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# 17 Aug 2012 Print Numerical / Quantitative Reasoning Best Practices #5: How Far is Half Way?

Whilst talking to people preparing for quantitative / numerical reasoning recruitment tests, I have heard several of them mention that they get confused by a particular type of problem.

It is closely related to percentage increases and decreases but it manifests itself in a slightly different way. I think, at this point, that an example could shorten this article considerably.

How to master % questions

Q1. A train travels from London for 1h 5m at 10% below its average speed for the overall journey. How much above its overall average speed must it now travel to arrive on time?

Q2. A train travels from London for 91 miles at 10% below its average speed for the overall journey. How much above its overall average speed must it now travel to arrive on time?

Are these two questions the same? In the first question we have travelled for half the time at a slower speed and in the second question we have travelled for half the distance at a slower speed. Surely the answer to both is 10% isn’t it?

Let’s solve question 1!

The overall average speed is the distance divided by the time. The time can be converted from 2h 10m to 13/6 hours and then the average speed is:

We now travel for 13/12 hours at 10% below this speed which is:

This means we now have to travel (182-81.9)=100.1 miles in the remaining 1h 5m which would require a speed of:

This is 8.4mph, or 10%, faster than the overall average speed.

Now let’s solve question 2!

The overall average speed is exactly the same at 84mph. We then travel for 91 miles at 10% below this speed which is:

We need to convert the overall time to decimal which is calculated as mins/60+hours:

Now we can see that we have 2.1666-1.2037=0.963 hours left to travel the remaining 91 miles. This needs an average speed of:

This is 10.5mph faster than the overall average which is 12.5% higher and not 10%. So, they are not the same. Why not?

The solution

We are travelling the first part of the journey below average speed so it will take more than half the time to travel half the distance. In question 1 we decide to speed up after half the time has elapsed as so have the other half to make up the deficit whereas in question 2 we speed up after half the distance has elapsed having used up more than half of the time. As we spent longer at the slower speed we must now average a higher speed to catch up.

Make sure you read the question carefully and you understand what it is that you need to solve.

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