Mahaviracharya/Ganita Sara Samgraha/Chapter VI/95-96

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Mahaviracharya: Ganita Sara Samgraha Chapter $\text {VI}$: Mixed Problems: Problem $95 \frac 1 2$ and $96 \frac 1 2$

Three puranas formed the pay of one man who is a mounted soldier.

At that rate there were $65$ soldiers in all.

Some among them broke down, and the amount of their pay was given to those who remained in the field.

Of this, each man obtained $10$ puranas.

You tell me, after thinking well, how many remained in the field and how many broke down.


Solution

$50$ soldiers broke down, while $15$ remained in the field.


Proof

Let $x$ denote the number of soldiers who remained in the field.

Let $y$ denote the number of soldiers who broke down.


We have:

\(\ds x + y\) \(=\) \(\ds 65\) that is, the total number of soldiers
\(\text {(1)}: \quad\) \(\ds 3 \paren {x + y}\) \(=\) \(\ds 195\) that is, the total amount of pay to be distributed
\(\text {(2)}: \quad\) \(\ds 3 \paren {x + y}\) \(=\) \(\ds \paren {3 + 10} x\) that is, the total pay was divided between the soldiers who remained
\(\ds \leadsto \ \ \) \(\ds 13 x\) \(=\) \(\ds 195\) substituting for $3 \paren {x + y}$ in $(2)$ from $(1)$
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds 15\)

which leaves $65 - 15 = 50$ soldiers who broke down.

$\blacksquare$


Sources

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