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Difficulty: MediumRatios, Rates, and Proportions

At a certain high school, the ratio of the number of students in the band to the number of students in the orchestra is 3:23:2. The ratio of the number of students in the orchestra to the number of students in the choir is 4:54:5. If there are 120120 students in the band, how many students are in the choir?

  1. A
    6464
  2. B
    8080
  3. 100100Answer
  4. D
    200200
  5. E
    225225

Answer

100 students are in the choir
To find the number of students in the choir, we first use the ratio of band students to orchestra students (3:23:2) to determine the number of orchestra students. Since there are 120120 band students, we set up the proportion 120orchestra=32\frac{120}{\text{orchestra}} = \frac{3}{2}, which yields 8080 orchestra students. Next, we use the ratio of orchestra students to choir students (4:54:5) to find the number of choir students. Setting up the proportion 80choir=45\frac{80}{\text{choir}} = \frac{4}{5} gives choir=80×54=100\text{choir} = 80 \times \frac{5}{4} = 100 students.

Step-by-Step Solution

1
Use the ratio of band students to orchestra students to find the number of orchestra students.
The number of orchestra students is 8080.
Since the ratio of band to orchestra is 3:23:2, and there are 120120 band students, we solve the proportion 120Orchestra=32\frac{120}{\text{Orchestra}} = \frac{3}{2}, giving Orchestra=120×23=80\text{Orchestra} = 120 \times \frac{2}{3} = 80.
2
Use the ratio of orchestra students to choir students to find the number of choir students.
The number of choir students is 100100.
Since the ratio of orchestra to choir is 4:54:5, and there are 8080 orchestra students, we solve the proportion 80Choir=45\frac{80}{\text{Choir}} = \frac{4}{5}, giving Choir=80×54=100\text{Choir} = 80 \times \frac{5}{4} = 100.

Key Concept

Solving multi-step ratio problems by finding a common linking quantity.
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Topics

Ratios, Rates, and Proportions

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