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.

Step 1: Understand the initial investment ratio
Let the initial investments of Rajan and Suresh be $10x$ and $7x$ respectively, where $x$ is a common multiplier.

Step 2: Calculate Rajan's total equivalent investment for 12 months
For the first $T$ months, Rajan's investment is $10x$.
After $T$ months, Rajan withdraws 50% of his capital.
Rajan's new capital = $10x-(0.50\times 10x)=5x$
This remaining capital stays for $(12-T)$ months.
Rajan's total investment-months = $(10x\times T)+[5x\times (12-T\left)\right]$

Simplifying Rajan's total investment:
$=10xT+60x-5xT=5xT+60x=5x(T+12)$

Step 3: Calculate Suresh's total equivalent investment for 12 months
For the first $T$ months, Suresh's investment is $7x$.
After $T$ months, Suresh increases his investment by 40%.
Suresh's new capital = $7x+(0.40\times 7x)=7x+2.8x=9.8x$
This updated capital stays for $(12-T)$ months.
Suresh's total investment-months = $(7x\times T)+[9.8x\times (12-T\left)\right]$

Simplifying Suresh's total investment:
$=7xT+117.6x-9.8xT=117.6x-2.8xT=x(117.6-2.8T)$

Step 4: Formulate the ratio equation
The ratio of total profit earned by Rajan and Suresh at the end of the year is given as 95 : 98.
$\frac{5x(T+12)}{x(117.6-2.8T)}=\frac{95}{98}$

Canceling $x$ from both the numerator and denominator:
$\frac{5(T+12)}{117.6-2.8T}=\frac{95}{98}$

Dividing the numerator on both sides by 5:
$\frac{T+12}{117.6-2.8T}=\frac{19}{98}$

Cross-multiplying and solving for $T$:
$98(T+12)=19(117.6-2.8T)$
$98T+1176=2234.4-53.2T$

Rearranging the terms:
$98T+53.2T=2234.4-1176$
$151.2T=1058.4$
$T=\frac{1058.4}{151.2}=7$

Thus, the value of $T$ is 7.