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Difficulty: MediumRatio and Proportion Word Problems

In a software development department, the numbers of Frontend developers, Backend developers, and DevOps engineers were initially in the ratio of 4:5:34 : 5 : 3, respectively. After the department hired 66 additional DevOps engineers, with no changes to the number of Frontend or Backend developers, the ratio of Backend developers to DevOps engineers became 1:11 : 1. What is the total number of developers and engineers in the department after these new hires?

  1. A
    30
  2. B
    36
  3. 42Answer
  4. D
    48
  5. E
    78

Answer

The total number of developers and engineers in the department after the new hires is 42.
By setting the initial counts of Frontend developers, Backend developers, and DevOps engineers as 4x4x, 5x5x, and 3x3x, adding 6 to DevOps yields 3x+63x + 6. Equating Backend and DevOps counts gives 5x=3x+65x = 3x + 6, so x=3x = 3. Substituting x=3x = 3 into the post-hire counts gives 12 Frontend, 15 Backend, and 15 DevOps, summing to a total of 42 employees.

Step-by-Step Solution

1
Define initial quantities using a common ratio multiplier.
Let the initial numbers of Frontend developers, Backend developers, and DevOps engineers be 4x4x, 5x5x, and 3x3x, respectively.
Ratios allow representing unknown totals in terms of a single multiplier xx.
2
Express the updated counts after hiring new DevOps engineers.
Frontend = 4x4x, Backend = 5x5x, and DevOps = 3x+63x + 6.
Only the DevOps group increases by 6 while other group sizes remain unchanged.
3
Set up an equation based on the new Backend-to-DevOps ratio of 1:11 : 1.
5x=3x+6  ⟹  2x=6  ⟹  x=35x = 3x + 6 \implies 2x = 6 \implies x = 3.
A 1:11 : 1 ratio means the number of Backend developers equals the new number of DevOps engineers.
4
Calculate the final total number of employees.
Frontend = 4(3)=124(3) = 12, Backend = 5(3)=155(3) = 15, DevOps = 3(3)+6=153(3) + 6 = 15. Total = 12+15+15=4212 + 15 + 15 = 42.
Summing all three roles after adding the 6 new hires gives the final count.

Key Concept

Ratio scaling and setting up algebraic equations from modified multi-part ratios.
Estimated Time:1m 30s
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Topics

Ratio and Proportion Word Problems

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