Money Quiz: Can you work it out? Poll results/discussion

MSE_Martin

This was originally posted in the 2007 quiz thread:

The answer is C.

I was going to write a long explanation about the commutative nature of maths and the fact that 25% off a value always had a bigger impact than 25% added to a value.

Yet as King Weasel did it so well I thought I would simply use his...

"So many numbers!

I hope the mathematicians didn't have to get out their spreadsheets to answer this one.

There are two elementary arithmetical principles here that tell us, FIRST, that the values of A and B must be equal and, SECONDLY, that the value of A and B must be less than the value of C. (And the logicians wll tell us that you don't need the first principle to arrive at the answer if you use the second principle twice.)

• FIRST PRINCIPLE: which is bigger - 7 x 6 or 6 x 7 ? You don't need to know your times tables to answer this: you can multiply numbers in any order you like without altering the result. The value of A is your original investment - say, £100 - times 1.05 ^ 4 (or 1.05 x 1.05 x 1.05 x 1.05) and then times 0.95 ^ 4. The value of B is £100 times 0.95 ^ 4 then times 1.05 ^ 4. These answers must be the same.

• SECOND PRINCIPLE: which is bigger - 10% of £10 or 10% of £20? (Again, you don't need to know what 10% of £10 or £20 actually is.) This is basically what alexjohnson was trying to tell us. I think he's right. So if you add a percentage - 5%, in this case - to your original investment and then deduct the SAME percentage off your new and higher value you are bound to finish up with less. (Take an extreme example if it helps: try increasing your investment by 100% and then reducing it by 100%.) So £100 x 1.05 x 0.95 must be less than £100. And of course you get the same result if you knock the 5% off first and then add it back, as in situation B.

If you can't follow this, get your spreadsheet out after all. You will get an answer, but may not know why."

For those who simply want a numerical answer, here it is:

• Option A. Rises 5%/year for 4 years then drops 5%/year for 4 years. END RESULT: 99% of the start value

• Option B. falls 5%/year for 4 years then rises 5%/year for 4 years. END RESULT: 99% of the start value

• Option C. Stays the same. END RESULT: 100% of the start value

• Option D. All of the above. NOT TRUE AS A, B & C produce different answers.

The fact so few people got this right (as in 2005) shows the worry about how financial marketeers use maths to confuse!

Martin

ScoobieGirl

Somewhere in my brain I knew this but I completly failed to apply the principle. So for idiots like me please can we have other practical applications where this principle applies - i.e. 25% off the price is better than 25% extra free?

50% off is better than bogof?

MSE_Martin

Lets do it very very simply....

Forget how many times you do it - lets just do it once.

You always buy a four pack of baked beans for £1...

The supermarket puts the price up by 50%

Now it costs you £1.50

Then six months later they say 50% off...

Now it only costs you 75p

This is because 50% off a bigger number (the £1.50) is always more than 50% on a smaller number (the original £1).

This would also work the other way round

If the original was a price cut of 50% it'd drop to 50p and then the price rises by 50% it'd then still cost 75p...

Martin

King_Weasel

ScoobieGirl wrote: »

Somewhere in my brain I knew this but I completly failed to apply the principle. So for idiots like me please can we have other practical applications where this principle applies - i.e. 25% off the price is better than 25% extra free?

50% off is better than bogof?

These questions aren't quite the same as Martin's: you are dealing with prices (P) and quantities (Q) and want to minimise the ratio of price to quantity (P/Q).

The general case you are asking is whether the ratio P2/Q1 is lower than the ratio P1/Q2 where

P2 = P1 * (1-a)
and
Q2 = Q1 * (1+a).

(In your example, a = 0.25.) OK so far?

In the simplest case you can set both P1 and Q1 as equal to 1.

So

P2/Q1 = 1-a

and

P1/Q2 = 1 / (1+a)

Which is smaller (and therefore cheapest)?

It's probably simplest to think of a simple arithmetical example, such as a = 0.5 :

P2/Q1 = 1/2 (0.5)

P1/Q2 = 1 / 1.5 = 2/3 (0.67)

Answer: P2/Q1 is lower.

So a given % off the price is cheaper than the same % increase in the quantity.

Maybe there's a simpler way of getting to this conclusion, but I hope you followed it.

In your second question the choice is between halving the price and doubling the quantity. Right? Both these halve the original unit price, but I suppose 50% off the price is to be preferred because you only need to buy one.

druss_the_axeman

To think Martin said "how financial marketeers use maths to confuse!"

Ratios, percentages, and algebra, are you trying to scare folks King Weasel ? ;-)

Druss

ScoobieGirl

King Weasel - thank you for taking the time to type that out. It does make sense although I did have to read it twice

King_Weasel

druss_the_axeman wrote: »

To think Martin said "how financial marketeers use maths to confuse!"

Ratios, percentages, and algebra, are you trying to scare folks King Weasel ? ;-)

Druss

You should be grateful I didn't use fluxiones.

Schamansky

Thanks to all who answered. It's all clear now.

I used to trap students with this:

You take out a loan of 100,000 at 10% and pay off 500 per month.
How long does it take to pay off the loan?

Answers ranged between 10 and 40 years. Oh well.

King_Weasel

Schamansky wrote: »

Thanks to all who answered. It's all clear now.

I used to trap students with this:

You take out a loan of 100,000 at 10% and pay off 500 per month.
How long does it take to pay off the loan?

Answers ranged between 10 and 40 years. Oh well.

10% a year? Depends how you apply the compounding, but it looks to me as though the debt grows with time. If so, can I take up your offer please?

And don't bother claiming against my estate because I shall plot the (growing) debt against time. A simple extrapolation would then demonstrate that I had cleared the debt before I took out the loan.

JimmyTheWig

Is that 100k loan secured or unsecured?