Two hundred years of algebra

In this section we recapitulate two hundred or so years of mathematical thought. Let us start with a familiar type of brain teaser.

Example 1.1.1 Sally and Martin go to The Olde Tea Shoppe. Sally buys three cream buns and two bottles of pop for thirteen shillings, whilst Martin buys two cream buns and four bottles of pop for fourteen shillings. How much does a cream bun cost and how much does a bottle of pop cost?

Solution. If Sally had bought six cream buns and four bottles of pop, then she would have bought twice as much and it would have cost her twenty six shillings. Similarly, if Martin had bought six cream buns and twelve bottles of pop, then he would have bought three times as much and it would have cost him forty two shillings. In this new situation, Sally and Martin would have bought the same number of cream buns, but Martin would have bought eight more bottles of pop than Sally. Since Martin would have paid sixteen shillings more, it follows that eight bottles of pop cost sixteen shillings and one bottle costs two shillings.

In our original problem, Sally bought three cream buns and two bottles of pop, which, we now know, must have cost her four shillings, for thirteen shillings. Thus her three cream buns cost nine shillings and each cream bun cost three shillings.