In a race of three horses, the first beat the second by $11$ metres and the third by $90$ metres. If the second beat the third by $80$ metres, what was the length, in metres, of the racecourse?

Entered answer:

✅ Correct Answer: 880

We have three horses in a race with the following results:

First horse beats the second horse by 11 metres

First horse beats the third horse by 90 metres

Second horse beats the third horse by 80 metres

We need to find the length of the racecourse.

Let's think about what happens when the first horse finishes the race:

When the first horse crosses the finish line:

The second horse is 11 metres behind the finish line

The third horse is 90 metres behind the finish line

This means the second horse is actually 79 metres ahead of the third horse at this moment.

Since the first horse beat the second by $11$ m and the third by $90$ m, the difference between second and third positions is: $90 - 11 = 79$ metres.

Here's where it gets interesting:

We know the second horse beats the third by 80 metres when the second horse finishes

But the second horse is already 79 metres ahead of the third horse

So the second horse needs to gain only 1 more metre on the third horse

Key Question: In what distance does the second horse gain 1 metre on the third horse?

From our earlier observation, when the first horse runs the final 11 metres to finish:

The second horse covers some distance less than 11 metres

The third horse covers even less distance

The second horse gains 1 metre on the third horse in this span

Since the first horse gains 11 metres on the second horse over the entire race, and in this same race the second horse gains 1 metre on the third horse, we can set up a proportion:

If the second horse gains 1 metre on the third horse when the first horse gains 11 metres on the second horse, then:

For the second horse to gain 80 metres on the third horse, the distance would be:

$80 \times 11 = 880$ metres

Therefore, the length of the racecourse is 880 metres.

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In a race of three horses, the first beat the second by $11$ metres and the third by $90$ metres. If the second beat the third by $80$ metres, what was the length, in metres, of the racecourse?