# Sunday Afternoon Maths VII

## Reverse bases

Find two digits \(a\) and \(b\) such that \(ab\) in base 10 is equal to \(ba\) in base 4.

Find two digits \(c\) and \(d\) such that \(cd\) in base 10 is equal to \(dc\) in base 7.

Find two digits \(e\) and \(f\) such that \(ef\) in base 9 is equal to \(fe\) in base 5.

#### Show answer & extension

#### Hide answer & extension

If \(ab\) in base 10 is equal to \(ba\) in base 4, then \(10a+b=4b+a\).

So, \(9a=3b\).

\(a\) and \(b\) must both be less than 4, as they are digits used in base 4, so \(a=1\) and \(b=3\).

So 13 in base 10 is equal to 31 in base 4.

By the same method, we find that:

- 23 in base 10 is equal to 32 in base 7.
- 46 in base 10 is equal to 64 in base 7.
- 12 in base 9 is equal to 21 in base 5.
- 24 in base 9 is equal to 42 in base 5.

#### Extension

For which pairs of bases \(A\) and \(B\) can you find two digits \(g\) and \(h\) such that \(gh\) in base \(A\) is equal to \(hg\) in base \(B\)?

## Ninety nine

In a 'ninety nine' shop, all items cost a number of pounds and 99 pence. Susanna spent £65.76. How many items did she buy?

#### Show answer & extension

#### Hide answer & extension

Every item bought will cause the pence in the total cost to fall by 1. So to spend £65.76, Susanna must have bought **24** items.

#### Extension

What is the smallest amount Susanna could spend for which we could not tell how many items she bought?