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Difficulty: Very hardRatio and Proportion Word Problems

An architectural restoration project allocates its workforce across three specialized teams: Masonry, Carpentry, and Glasswork. Initially, the ratio of Masonry workers to Carpentry workers is 5:65 : 6, and the ratio of Carpentry workers to Glasswork workers is 4:54 : 5.

During phase two of the project:
• The number of Masonry workers is increased by 50%50\%.
• The number of Carpentry workers is decreased by 25%25\%.
• The number of Glasswork workers is increased by X%X\%.

If the new ratio of Masonry workers to the total number of workers across all three teams is 1:31 : 3, what is the value of XX?

Answer: 40

Answer

The value of XX is 40.
To find XX, first combine the initial ratios by equating the common Carpentry term: Masonry : Carpentry = 10 : 12 and Carpentry : Glasswork = 12 : 15, yielding a joint ratio of 10 : 12 : 15. Representing the initial teams as 10k10k, 12k12k, and 15k15k, the updated Masonry team is 10k×1.5=15k10k \times 1.5 = 15k and Carpentry is 12k×0.75=9k12k \times 0.75 = 9k. Using the part-to-total ratio New MasonryNew Total=13\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}, we set up 15k15k+9k+New Glasswork=13\frac{15k}{15k + 9k + \text{New Glasswork}} = \frac{1}{3}, which yields New Glasswork=21k\text{New Glasswork} = 21k. The percentage increase from 15k15k to 21k21k is 21k−15k15k×100%=40%\frac{21k - 15k}{15k} \times 100\% = 40\%. Thus, X=40X = 40.

Step-by-Step Solution

1
Unify the two given ratios into a single three-part ratio.
The ratio Masonry : Carpentry : Glasswork is 10:12:1510 : 12 : 15.
Carpentry is common to both given ratios (5:65 : 6 and 4:54 : 5). The least common multiple of 6 and 4 is 12. Scaling 5:65 : 6 by 2 yields 10:1210 : 12, and scaling 4:54 : 5 by 3 yields 12:1512 : 15.
2
Define initial workforce quantities using a multiplier kk.
Initial Masonry = 10k10k, Carpentry = 12k12k, Glasswork = 15k15k.
Using an algebraic variable preserves the proportional relationships across all teams.
3
Calculate the updated worker counts for Masonry and Carpentry.
New Masonry = 15k15k, New Carpentry = 9k9k.
Masonry increases by 50%50\% (10k×1.50=15k10k \times 1.50 = 15k). Carpentry decreases by 25%25\% (12k×0.75=9k12k \times 0.75 = 9k).
4
Set up the part-to-total ratio equation to solve for the new Glasswork team size.
New Glasswork worker count = 21k21k.
The total workforce is 15k+9k+New Glasswork=24k+New Glasswork15k + 9k + \text{New Glasswork} = 24k + \text{New Glasswork}. Given New MasonryNew Total=13\frac{\text{New Masonry}}{\text{New Total}} = \frac{1}{3}, we have 15k24k+New Glasswork=13  ⟹  45k=24k+New Glasswork  ⟹  New Glasswork=21k\frac{15k}{24k + \text{New Glasswork}} = \frac{1}{3} \implies 45k = 24k + \text{New Glasswork} \implies \text{New Glasswork} = 21k.
5
Compute the percentage increase XX for the Glasswork team.
X=40X = 40.
The initial Glasswork count was 15k15k and the new count is 21k21k. The percentage increase is 21k−15k15k×100=615×100=40%\frac{21k - 15k}{15k} \times 100 = \frac{6}{15} \times 100 = 40\%.

Key Concept

Combining compound ratios and solving multi-step part-to-whole proportion equations.
Estimated Time:2m 30s
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Topics

Ratio and Proportion Word Problems

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