Challenge Level

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.

£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

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.