H3. Alex is given $£ 1$ by his grandfather and decides:
(i) to spend at least one third of the $£ 1$ on toffees at 5 p each;
(ii) to spend at least one quarter of the $£ 1$ on packs of bubblegum at 3 p each; and
(iii) to spend at least one tenth of the $£ 1$ on jellybeans at 2 p each.
He only decides how to spend the rest of the money when he gets to the shop, but he spends all of the $£ 1$ on toffees, packs of bubblegum and jellybeans.
What are the possibilities for the number of jellybeans that he buys?
Solution
It follows from decision (i) that Alex spends at least 35 p on toffees; it follows from decision (ii) that he spends at least 27 p on bubblegum; and it follows from decision (iii) that he spends at least 10 p on jellybeans. Therefore, out of the total $£ 1$ that he will spend, he has to decide how to spend 28 p.
He may spend the whole 28 p on jellybeans, which is an extra 14 jellybeans.
He cannot spend 26 p or 24 p on jellybeans, because he cannot spend the remaining money ( $2 \mathrm{p}$ or $4 \mathrm{p}$ ) on the other items.
But he may spend any even amount from 22p downwards on jellybeans, since the remaining money would then be an even amount from 6 p to 28 p, and he is able to spend this on toffees or bubblegum (or both), as the following table shows.
\begin{tabular}{ccc}
\hline \begin{tabular}{c} 
Remaining \\
money
\end{tabular} & \begin{tabular}{c} 
Toffees \\
at 5p
\end{tabular} & \begin{tabular}{c} 
Bubblegum \\
at 3p
\end{tabular} \\
\hline $6 \mathrm{p}$ & 0 & 2 \\
$8 \mathrm{p}$ & 1 & 1 \\
$10 \mathrm{p}$ & 2 & 0 \\
$12 \mathrm{p}$ & 0 & 4 \\
$14 \mathrm{p}$ & 1 & 3 \\
$16 \mathrm{p}$ & 2 & 2 \\
$18 \mathrm{p}$ & 0 & 6 \\
$20 \mathrm{p}$ & 1 & 5 \\
$22 \mathrm{p}$ & 2 & 4 \\
$24 \mathrm{p}$ & 0 & 8 \\
$26 \mathrm{p}$ & 1 & 7 \\
$28 \mathrm{p}$ & 2 & 6 \\
\hline
\end{tabular}

Note that in some cases there are other ways to spend the money.
Thus the number of additional jellybeans that he may buy is a number from 0 to 11 , or is 14 .
But these are in addition to the five he buys as a result of decision (iii). Therefore the number of jellybeans that he buys is a number from 5 to 16 , or is 19 .