. How far from London do they meet?
5. What if Lauren started 3 hours later instead of 2 hours?

**Tip:** Always set up equations by carefully defining each variable and step, especially in relative motion problems like this.

Hannah travels by train from London to Glasgow. The train has an average speed of 40 mph. Lauren catches a train two hours later from the same station to Glasgow. Her train travels at an average speed of 60 mph. How long will it be before Lauren’s train overtakes Hannah’s train?

Solve a real-world math problem involving relative speed, where two trains travel from the same station at different times and speeds. Learn how long it will take for the faster train to overtake the slower one. Suitable for Grades 8-10.

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