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**sam1234****Guest**

Question is:

If you produce 20 percent more milk per cow, you can decrease your herd by 20% to produce the same amount of milk.

(i) Explain how the error was made

(ii)Show how you would put the error right.

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

(i) One way to prove the statement wrong is with a counterexample.

Suppose you have 100 cows each producing 10 litres of milk. So you get 1000 litres of milk in total. If each cow produces 20% more milk, you would get 12 litres of milk per cow, giving 1200 litres in total.

But if you reduce your herd by 20%, you would have 80 cows; if each cow continues to produce 12 litres of milk, that would only yield 960 litres less than the 1000 litres previously.

(ii) To put the error right, simply change the statement to the following one:

"If you produce 20 percent more milk per cow, you **cannot** decrease your herd by 20% to produce the same amount of milk."

(If you want to decrease your herd and produce at least the same amount of milk as before, you should decrease your herd by no more than (200⁄3)%. This I leave for you to work out as an exercise. )

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**sam1234****Guest**

thanks so much

by the way (200/3) equals 66 and 2/3?

**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Yes it does, but Jane made a typo so that's wrong.

You can actually decrease your herd by no more than (20/3)%.

Why did the vector cross the road?

It wanted to be normal.

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**sam1234****Guest**

how did u get 20/3 in the first place. Please explain how you found out how to get the exact percentage you need to decrease your herd by to make the statement correct!

**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Umm, sorry, but I got it wrong as well. I can absolutely guarantee that I've got it right this time though.

First we increased the percentage by 20%, so it's at 120%.

For it to go back down to 100, there needs to be a decrease of 20 percentage points, so we need to work out what percentage of 120 is 20. This is found by 20/120 *100%, which is (50/3)%.

Why did the vector cross the road?

It wanted to be normal.

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