Distance, rate, time word problems Worksheets - Free Printable

Let's solve the problem step by step.

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Problem Statement:

- A passenger train leaves the station and travels at 29.2 km/h.
- 2.4 hours later, a cattle train leaves the same station, traveling in the same direction at 36.5 km/h.
- We are to find: How long did the passenger train travel before the cattle train caught up?

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Step 1: Define Variables

Let:
- $ t $ = time (in hours) that the cattle train traveled until it caught up with the passenger train.

Then, since the passenger train left 2.4 hours earlier, it has been traveling for:
- $ t + 2.4 $ hours when the cattle train catches up.

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Step 2: Use Distance Formula

Distance = Speed × Time

When the cattle train catches up, both trains have traveled the same distance from the station.

So:

$$
\text{Distance}_{\text{passenger}} = \text{Distance}_{\text{cattle}}
$$

$$
29.2 \times (t + 2.4) = 36.5 \times t
$$

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Step 3: Solve the Equation

$$
29.2(t + 2.4) = 36.5t
$$

Expand the left side:

$$
29.2t + 29.2 \times 2.4 = 36.5t
$$

Calculate $ 29.2 \times 2.4 $:

$$
29.2 \times 2.4 = (29.2 \times 2) + (29.2 \times 0.4) = 58.4 + 11.68 = 70.08
$$

So:

$$
29.2t + 70.08 = 36.5t
$$

Subtract $ 29.2t $ from both sides:

$$
70.08 = 36.5t - 29.2t = 7.3t
$$

Now divide:

$$
t = \frac{70.08}{7.3}
$$

Calculate:

$$
t = 9.6 \text{ hours}
$$

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Step 4: Find How Long the Passenger Train Traveled

The passenger train had a 2.4-hour head start, and the cattle train took 9.6 hours to catch up.

So, total time the passenger train traveled:

$$
t + 2.4 = 9.6 + 2.4 = 12 \text{ hours}
$$

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✔ Final Answer:

$$
\boxed{12} \text{ hours}
$$

The passenger train traveled for 12 hours before the cattle train caught up.

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🔍 Explanation:

Even though the cattle train is faster, it started later. The passenger train had a 2.4-hour head start, covering some distance first. The cattle train closes the gap at a rate of $ 36.5 - 29.2 = 7.3 $ km/h. The initial lead was $ 29.2 \times 2.4 = 70.08 $ km. At a closing speed of 7.3 km/h, it takes $ 70.08 / 7.3 = 9.6 $ hours for the cattle train to catch up. So the passenger train traveled $ 9.6 + 2.4 = 12 $ hours total.