Little Maths Problem - Your Help Required Please

[tubeguru]

Now, I know I should be able to solve this problem, but it's doing my head in, and I can't do it.

So, I thought I'd put the maths skills of the forum to the test by presenting you with this little teaser:

Annie, Barbie and Carl Searcher are a family of treasure-hunters. They have been comparing how many coins they have found.

Annie has discovered a total of 12 coins
Barbie has also discovered a total of 12 coins
Their brother Carl has discovered a total of 15 coins

1 coin has been discovered by all three of them
4 coins have been discovered by just Annie and Carl and 2 by just Annie and Barbie
28 coins have been discovered by at least one of the three siblings (I found this piece of info to be a little confusing, but this is how it was written)

How many coins were discovered by ONLY Annie?
How many were discovered by ONLY Barbie?
How many were discovered by ONLY Carl?

I know that this problem requires a Venn diagram, but there is a piece of missing information - the number of coins found by both Barbie and Carl, which confuses me.

Can anyone solve this puzzle for me? Thankyou.

[greatkingrat]

28 coins have been discovered by at least one of the three siblings (I found this piece of info to be a little confusing, but this is how it was written)

This is the sum of each of the individual segments of your Venn diagram. Although you don't have the number of coins found by both B and C you can work this out by working backwards from the overall total.

Assume the number of coins found by B and C is x, then work out what the overall total is in terms of x and then find what value of x gives an overall total of 28.

[tubeguru]

I've drawn this Venn diagram, which I think shows what's going on. With the B and C circles, I am not sure what the formula should be, hence the two possibilities I've entered. Which of the two is right?

[snudge27]

I think that 12-x-1-1 and 15-x-3-1 are probably correct, as you'll need to take into account the other 'joint finds' as part of their total.

[tubeguru]

Exactly, as I took 3, 1 and 1 away from 12 in A's circle.

[greatkingrat]

That is the right idea although I think some of the numbers are wrong.

There are 4 coins found by A+C, 2 coins found by A+B, 1 coin found by A+B+C, which leaves 5 coins found only by A (12-4-2-1).

[tubeguru]

But the 3 that I have in the shared section between A and C is 4 minus the ONE in the middle, as that's already been accounted for, no? This is what confuses me about Venn diagrams - accounting for everything more than once, or not as the case may be.

Using my diagram above, I worked out that x=5, but is that right?

[greatkingrat]

It says "4 coins have been discovered by JUST Annie and Carl", so this does not include the 1 coin discovered by all three.

[tubeguru]

Good point. So in fact, we SHOULD have this, no?

[tubeguru]

Giving x=3.

[tubeguru]

And x=s is wrong too. Should A + B + C + (A+B) + (A+C) + (B+C) + (A+B+C) = 28 or 35?

[greatkingrat]

Yes, x=3 is correct. The sum of all seven segments is 28.

[tubeguru]

Yay, thanks for clearing that one up.

I've just noticed that if I were to add everything in each segment up, I could sum it up as:

1+2+4+5+x+(12-x-2-1)+(15-x-4-1)=28,

Which works out as 31-x=28, hence x=3.

[RobbieM]

Yes, I agree, so I guess we can answer the original questions now:

How many coins were discovered by ONLY Annie? 5
How many were discovered by ONLY Barbie? 6
How many were discovered by ONLY Carl? 5

By my reckoning.

[tubeguru]

Carl has found seven, not five.

15-4-3-1 = 7, not 5.