Which of the following ratios is the greatest?

Let us convert each given ratio into fraction form and evaluate their approximate decimal values or compare them structurally:
* Option A: $\frac{7}{15}\approx 0.4667$
* Option B: $\frac{15}{23}\approx 0.6521$
* Option C: $\frac{17}{25}=0.6800$
* Option D: $\frac{21}{29}\approx 0.7241$

Alternative Property Method:
Notice that for options B, C, and D, the difference between the denominator and the numerator is a constant value of $8$:

* $23-15=8$
* $25-17=8$
* $29-21=8$

For proper fractions of the form $\frac{x}{x+k}$ where $k>0$, as the value of $x$ increases, the value of the fraction also increases.
Comparing the numerators: $21>17>15$.
Therefore, $\frac{21}{29}$ is the greatest fraction.