A new drug to treat the symptoms of a cold brought relief to 9/10 of the patients who took it, and 5/6 of those patients felt improvement within 3 days. What fraction of the total number of patients in the study felt improvement within 3 days of taking the drug? Write the answer in simplest form.
The fraction of patients who felt improvement within 3 days is the fraction of patients who both took the drug and felt improvement within 3 days. First we must compute the total fraction of patients who both took the drug and felt improvement, which is $\frac{9}{10} \times \frac{5}{6} = \frac{45}{60} = \frac{3}{4}$. Only three of the four parts are patients who took the drug, so $\frac{3}{4}$ of the patients felt improvement within 3 days. Simplest form yields a fraction of $\boxed{\frac{3}{4}}$.
