College

A new drug to treat the symptoms of a cold brought relief to [tex]\frac{9}{10}[/tex] of the patients who took it, and [tex]\frac{5}{6}[/tex] 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.

A. [tex]\frac{3}{4}[/tex]
B. [tex]\frac{5}{6}[/tex]
C. [tex]\frac{9}{12}[/tex]
D. [tex]\frac{45}{60}[/tex]

Answer :

To solve this problem, we need to find the fraction of the total number of patients in the study who felt improvement within 3 days of taking the drug.

1. Understand the problem:
- We know that 9 out of 10 patients (which is [tex]\(\frac{9}{10}\)[/tex]) who took the drug found relief.
- Out of these patients who found relief, 5 out of 6 of them (which is [tex]\(\frac{5}{6}\)[/tex]) felt improvement within 3 days.

2. Calculate the overall fraction of patients who felt improvement within 3 days:
- We need to multiply the two fractions to find the fraction of the total number of patients who felt improvement within 3 days:

[tex]\[
\text{Fraction of total patients with improvement} = \frac{9}{10} \times \frac{5}{6}
\][/tex]

3. Perform the multiplication:
- Multiply the numerators: [tex]\(9 \times 5 = 45\)[/tex]
- Multiply the denominators: [tex]\(10 \times 6 = 60\)[/tex]
- So, the fraction is [tex]\(\frac{45}{60}\)[/tex].

4. Simplify the fraction:
- To simplify [tex]\(\frac{45}{60}\)[/tex], find the greatest common divisor (GCD) of 45 and 60, which is 15.
- Divide both the numerator and the denominator by 15:

[tex]\[
\frac{45 ÷ 15}{60 ÷ 15} = \frac{3}{4}
\][/tex]

Therefore, the fraction of the total number of patients in the study who felt improvement within 3 days of taking the drug is [tex]\(\frac{3}{4}\)[/tex].

The correct answer is [tex]\(\frac{3}{4}\)[/tex].