Word Problems: Cars overtaking: 2-hour delay

Draw a diagram of the start of the slower car.

Draw a later diagram showing the situation when the faster car starts.

Draw a diagram of the final positions of the cars.

Define the variables for this problem.

The problem asks for a time, so an unknown is t, the time between the start of the slow car and the overtaking.

Is this the only variable we need?

No.

What else is unknown?

The distance, D, from the start to the overtake point.

Relate D to the distances traveled by each car.

D is equal to the distance travelled by each car.

How is each car's distance from the start related to the time?

For each car,
its distance = rate * time, or
its distance = (its speed) * (its time).

Is the time the same for each car?

No. As we have defined t, it is the time for the slower car.

Write an equation summarizing this information.

D = (30 * t)

Will the time for the faster car be greater or less than t?

Since the second car leaves 2 hours later, its time of travel will be 2 hours less than t.

Express the time for the second, faster car.

The time for the second car is t – 2 hours.

Relate this time, D, and the speed of the second car.

Its distance = rate * its time, or 40 * (t – 2) = D

So, we have two equations for our two unknowns:
Equation 1, and

Equation 2

How can we solve these?

We can eliminate D.

Do that.

We get

Distribute the multiplication.

Collect like terms.

Solve for t.

We get t = 8

What should be the units on this result?

Since the rates are in km per hour, the time we calculate from the rates should be in hours.

Using t = 8 hours, how can we find D?

We can use either equation 1 or equation 2.

Use equation 1 to find D.

Check by using equation 2.

Summarize these results.

8 hours after the slower car starts, the faster car has overtaken it, in spite of starting 2 hours later. Each car travels 240 km from the starting spot.

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