Three friends contribute equal amounts of money and buy jars of cream together. When they divide the jars, the first friend takes 10 more jars than the second, and the second takes 4 more than the third. To make things fair, the first friend pays 144 to the third friend. What is the price of one jar?

Step-by-step solution

1. 1

   Express all three shares with one variable. Let the third friend take $a$ jars. Then the second takes $a+4$ and the first takes $(a+4)+10=a+14$. Anchoring on the smallest share keeps every quantity non-negative and makes the differences easy to read.

2. 2

   Find the fair share. Because the three paid equally, each is entitled to a third of the jars:

   $\frac{a+(a+4)+(a+14)}{3}=\frac{3a+18}{3}=a+6$

   The unknown $a$ cancels out here, which is why the problem is solvable without knowing the total.

3. 3

   Measure each person's surplus or shortfall against that share.

   $\text{first: }(a+14)-(a+6)=+8,\qquad \text{second: }(a+4)-(a+6)=-2,\qquad \text{third: }a-(a+6)=-6$

   So the first friend has $8$ jars too many, the second is $2$ short and the third is $6$ short. The surpluses and shortfalls balance: $8=2+6$.

4. 4

   Match the payment to the right shortfall. The $144$ goes to the third friend, so it compensates that friend's shortfall alone — $6$ jars, not the first friend's whole $8$-jar surplus. Letting $x$ be the price of a jar:

   $6x=144 \quad\Longrightarrow\quad x=24$

5. 5

   Check that the whole settlement closes. At $24$ per jar the first friend's $8$ extra jars are worth $8\times24=192$. Paying $144$ to the third friend and $2\times24=48$ to the second accounts for exactly $144+48=192$. Everyone ends up with value equal to a one-third share, so $24$ is consistent.