# Average speed and percentage

By Murray Bourne, 08 May 2009

This kind of puzzle always catch out students.

I travel to work at 30 km/h, but on the way home there is less traffic so I can travel at 60 km/h. What is my average speed?

And the second one is also an old favourite.

I have a jar of water and I remove 10% of it. Then my friend comes along and increases the amount of water by 10%. Is there more, the same or less water in the jar than when we started?

I'll offer a prize to the first correct answers with good reasoning. What prize?

You will get a warm inner glow for having achieved something. π

See the 10 Comments below.

8 May 2009 at 2:57 am [Comment permalink]

Those are great examples of simple problems that are commonly misunderstood.

Here's a blog post by Jason Cohen that answers your first question.

This doesn't exactly answer your second question, but here's a post on a similar problem with percentages, explaining why a wrong approach often gives pretty good answers.

8 May 2009 at 6:13 am [Comment permalink]

The thing I always find surprising is that, in problem #2, if your friend had added his 10% first, and then you took away 10%, you would still end up with the same amount. It doesn't matter if the increase (or sales tax) or the decrease (discount) is calculated first --- the final total is still the same.

8 May 2009 at 6:39 am [Comment permalink]

First question. Obvious but wrong answer is 45 km/h. Why? Let's work it out so it's obvious. Let's say the distance to work is 60 km. Thus the way to work and back is 120 km. How many hours would this take, as stated?

At 30 km/h, we can cover the way to work in 2 hours. At 60 km/h, it takes 1 hour. So we take 3 hours to cover 120 km. Thus our average speed is 120 km / 3 h = 40 km/h.

We can work this out with any distance and get the same result. Say work is only 30 km away. We get to work in 1 hour, and back in 1/2 hour. Thus, 60 km / 1.5 hours = 40 km/h.

I think that's all that's needed to justify the answer of 40 km/h average speed. We can't just add the speeds and divide by two, we must take into account the total amount of time taken.

Second one is simple to solve if you think it through. There is less water than what we started with. Say we start with 100 mL of water. Remove 10%, which is 10 mL, resulting in 90 mL. Now, add 10% *of that*. 10% of 90 mL is 9 mL. 90 mL + 9 mL = 99 mL, which is less than the 100 mL we started with.

Hope it's ok that I did this numerically in this way, but I thought this was the clearest way to demonstrate what's going on so anyone can understand.

8 May 2009 at 8:28 am [Comment permalink]

1>If we assume that distance from home to office is 30 kms, than the u have taken 1.5 hours to cross 60kms.

average speed=(60kms/1.5 hours)=40km/h

2>less.because 10% of 90% filled jar is lesser than 10% of the full filled jar

9 May 2009 at 2:07 am [Comment permalink]

Thanks for your answers!

Note:I published all these replies at once, so the later responders had not seen the earlier replies.John:Thanks for the interesting links.Denise, Mike and Chethan:Thanks also for your inputs.Just to clarify Denise's answer (because I mis-read it first time):

As Mike correctly said, you'll end up with 99% of the original amount if you take 10% off first and then add 10% to the remaining amount later.

Denise meant that if you add 10% first you have 110 mL (to use Mike's figures) and then reduce that by 10%, you will be left with 99 mL (or 99% of what you started with). So it's the same whether you subtract first then add or add first then subtract.

Hope each of you are enjoying your warm inner glows! π

9 May 2009 at 3:28 pm [Comment permalink]

The second problem can be factored into the form

x * (1+p1) * (1+p2) * ...

Where x is the original amount, p1, p2 are any fraction you add (positive) or subtract (negative). It doesn't take much to see from commutativity of multiplication that the order of adding or subtracting doesn't matter. In fact it doesn't matter how many times you do these operations either.

25 Jun 2009 at 11:53 pm [Comment permalink]

For The second problem i can give you a formula....

consider'-'for the removal of quantity and

'+'for the increasing

-x+y-x*y/100

-10+10-10*10/100

-1%

this means you have a loss of 1% of

water in your jar....ok...

19 Aug 2009 at 2:35 pm [Comment permalink]

1.60=30=90 divided by 2 which is 45.

2.X-10%+10%=X so you end up with the same amount of water in the jar.

How old are the kids your teaching this,isn't advanced enough for them.

19 Aug 2009 at 5:38 pm [Comment permalink]

Tassneem: Presumably you mean:

60+30=90

90 divided by 2 which is 45. However, this is not the correct answer.

For the second one, the answer is not "the same amount".

Seems that the questions are indeed "advanced enough"!

8 Feb 2012 at 7:56 am [Comment permalink]

Let t be the time taken to cover the distance with 60 kmph.

Then 2t is the time taken to cover the distance with 30 kmph.

The total distance covered is 60t km, t being in hours.

(At one time only)

When you come and go back, it is 120t km.

You spent 3t time (2t+t) covering a distance of 120t km. Hence, your average velocity is 120t km/3t h= 40 kmph.