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# Stamp Collecting

##### Age 11 to 14 ShortChallenge Level

Method 1:
If Tom bought 9 stamps he would spend £9.90 and there would be 10p left.
If Tom bought 8 stamps he would spend £8.80 and there would be £1.20 left (not a multiple of 70p).
If Tom bought 7 stamps he would spend £7.70 and there would be £2.30 left (not a multiple of 70p).
If Tom bought 6 stamps he would spend £6.60 and there would be £3.40 left (not a multiple of 70p).
If Tom bought 5 stamps he would spend £5.50 and there would be £4.50 left (not a multiple of 70p).
If Tom bought 4 stamps he would spend £4.40 and there would be £5.60 left (a multiple of 70p).

So if Tom bought 4 stamps, Sophie could buy 8 stamps.

Method 2:
£1.10 has 10p so can fit around 70p s to make whole numbers of pounds
Sophie 1 stamp  + Tom 3 stamps = 70p + £3.30 = £4 not a factor of £10
Sophie 2 stamps + Tom 6 stamps = 2 $\times$ £4 = £8
Sophie 3 stamps + Tom 9 stamps = 3 $\times$ £4 = £12
Sophie 4 stamps + Tom 2 stamps = £2.80 + £2.20 = £5 half of £10
Sophie 8 stamps + Tom 4 stamps = £10
8 + 4 = 12 stamps

Method 3:
Suppose Tom bought $x$ stamps and Sophie bought $y$ stamps. Then, $1.1x + 0.7y = 10$.

Multiplying this by $10$, $11x + 7y = 100$.

$100$ has a remainder of $2$ when divided by $7$, so $11x$ must have the same remainder. Since $11x \leq 100$, we need only to check values of $x$ that are at most $9$.
The table below shows the values:

$x$ $11x$ Remainder when divided by $7$
0 0 0
1 11 4
2 22 1
3 33 5
4 44 2
5 55 6
6 66 3
7 77 0
8 88 4
9 99 1

This means the only possible value of $x$ is $4$.

Substituting this in, $44 + 7y = 100$. Subtracting 44 gives $7y = 56$, and dividing by $7$ gives $y = 8$.

Therefore Tom bought $4$ stamps and Sophie bought $8$ stamps.

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.