Working together, Liam and Noah can complete a project in 3T days. Liam alone can do the same project in 24 days. If Noah alone can complete the project in 8T days, then find the value of T.

The correct answer is 5.

To find the value of $T$, we can use the concept of work rates, where work rate is defined as the fraction of work completed in one day.

Step 1: Express the individual work rates of Liam and Noah.

Liam alone can complete the entire project in 24 days.
Liam's 1-day work rate = $\frac{1}{24}$

Noah alone can complete the entire project in $8T$ days.
Noah's 1-day work rate = $\frac{1}{8T}$

Step 2: Express their combined work rate.

Working together, Liam and Noah can complete the project in $3T$ days.
Their combined 1-day work rate = $\frac{1}{3T}$

Step 3: Set up the equation.

The sum of their individual 1-day work rates is equal to their combined 1-day work rate:

$\frac{1}{24}+\frac{1}{8T}=\frac{1}{3T}$

Step 4: Solve for $T$.

Subtract $\frac{1}{8T}$ from both sides of the equation:

$\frac{1}{24}=\frac{1}{3T}-\frac{1}{8T}$

Find a common denominator for the fractions on the right side. The common denominator of $3T$ and $8T$ is $24T$:

$\frac{1}{3T}=\frac{8}{24T}$

$\frac{1}{8T}=\frac{3}{24T}$

Substitute these fractions back into the equation:

$\frac{1}{24}=\frac{8-3}{24T}$

$\frac{1}{24}=\frac{5}{24T}$

Multiply both sides of the equation by 24:

$1=\frac{5}{T}$

Multiply both sides by $T$:

$T=5$

Therefore, the value of $T$ is 5.