Let's solve the equation \(\frac{q-r}{t} = \frac{v-q}{v}\) to make \(q\) the subject.

1. Start by cross-multiplying to eliminate the fractions:
\[
v(q - r) = t(v - q)
\]

2. Distribute both sides:
\[
vq - vr = tv - tq
\]

3. Collect all terms involving \(q\) on one side of the equation:
\[
vq + tq = tv + vr
\]

4. Factor out \(q\) from the left side:
\[
q(v + t) = tv + vr
\]

5. Finally, divide both sides by \((v + t)\) to solve for \(q\):
\[
q = \frac{tv + vr}{v + t}
\]

6. Factor out \(v\) from the numerator:
\[
q = \frac{v(t + r)}{v + t}
\]

So, the correct answer is A. \(q = \frac{v(t + r)}{v + t}\).