Three partners, A, B, and C, launched a joint venture by investing capital in the ratio of 3 : 5 : 4, respectively. After 6 months, A added Rs 600 to his investment, whereas B withdrew Rs 600 from his investment. If A's share of the annual net profit is Rs 12,000 out of a total profit of Rs 40,000, calculate the initial investment (in Rs) made by C.

The correct answer is 2000.

Step-by-step explanation:

1. Express initial investments using a variable:

Let the initial investments of partners A, B, and C be in the ratio 3 : 5 : 4.
We can express their initial investments as:
Initial investment of A =
$3x$
Initial investment of B =
$5x$
Initial investment of C =
$4x$

2. Calculate the total investment-months for each partner over 1 year (12 months):

For Partner A:
For the first 6 months, A invested
$3x$
. After 6 months, A added Rs 600, making the investment
$3x+600$
for the remaining 6 months.
Total investment for A =
$(3x\times 6)+(3x+600)\times 6$
=
$18x+18x+3600=36x+3600$

For Partner B:
For the first 6 months, B invested
$5x$
. After 6 months, B withdrew Rs 600, making the investment
$5x-600$
for the remaining 6 months.
Total investment for B =
$(5x\times 6)+(5x-600)\times 6$
=
$30x+30x-3600=60x-3600$

For Partner C:
C kept
$4x$
invested for the entire 12 months without any change.
Total investment for C =
$4x\times 12=48x$

3. Calculate total combined investment of all three partners:
Total investment =
$(36x+3600)+(60x-3600)+48x$
=
$144x$

4. Determine value of x using A's share of profit:
A's share of profit is Rs 12,000 out of total profit of Rs 40,000.
The ratio of A's investment to total investment is equal to the ratio of A's profit to total profit:

$\frac{36x+3600}{144x}=\frac{12000}{40000}$

Simplifying the profit fraction:

$\frac{12000}{40000}=\frac{3}{10}$

Equating the two fractions:

$\frac{36x+3600}{144x}=\frac{3}{10}$

Cross-multiplying to solve for
$x$
:

$10\times (36x+3600)=3\times 144x$

$360x+36000=432x$

$432x-360x=36000$

$72x=36000$

$x=\frac{36000}{72}=500$

5. Calculate C's initial investment:
Initial investment of C =
$4x=4\times 500=2000$

Hence, the initial investment made by C is Rs 2000.