Thread: Primary school problem!

My son who is at primary school was given the following homework:

"Finding all possibilities Logic Problem"

Jessica and Simon were blowing up balloons for Garreths birthday. There were at least two of each. Some balloons had three spots and some had 5 spots.

There were 31 spots altogether.

Q: How many balloons had three spots and how many had 5 spots?

What if there were 24 spots?
What if there were 65 spots?

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OK, so knowing that there were two of each to start with (5+5+3+3=16 spots) we only need to establish the possible permutations for the remaining 15 ie 3 x 5 spot balloons or 5 x 3 spot balloons

For 24 spots there is only one possible answer (24 - 16 = 8 spots = 1 x 5 spot balloon and 1 x 3 spot balloon

For 65 there are a few permutations 65-16=49

8 x 5 + 3 x 3 = 49
2 x 5 + 13 x 3 = 49
5 x 5 + 8 x 3 = 49

But can we be sure that we have found all the possible answers? Is there a formulae for testing and am I posting this question in the right place!

This is how I would go about it:

Let be the number of balloons with 3 spots and be the number of balloons with 5 spots. We then have:

where in the total number of spots.

Next, I would let:

and and so there results:





Now, we want to find a specific instance that works for a particular .

In the case of , I would begin taking multiples of 5 away from 31 until I have a multiple of 3. In this case:

and so we find:

hence:

and thus:

and

We know produces a valid solution , but does not, so going in the other direction we find:



But, produces an invalid solution, so we know we have found the only two solutions that work.

A similar process can be done for the other two values of .