Question

Three merchants find a purse lying in the road. One merchant says, “If I keep the purse, I will have twice as much money as the two of you together.” “Give me the purse and I will have three times as much as the two of you together,” said the second merchant. The third merchant said, “I will be much better off than either of you if I keep the purse, I will have five times as much as the two of you together.” If there are coins (of equal value) in the purse, how much money does each merchant have? (From Mahavira)

Short answer

The first merchant, the second merchant and the third merchant will have money of,4 , 12 and 20 coins respectively.

Step-by-step solution

Represent the number of coins had by first, second and third merchant in terms of system of equations.

The number of coins each merchant can have can be found using the augmented matrix.

Let ‘F’ represent the first merchant, ‘S’ represent the second merchant and ‘T’ represent the third merchant.

The first merchant gets twice the money as much as two of others get.

$$\begin{gathered} F+60=2(S+T) \\ \Rightarrow -F+2S+2T=60.........\left(1\right) \end{gathered}$$

The second merchant gets thrice the money as much as two of others get.

$$\begin{gathered} S+60=3(F+T) \\ \Rightarrow 3F-S+3T=60..........\left(2\right) \end{gathered}$$

The third merchant gets five times the money as much as two of others get.

$$\begin{gathered} T+60=5(S+F) \\ \Rightarrow 5F+5S-T=60.........\left(3\right) \end{gathered}$$

The system of equations is,

role="math" localid="1659342704513" $\left|\begin{array}{c}-F+2S+2T=60\\ 3F-S+3T=60\\ 5F+5S-T=60\end{array}\right|$

Represent the above equations in terms of a matrix.

role="math" localid="1659342732834" $\left|\begin{array}{cccc}-1& 2& 2& 60\\ 3& -1& 3& 60\\ 5& 5& -1& 60\end{array}\right|$

Perform the row operations on the matrix.

Consider the matrix.

$$\begin{gathered} \left|\begin{array}{cccc}-1& 2& 2& 60\\ 3& -1& 3& 60\\ 5& 5& -1& 60\end{array}\right| \\ =\left|\begin{array}{cccc}5& 5& -1& 60\\ 0& -4& \frac{18}{5}& 24\\ 0& 0& \frac{9}{2}& 60\end{array}\right| \\ =\left|\begin{array}{cccc}1& 0& 0& 4\\ 0& 1& 0& 12\\ 0& 0& 1& 20\end{array}\right| \end{gathered}$$

The values are, $F=4,S=12,T=20$

Final answer

The first merchant will have money of 4 coins, the second merchant will have money of 12 coins and the third merchant will have money of 20 coins.