Finding Capital of Two Persons

Video Explanation

Question

One person says: “Give me ₹100, I shall become twice as rich as you.” The other replies: “If you give me ₹10, I shall be six times as rich as you.” Find their respective capitals.

Solution

Step 1: Let Variables

Let first person’s capital = \(x\)

Let second person’s capital = \(y\)

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Step 2: Form Equations

First condition: After transfer:

First → \(x + 100\), Second → \(y – 100\)

\[ x + 100 = 2(y – 100) \quad (1) \]

— Second condition: After transfer:

First → \(x – 10\), Second → \(y + 10\)

\[ y + 10 = 6(x – 10) \quad (2) \]

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Step 3: Simplify

From (1):

\[ x + 100 = 2y – 200 \]

\[ x = 2y – 300 \quad (3) \]

From (2):

\[ y + 10 = 6x – 60 \]

\[ y = 6x – 70 \quad (4) \]

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Step 4: Solve Linear Equations

Substitute (4) into (3):

\[ x = 2(6x – 70) – 300 \]

\[ x = 12x – 140 – 300 \]

\[ x = 12x – 440 \]

\[ 11x = 440 \Rightarrow x = 40 \]

Then:

\[ y = 6(40) – 70 = 170 \]

—

Conclusion

\[ \text{First person’s capital} = ₹40,\quad \text{Second person’s capital} = ₹170 \]

Verification

Check 1: \(140 = 2 × 70\) ✔

Check 2: \(180 = 6 × 30\) ✔