Alright, let's solve the problem step-by-step using the six-step method.
### Six-Step Method
1. Understand the Problem:
- Given:
- The number of antibiotic prescriptions is [tex]\(\frac{7}{4}\)[/tex] times the number of tranquilizer prescriptions.
- The total number of prescriptions (both tranquilizers and antibiotics) is 55.
- Find: The number of tranquilizer prescriptions.
2. Devise a Plan:
- Let [tex]\(x\)[/tex] represent the number of tranquilizer prescriptions.
- Let [tex]\(y\)[/tex] represent the number of antibiotic prescriptions.
- Set up two equations based on the given information:
1. [tex]\(y = \frac{7}{4}x\)[/tex]
2. [tex]\(x + y = 55\)[/tex]
3. Carry Out the Plan:
- Substitute [tex]\(y = \frac{7}{4}x\)[/tex] from the first equation into the second equation:
[tex]\[
x + \frac{7}{4}x = 55
\][/tex]
- Combine like terms:
[tex]\[
\left(1 + \frac{7}{4}\right)x = 55
\][/tex]
[tex]\[
\left(\frac{4}{4} + \frac{7}{4}\right)x = 55
\][/tex]
[tex]\[
\frac{11}{4}x = 55
\][/tex]
4. Solve the Equation:
- Multiply both sides by [tex]\(\frac{4}{11}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 55 \times \frac{4}{11}
\][/tex]
- Simplify:
[tex]\[
x = 20
\][/tex]
5. Check the Solution:
- Calculate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{7}{4} \times 20 = 35
\][/tex]
- Verify the total number of prescriptions:
[tex]\[
x + y = 20 + 35 = 55
\][/tex]
The total does match the given condition of 55 prescriptions.
6. State the Answer:
- The number of tranquilizer prescriptions is [tex]\( \boxed{20} \)[/tex].
