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Difficulty: MediumRatios, Rates, and Proportions

In a science department, the ratio of the number of students enrolled in Chemistry to the number of students enrolled in Physics is 55 to 33. The ratio of the number of students enrolled in Physics to the number of students enrolled in Biology is 22 to 77. If no student is enrolled in more than one of these courses and there are 105105 students enrolled in Biology, what is the total number of students enrolled in these three courses?

  1. A
    135
  2. B
    153
  3. 185Answer
  4. D
    225

Answer

185
To find the total number of students, we must determine the enrollment in each course. The number of Biology students is given as 105105. The ratio of Physics to Biology students is 22 to 77, which means the number of Physics students is 105×27=30105 \times \frac{2}{7} = 30. The ratio of Chemistry to Physics students is 55 to 33, so the number of Chemistry students is 30×53=5030 \times \frac{5}{3} = 50. Summing these values gives the total enrollment: 50+30+105=18550 + 30 + 105 = 185.

Step-by-Step Solution

1
Calculate the number of students enrolled in Physics
30 students
Since the ratio of Physics to Biology is 22 to 77 and there are 105105 students in Biology, we set up the proportion Physics105=27\frac{\text{Physics}}{105} = \frac{2}{7}, which yields Physics=105×27=30\text{Physics} = 105 \times \frac{2}{7} = 30.
2
Calculate the number of students enrolled in Chemistry
50 students
Since the ratio of Chemistry to Physics is 55 to 33 and there are 3030 students in Physics, we set up the proportion Chemistry30=53\frac{\text{Chemistry}}{30} = \frac{5}{3}, which yields Chemistry=30×53=50\text{Chemistry} = 30 \times \frac{5}{3} = 50.
3
Calculate the total number of students enrolled in the three courses
185 students
Add the students enrolled in all three courses: 50 (Chemistry)+30 (Physics)+105 (Biology)=18550 \text{ (Chemistry)} + 30 \text{ (Physics)} + 105 \text{ (Biology)} = 185.

Key Concept

Solving multi-step ratio problems by finding a common variable to link multiple ratios.
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Topics

Ratios, Rates, and Proportions

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