High School

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------------------------------------------------ A and B started a business with investments of \((x + 4000)\) and \((x + 2000)\), respectively. C joined 6 months later with an investment of \((x + 6000)\). The total profit earned by C at the end of the year was Rs. 6000 out of a total profit of Rs. 28000. Find the value of \(x\).

1) 30000
2) 56000
3) 60000
4) 28000
5) 54000

Answer :

To solve this problem, we need to understand the distribution of profit in the partnership based on the time and amount of investment.

First, let's break down the investments:

  • A started with an investment of [tex]x + 4000[/tex].
  • B started with an investment of [tex]x + 2000[/tex].
  • C joined 6 months later with an investment of [tex]x + 6000[/tex].

Since C joined 6 months later, C's investment was only active for 6 months, while A and B's investments were active for the whole year. Therefore, the effective investment time in terms of months becomes important:

  • A's investment for the whole year becomes [tex]12(x + 4000)[/tex]
  • B's investment for the whole year becomes [tex]12(x + 2000)[/tex]
  • C's investment for 6 months becomes [tex]6(x + 6000)[/tex]

Now, we can calculate the total profit distribution based on these contributions. Let's denote the shares as follows:

[tex]\text{A's share: } 12(x + 4000) \\
\text{B's share: } 12(x + 2000) \\
\text{C's share: } 6(x + 6000)[/tex]

Given that C's profit is Rs. 6000 and the total profit is Rs. 28000, the profit ratio can be set up with these shares:

[tex]\text{Ratio of shares: } \frac{6(x + 6000)}{12(x + 4000) + 12(x + 2000) + 6(x + 6000)}[/tex]

Plugging in the known profit amounts for C:

[tex]\frac{6(x + 6000)}{12(x + 4000) + 12(x + 2000) + 6(x + 6000)} \times 28000 = 6000[/tex]

Let's simplify:

  1. Calculate person's C share of profit:
    [tex]\frac{6(x + 6000)}{12(x + 4000) + 12(x + 2000) + 6(x + 6000)} = \frac{6000}{28000}[/tex]

  2. Simplifying the equation:
    [tex]\frac{6(x + 6000)}{24(x + 3000) + 6(x + 6000)} = \frac{3}{14}[/tex]

  3. We manage from:
    [tex]3(24(x + 3000) + 6(x + 6000)) = 6(x + 6000) \times 14[/tex]

  4. Solving furthers give:
    [tex]72(x + 3000) + 18(x + 6000) = 84(x + 6000)[/tex]

  5. After simplifying, solve for [tex]x[/tex]:
    [tex]72x + 216000 + 18x + 108000 = 84x + 504000[/tex]

  6. Combining like terms:
    [tex]90x + 324000 = 84x + 504000[/tex]

  7. Isolating [tex]x[/tex]:
    [tex]6x = 180000[/tex]

  8. Solving for [tex]x[/tex]:
    [tex]x = 30000[/tex]

So, the value of [tex]x[/tex] is 30,000.

The correct option is: 1) 30000.

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