Rajan and Suresh invest in a food truck venture in the respective ratio of 10 : 7. After T months, Suresh increases his investment by 40% while Rajan withdraws 50% of his capital. If at the end of the year, the respective ratio of profit earned by Rajan and Suresh is 95 : 98, then find the value of T.

The correct answer is 7.

In business partnerships, the ratio of profit shared between partners is directly proportional to the product of their investment amounts and the duration (time) for which the investments were made.

Step 1: Define the initial investments
The ratio of initial investments of Rajan and Suresh is given as $10:7$.
Let Rajan's initial investment = $10x$
Let Suresh's initial investment = $7x$
The total investment period is 1 year, which equals 12 months.

Step 2: Calculate Rajan's total equivalent investment
For the first $T$ months, Rajan's capital is $10x$.
After $T$ months, he withdraws 50% of his capital.
Remaining capital = $10x-(50\%\times 10x)=5x$
This remaining capital stays invested for the rest of the year, which is $(12-T)$ months.

Rajan's total equivalent investment = $(10x\times T)+[5x\times (12-T\left)\right]$

$\text{Rajan's total investment}=10xT+60x-5xT=5xT+60x=5x(T+12)$

Step 3: Calculate Suresh's total equivalent investment
For the first $T$ months, Suresh's capital is $7x$.
After $T$ months, he increases his capital by 40%.
Increase amount = $40\%\times 7x=2.8x$
New capital = $7x+2.8x=9.8x$
This new capital stays invested for the remaining $(12-T)$ months.

Suresh's total equivalent investment = $(7x\times T)+[9.8x\times (12-T\left)\right]$

$\text{Suresh's total investment}=7xT+117.6x-9.8xT=117.6x-2.8xT=2.8x(42-T)$

Step 4: Set up the ratio equation
The profit sharing ratio of Rajan to Suresh at the end of the year is given as $95:98$.

$\frac{5x(T+12)}{2.8x(42-T)}=\frac{95}{98}$

Canceling out $x$ from the numerator and denominator:

$\frac{5(T+12)}{2.8(42-T)}=\frac{95}{98}$

Dividing both numerators by 5:

$\frac{T+12}{2.8(42-T)}=\frac{19}{98}$

Cross-multiplying to solve for $T$:

$98(T+12)=19\times 2.8(42-T)$

$98T+1176=53.2(42-T)$

$98T+1176=2234.4-53.2T$

Grouping the terms containing $T$ on one side:

$98T+53.2T=2234.4-1176$

$151.2T=1058.4$

$T=\frac{1058.4}{151.2}=7$

Therefore, the value of $T$ is 7 months.