Finding Stake Money of Two Cock Owners

Video Explanation

Question

A wizard makes offers to two cock owners. To one: if he wins, the wizard takes his stake; if he loses, the wizard pays \( \frac{2}{3} \) of the stake. To the other: if he wins, the wizard takes his stake; if he loses, the wizard pays \( \frac{3}{4} \) of the stake. In either case, the wizard gains 12 gold coins. Find the stakes of both owners.

Solution

Step 1: Let Variables

Let stake of first owner = \(x\)

Let stake of second owner = \(y\)

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Step 2: Form Equations

Case 1: First wins, second loses Wizard gets \(x\) and pays \( \frac{3}{4}y \)

\[ x – \frac{3}{4}y = 12 \quad (1) \]

— Case 2: Second wins, first loses Wizard gets \(y\) and pays \( \frac{2}{3}x \)

\[ y – \frac{2}{3}x = 12 \quad (2) \]

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Step 3: Remove Fractions

Multiply (1) by 4:

\[ 4x – 3y = 48 \quad (3) \]

Multiply (2) by 3:

\[ 3y – 2x = 36 \quad (4) \]

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Step 4: Solve Linear Equations

Add (3) and (4):

\[ (4x – 3y) + (3y – 2x) = 48 + 36 \]

\[ 2x = 84 \Rightarrow x = 42 \]

Substitute into (4):

\[ 3y – 2(42) = 36 \]

\[ 3y – 84 = 36 \]

\[ 3y = 120 \Rightarrow y = 40 \]

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Conclusion

\[ \text{First owner’s stake} = 42,\quad \text{Second owner’s stake} = 40 \]

Verification

Case 1: \(42 – 30 = 12\) ✔

Case 2: \(40 – 28 = 12\) ✔