Directions: Solve the following word problem.

A team of 9 senior developers and 8 junior developers can finish a software module in 12 days, while 6 senior developers and 8 junior developers can complete the same module in 15 days. If K senior developers and 10 junior developers can complete the module in 8 days, find the value of K.

The correct answer is 15.

Step 1: Define variables for daily work rate.

Let $S$ be the work done by one senior developer in one day.

Let $J$ be the work done by one junior developer in one day.

Let $W$ be the total work required to finish the software module.

Step 2: Formulate equations based on the given conditions.

From the first condition, 9 senior developers and 8 junior developers complete the module in 12 days:

$(9S+8J)\times 12=W$

$108S+96J=W$   --- (Equation 1)

From the second condition, 6 senior developers and 8 junior developers complete the module in 15 days:

$(6S+8J)\times 15=W$

$90S+120J=W$   --- (Equation 2)

Step 3: Relate senior developer work rate to junior developer work rate.

Equating Equation 1 and Equation 2:

$108S+96J=90S+120J$

Subtracting $90S$ and $96J$ from both sides:

$18S=24J$

Dividing both sides by 6:

$3S=4J$

$J=\frac{3}{4}S$

Step 4: Calculate total work $W$ in terms of $S$.

Substituting $J=\frac{3}{4}S$ into Equation 1:

$W=108S+96\left(\frac{3}{4}S\right)$

$W=108S+72S$

$W=180S$

Step 5: Solve for $K$.

We are given that $K$ senior developers and 10 junior developers complete the module in 8 days:

$(KS+10J)\times 8=W$

Substitute $J=\frac{3}{4}S$ and $W=180S$:

$(KS+10\times \frac{3}{4}S)\times 8=180S$

$(KS+7.5S)\times 8=180S$

Divide both sides by $S$:

$(K+7.5)\times 8=180$

$8K+60=180$

$8K=120$

$K=15$

Therefore, the value of $K$ is 15.