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Difficulty: HardWeighted Average and Combined Sets

In a university computer science department, students belong to one of three groups: Undergraduates, Master's students, or Doctoral candidates. The ratio of the number of Undergraduates to Master's students is 3:23 : 2, and the ratio of the number of Master's students to Doctoral candidates is 4:14 : 1. The average number of weekly research hours is 66 hours for Undergraduates and 1414 hours for Master's students. If the combined average number of weekly research hours across all three groups is 1111 hours, what is the average number of weekly research hours for a Doctoral candidate?

  1. A
    13
  2. B
    18
  3. C
    20
  4. D
    21
  5. 29Answer

Answer

The average number of weekly research hours for a Doctoral candidate is 29 hours.
To find the average weekly research hours for Doctoral candidates, first combine the group ratios. Given U:M=3:2U : M = 3 : 2 and M:D=4:1M : D = 4 : 1, scale U:MU : M to 6:46 : 4 so that the MM term is identical in both ratios. This yields a single combined ratio U:M:D=6:4:1U : M : D = 6 : 4 : 1, giving a total of 6+4+1=116 + 4 + 1 = 11 equal parts.

Next, calculate the total research hours across all parts: 6(6)+4(14)+1(x)=36+56+x=92+x6(6) + 4(14) + 1(x) = 36 + 56 + x = 92 + x. Setting the overall weighted average to 1111 yields 92+x11=11\frac{92 + x}{11} = 11, which simplifies to 92+x=12192 + x = 121, so x=29x = 29.

Step-by-Step Solution

1
Determine the combined ratio of students across the three groups.
Undergraduates : Master's : Doctoral = 6:4:16 : 4 : 1.
We are given U:M=3:2U : M = 3 : 2 and M:D=4:1M : D = 4 : 1. To combine these into a single ratio U:M:DU : M : D, scale U:MU : M by multiplying by 22 so that the MM term matches: U:M=6:4U : M = 6 : 4. Thus, U:M:D=6:4:1U : M : D = 6 : 4 : 1.
2
Express total students and total research hours in terms of a multiplier kk.
Total students = 11k11k; Total hours = 36k+56k+kx=92k+kx36k + 56k + kx = 92k + kx.
Let the number of students be U=6kU = 6k, M=4kM = 4k, and D=1kD = 1k. The total research hours contributed by Undergraduates is 6k×6=36k6k \times 6 = 36k, by Master's students is 4k×14=56k4k \times 14 = 56k, and by Doctoral candidates is 1k×x=kx1k \times x = kx.
3
Set up the weighted average formula and solve for xx.
x=29x = 29.
The combined average is given by Total HoursTotal Students=11\frac{\text{Total Hours}}{\text{Total Students}} = 11. Substituting our values gives 92k+kx11k=11  ⟹  92+x11=11  ⟹  92+x=121  ⟹  x=29\frac{92k + kx}{11k} = 11 \implies \frac{92 + x}{11} = 11 \implies 92 + x = 121 \implies x = 29.

Key Concept

Weighted Average and Combined Sets with Compound Ratios
Estimated Time:2m 0s
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Topics

Weighted Average and Combined Sets

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