Three friends each paid the same amount to buy tins of cream together. When they divided the tins, the first took 10 more than the second, and the second took 4 more than the third. To make things fair the first had to refund 144 dollars to the third. What is the price of one tin?

Step-by-step solution

1. 1

   Name the unknown you actually want, then a second one for the counts. Let $p$ be the price of one tin and let $a$ be the number of tins the third friend received. The two comparisons then give

   $\text{third}=a,\qquad \text{second}=a+4,\qquad \text{first}=a+4+10=a+14.$

2. 2

   Turn 'they each paid the same' into a statement about tins. Equal money means each friend is entitled to one third of everything bought:

   $\text{total}=a+(a+4)+(a+14)=3a+18,\qquad \text{fair share}=\frac{3a+18}{3}=a+6.$

   Notice that $a$ survives here but will cancel in the next step — that is why the problem is solvable even though we never learn how many tins there were.

3. 3

   Measure the overshoot against the fair share. The first friend holds $a+14$ tins but is entitled to $a+6$, so the surplus is

   $(a+14)-(a+6)=8\text{ tins},$

   a fixed number, independent of $a$.

4. 4

   Read what the 144 dollars is paying for. The refund exactly buys back the tins the first friend took beyond the fair share, so $144$ dollars is the value of $8$ tins:

   $8p=144.$

5. 5

   Solve for the price.

   $p=\frac{144}{8}=18.$

   Check: $8$ surplus tins at $18$ each is $8\times 18=144$, matching the refund, so the price of one tin is $18$ dollars.