**Difficulty: Tricky**

A team of geologists are using the latest technology to drill into the Earth’s ocean floor and look at the rock underneath! They manage to drill a total of 1250 metres (based on a recent true story!). But the drilling gets harder the deeper they get. In this problem, the team tries to drill at a constant speed, but they find that every 500 metres, the drill suddenly goes 50% slower.

If it took them 75 days to complete the 1250-metre drill, how long did it take to drill to 500 metres?

## Need a hint?

We recommend drawing a picture of the drill hole and labelling the 500 metre segments. You’ll find that there are three different segments. What distance do they cover at the original speed, at half speed and at quarter speed?

Remember that if you’re going half the speed, it takes twice as long to go the same distance. And if you halve it again, it takes four times as long as at the start!

## Brainteaser answer

It took 15 days to reach the 500 metre mark.

We started by drawing a picture of the drill hole and dividing it into sections every 500 metres, measuring from the top:

Now we know that there are three segments with three different drill speeds, each ½ the speed of the previous segment on top.

You might have seen the equation speed equals distance divided by time (sometimes people use the word “rate” instead of “speed”, but they mean the same thing):

Speed = distance / time

Since we know the time (75 days), this equation tells us we need to find the total distance to find speed. The only problem is the speed changes between segments. You can find the original speed using algebra (see below), but we solved it with a little trick involving distances.

The first two segments are the same distance: 500 metres. We know that at half speed, the second segment will take twice as long. Another way to think about it is, keeping the original speed the same, it’s like the drill had to go twice as far or 1000 metres in the second segment.

This means that so far, at original speed, it’s like the drill has gone 500 + 1000 = 1500 metres.

By the same logic for the quarter speed segment, it’s as if the drill going the original speed had to go four times as far. The last segment is 250 metres, multiplied by four gives us 1000 metres.

This means that all up, at original speed, it’s like the drill has gone 1500 + 1000 metres, which is 2500 metres total.

The team was drilling at full speed for the first 500 metres, so that’s 500 ÷ 2500 = 1/5 of the total drilling time.

They drilled for 75 days, so that first 500 metres took:

1/5 of 75 = 15 days

We can double check our maths by calculating the total time and making sure it is 75 days.

Time = time of first segment + time of second segment + time of third segment

The second segment is twice as slow as the first, and it’s the same distance. The third segments is four times as slow, but it’s only half the distance. So our calculation becomes:

Time = 15 days + 15 days x 2 + 15 days x 4 ÷ 2

Time = 15 + 30 + 30 = 75 days

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