Beef and Sausages

"A neighbour of mine," said Aunt Jane, "bought a certain quantity of beef at two shillings a pound [i.e. at 24 pence a pound], and the same quantity of sausages at eighteenpence a pound. I pointed out to her that if she had divided the same money equally between beef and sausages she would have gained two pounds in the total weight. Can you tell me exactly how much she spent?"

"Of course, it is no business of mine," said Mrs. Sunniborne; "but a lady who could pay such prices must be somewhat inexperienced in domestic economy."

"I quite agree, my dear," Aunt Jane replied, "but you see that is not the precise point under discussion, any more than the name and morals of the tradesman."

(Source: Amusements in Mathematics by Henry Ernest Dudeney)

(This background information on currency may be useful.) 

The key here is to focus on a variable, let's call it $b$ (for budget, denominated in pence) and then relate that to the amounts of each product bought. It should be clear that, if she had bought one pound of beef and another of sausages, she would have spent a total of 3s.6d. (or 42 pence). If it had been two pounds of each, she would have spent 7s. even (or 84 pence) and so on.

It is also important to note that the amount of beef bought is equal to $b \times \frac{24}{42} \times \frac{1}{24}$, or just $\frac{b}{42}$. This means to say that the amount of beef bought is equal to the fraction of the budget spent on beef, divided by the price per pound. Similarly, the amount of sausages bought is also $\frac{b}{42}$. Therefore the total amount of pounds of meat bought all told is $\frac{b}{21}$.

If, on the other hand, the neighbor had spent an equal amount of her budget on both of the two foodstuffs, then the amount of beef bought would have been $b \times \frac{1}{2} \times \frac{1}{24}$ and, of sausages, $b \times \frac{1}{2} \times \frac{1}{18}$. All that remains now is to add two to the poundage of the items that she did in fact buy and equate it with the poundage that she might have bought as described here. Simplifying a bit:

\[ \frac{b}{21} + 2 = \frac{b}{2} \times \left(\frac{1}{24} + \frac{1}{18}\right) \]

Solving for $b$ reveals that it is equal to £8.8s. (or 2016 pence). Because the total poundage of meat bought is equal to the budget divided by 21, she bought 96 total, or 48 of each. Had she spent equal amounts on each of the two foodstuffs she would have bought 42 pounds of beef and 56 of sausages for a total of 98.