Problem D and Solution

Two Birds

Katya owns two cockatoos, an older white cockatoo and a younger Galah cockatoo. At present, the sum of the cockatoos’ ages is \(44\) years. In \(n\) years, where \(n >0\), the white cockatoo’s age will be four times the Galah cockatoo’s age. If \(n\) is an integer, determine the possible present ages of each cockatoo.

Let \(g\) represent the present age of the Galah cockatoo and \(w\) represent the present age of the white cockatoo. Since the sum of their present ages is \(44\), we have \(g+w=44\) or \(w=44-g\).

In \(n\) years, the Galah cockatoo will be \((g+n)\) years old and the white cockatoo will be \((44-g+n)\) years old. At that time the white cockatoo will be four times older than the Galah cockatoo. Therefore, $$\begin{align} 4(g+n)&=44-g+n\\ 4g+4n&=44-g+n\\ 5g+3n&=44\\ g&=\frac{44-3n}{5} \end{align}$$ We are looking for integer values of \(n\) so that \(44-3n\) is divisible by \(5\).

When \(n=3\), \(g=\frac{44-3n}{5}=\frac{44-3(3)}{5}=\frac{35}{5}=7\). When \(g=7\), \(w=44-g=44-7=37\).

When \(n=8\), \(g=\frac{44-3n}{5}=\frac{44-3(8)}{5}=\frac{20}{5}=4\). When \(g=4\), \(w=44-g=44-4=40\).

When \(n=13\), \(g=\frac{44-3n}{5}=\frac{44-3(13)}{5}=\frac{5}{5}=1\). When \(g=1\), \(w=44-g=44-1=43\).

When \(n=18\), \(g=\frac{44-3n}{5}=\frac{44-3(18)}{5}=\frac{-10}{5}=-2\). Since \(g<0\), \(n=16\) does not produce a valid age for the Galah cockatoo. No higher value of \(n\) would produce a value of \(g>0\).

No integer values of \(n\) between \(0\) and \(18\), other than \(3\), \(8\), and \(13\), produce a multiple of \(5\) when substituted into \(44-3n\).

If today the white cockatoo is \(37\) and the Galah cockatoo is \(7\), then in \(3\) years the white cockatoo will be \(40\) and the Galah cockatoo will be \(10\). The white cockatoo will be four times older than the Galah cockatoo since \(4\times 10=40\).

If today the white cockatoo is \(40\) and the Galah cockatoo is \(4\), then in \(8\) years the white cockatoo will be \(48\) and the Galah cockatoo will be \(12\). The white cockatoo will be four times older than the Galah cockatoo since \(4\times 12=48\).

If today the white cockatoo is \(43\) and the Galah cockatoo is \(1\), then in \(13\) years the white cockatoo will be \(56\) and the Galah cockatoo will be \(14\). The white cockatoo will be four times older than the Galah cockatoo since \(4\times 14=56\).

Therefore, the possible present ages for the white cockatoo and Galah cockatoo are \(37\) and \(7\), or \(40\) and \(4\), or \(43\) and \(1\).