Directions: Solve the problem below and select the correct option.

Liam and Noah entered into a business partnership investing $X and $(X + 3000) respectively. After 4 months, Liam withdrew $2000 and Noah invested $3000 more. If the ratio of their profit shares at the end of one year is 20:39, find the initial amount invested by Noah.

The correct option is $11,000.

To find the initial amount invested by Noah, we need to calculate the equivalent total investment for both Liam and Noah over the entire 1-year (12-month) period, taking into account their mid-term withdrawals and additional investments.

Step 1: Define initial investments
Let Liam's initial investment be $X.
Since Noah invested $3000 more than Liam, Noah's initial investment is $(X + 3000).

Step 2: Calculate Liam's total effective investment
For the first 4 months, Liam's investment is $X.
After 4 months, Liam withdraws $2000, so his investment for the remaining 8 months (12 - 4 = 8 months) becomes $(X - 2000).
Liam's total investment in month-units is:

$\text{Liam's Total}=4\times X+8\times (X-2000)$
$\text{Liam's Total}=4X+8X-16000=12X-16000$

Step 3: Calculate Noah's total effective investment
For the first 4 months, Noah's investment is $(X + 3000).
After 4 months, Noah adds $3000 more, making his investment $(X + 3000 + 3000) = $(X + 6000) for the remaining 8 months.
Noah's total investment in month-units is:

$\text{Noah's Total}=4\times (X+3000)+8\times (X+6000)$
$\text{Noah's Total}=4X+12000+8X+48000=12X+60000$

Step 4: Set up the profit ratio equation
The ratio of their profit shares at the end of the year is given as 20:39:

$\frac{12X-16000}{12X+60000}=\frac{20}{39}$

Simplify the fraction on the left by dividing the numerator and denominator by 4:

$\frac{3X-4000}{3X+15000}=\frac{20}{39}$

Now, cross-multiply to solve for X:

$39\times (3X-4000)=20\times (3X+15000)$
$117X-156000=60X+300000$
$117X-60X=300000+156000$
$57X=456000$
$X=\frac{456000}{57}=8000$

Step 5: Determine Noah's initial investment
Since Noah's initial investment was $(X + 3000):

$\text{Noah's Initial Investment}=8000+3000=11000$

Therefore, the initial amount invested by Noah is $11,000.