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Difficulty: MediumRatio and Proportion Word Problems

An architectural firm initially allocates its annual project hours among three divisions—Residential, Commercial, and Urban Planning—in the ratio of 7:5:37 : 5 : 3, respectively. Mid-year, to meet new project demands, 300300 hours from Residential and 100100 hours from Commercial are transferred to Urban Planning. Following this transfer, the ratio of Commercial hours to Urban Planning hours becomes 1:11 : 1. What was the total number of project hours initially allocated across all three divisions?

Answer: 3750 hours

Answer

The total number of project hours initially allocated across all three divisions was 3,750 hours.
Representing the initial hours as 7x7x, 5x5x, and 3x3x gives a total of 15x15x hours. After the transfers, Commercial hours equal 5x−1005x - 100 and Urban Planning hours equal 3x+300+100=3x+4003x + 300 + 100 = 3x + 400. Setting these equal gives 5x−100=3x+4005x - 100 = 3x + 400, which simplifies to 2x=5002x = 500, or x=250x = 250. The initial total hours were 15×250=3,75015 \times 250 = 3,750.

Step-by-Step Solution

1
Define initial quantities using a multiplier variable
Let xx represent the multiplier. Initial hours are Residential = 7x7x, Commercial = 5x5x, and Urban Planning = 3x3x. Total initial hours = 15x15x.
Ratios define the proportional relationship among the three divisions.
2
Determine updated hours after the transfers
Commercial hours become 5x−1005x - 100. Urban Planning receives 300300 hours from Residential and 100100 hours from Commercial, becoming 3x+300+100=3x+4003x + 300 + 100 = 3x + 400.
Quantities added to Urban Planning must equal the sum of hours subtracted from the other two divisions.
3
Set up and solve the equation for the new ratio
Since Commercial and Urban Planning hours are now in a 1:11 : 1 ratio, 5x−100=3x+400  ⟹  2x=500  ⟹  x=2505x - 100 = 3x + 400 \implies 2x = 500 \implies x = 250.
A 1:11 : 1 ratio means both quantities are equal.
4
Calculate the initial total project hours
Total initial hours = 15×250=3,75015 \times 250 = 3,750.
Substitute x=250x = 250 into the expression for total initial hours (15x15x).

Key Concept

Solving multi-part ratio word problems involving internal transfers by setting up algebraic equations with a single unknown multiplier.
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Topics

Ratio and Proportion Word Problems

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