Capital of Two Friends

Video Explanation

Question

One says, “Give me a hundred, friend! I shall then become twice as rich as you.” The other replies, “If you give me ten, I shall be six times as rich as you.” Find their respective capitals.

Solution

Step 1: Let the Variables

Let the first person’s capital = ₹ \(x\)

Let the second person’s capital = ₹ \(y\)

Step 2: Form the Equations

If the second gives ₹100 to the first:

First person’s amount = \(x + 100\)

Second person’s amount = \(y – 100\)

According to the condition:

\[ x + 100 = 2(y – 100) \]

\[ x + 100 = 2y – 200 \]

\[ x – 2y = -300 \quad (1) \]

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If the first gives ₹10 to the second:

First person’s amount = \(x – 10\)

Second person’s amount = \(y + 10\)

According to the condition:

\[ y + 10 = 6(x – 10) \]

\[ y + 10 = 6x – 60 \]

\[ -6x + y = -70 \quad (2) \]

Step 3: Solve by Elimination Method

Multiply equation (1) by 6:

\[ 6x – 12y = -1800 \quad (3) \]

Add equation (2) and (3):

\[ (6x – 12y) + (-6x + y) = -1800 – 70 \]

\[ -11y = -1870 \]

\[ y = 170 \]

Step 4: Find the Value of x

Substitute \(y = 170\) in equation (1):

\[ x – 2(170) = -300 \]

\[ x – 340 = -300 \]

\[ x = 40 \]

Final Answer (For Exam)

First person’s capital = ₹40
Second person’s capital = ₹170