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Difficulty: MediumRatio and Proportion Word Problems

At a research institute, the ratio of senior scientists to associate scientists to junior researchers was initially 2:3:52 : 3 : 5, respectively. During an organizational expansion, 8 new senior scientists were hired, and 4 junior researchers were promoted to associate scientists, while no other personnel changes occurred. If the new ratio of senior scientists to associate scientists became 4:54 : 5, what was the initial total number of researchers at the institute?

  1. A
    40
  2. B
    60
  3. C
    100
  4. 120Answer
  5. E
    128

Answer

120
By assigning a multiplier xx to the initial ratio 2:3:52 : 3 : 5, the counts are 2x2x senior scientists, 3x3x associate scientists, and 5x5x junior researchers, making the initial total 10x10x. Adding 8 senior scientists gives 2x+82x + 8, and adding 4 promoted associate scientists gives 3x+43x + 4. Equating their ratio to 45\frac{4}{5} yields 2x+83x+4=45\frac{2x + 8}{3x + 4} = \frac{4}{5}, which simplifies to 2x=242x = 24, or x=12x = 12. Multiplying x=12x = 12 by the total 10 parts gives the correct initial total of 120.

Step-by-Step Solution

1
Define initial quantities using a common ratio multiplier xx.
Senior scientists = 2x2x, Associate scientists = 3x3x, Junior researchers = 5x5x. Initial total researchers = 2x+3x+5x=10x2x + 3x + 5x = 10x.
Expressing quantities in terms of xx allows setting up algebraic equations based on personnel updates.
2
Apply the specified personnel changes to find the new counts of senior and associate scientists.
New Senior scientists = 2x+82x + 8. New Associate scientists = 3x+43x + 4 (since 4 junior researchers were promoted to associate scientists).
Promoting 4 junior researchers increases the associate scientist count by 4.
3
Set up the proportion equation for the new ratio of senior scientists to associate scientists.
\frac{2x + 8}{3x + 4} = \frac{4}{5}
The problem states the new ratio between senior and associate scientists is 4:54 : 5.
4
Cross-multiply and solve for xx.
5(2x + 8) = 4(3x + 4) \implies 10x + 40 = 12x + 16 \implies 2x = 24 \implies x = 12.
Solving for xx provides the multiplier required to compute the initial total.
5
Calculate the initial total number of researchers.
Initial total = 10x=10×12=12010x = 10 \times 12 = 120.
The total number of initial researchers corresponds to 10x10x parts.

Key Concept

Setting up and solving algebraic proportions involving multi-part ratios after internal transfers and external additions
Estimated Time:2m 0s
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Topics

Ratio and Proportion Word Problems

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