Stamp collecting
Last week, Tom and Sophie bought some stamps and they spent exactly £10. Can you work out how many stamps they bought?
Age
11 to 14
Challenge level
Problem
Last week, Tom and Sophie bought some stamps for their collections.
Each stamp Tom bought cost him £1.10, whilst Sophie paid 70p for each of her stamps.
Between them they spent exactly £10.
How many stamps did they buy in total?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 12 stamps
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.