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?
