Which of the following ratios is greatest?

Let us convert each ratio into its decimal value to find the greatest one:
* $\frac{7}{15}\approx 0.4667$
* $\frac{15}{23}\approx 0.6521$
* $\frac{17}{25}=0.68$
* $\frac{21}{29}\approx 0.7241$

Alternatively, notice that for the proper fractions $\frac{15}{23}$, $\frac{17}{25}$, and $\frac{21}{29}$, the difference between the denominator and numerator is constant ($23-15=8$, $25-17=8$, $29-21=8$). For fractions where the difference $\text{Denominator}-\text{Numerator}=k>0$, the fraction with the larger numbers is greater. Thus, $\frac{21}{29}$ is the greatest.