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Difficulty: HardRatio and Proportion Word Problems

A freight logistics warehouse initially stores Standard, Refrigerated, and Oversized cargo containers in the ratio 4:3:24 : 3 : 2, respectively. During a morning shift, 1212 Standard containers are shipped out and 66 Refrigerated containers arrive at the warehouse, while no Oversized containers are moved. Following these changes, the ratio of Standard containers to Refrigerated containers in the warehouse becomes 1:11 : 1. What was the total number of containers (Standard, Refrigerated, and Oversized) initially stored in the warehouse?

  1. A
    18
  2. B
    72
  3. C
    126
  4. D
    156
  5. 162Answer

Answer

162
The correct answer 162 is obtained by representing the initial numbers of containers as 4x4x, 3x3x, and 2x2x. Equating the modified quantities of Standard (4x−124x - 12) and Refrigerated (3x+63x + 6) yields x=18x = 18. Summing all three parts gives 9x=9×18=1629x = 9 \times 18 = 162.

Step-by-Step Solution

1
Define initial quantities using a common variable based on the given three-part ratio.
Let the initial numbers of Standard, Refrigerated, and Oversized containers be 4x4x, 3x3x, and 2x2x, respectively. The initial total count of containers is 4x+3x+2x=9x4x + 3x + 2x = 9x.
Ratios represent relative parts, so multiplying each term by a constant multiplier xx yields the actual quantities.
2
Set up an equation reflecting the change in container quantities.
After shipping 1212 Standard containers and adding 66 Refrigerated containers, the new quantities are (4x−12)(4x - 12) Standard containers and (3x+6)(3x + 6) Refrigerated containers. Since their new ratio is 1:11 : 1, we write 4x−12=3x+64x - 12 = 3x + 6.
A 1:11 : 1 ratio means the two quantities are equal.
3
Solve for the multiplier xx.
4x−3x=6+12  ⟹  x=184x - 3x = 6 + 12 \implies x = 18.
Isolating xx gives the ratio scaling factor.
4
Calculate the initial total number of containers.
Initial total =9x=9×18=162= 9x = 9 \times 18 = 162.
The question asks for the total initial count across all three container categories (4x+3x+2x=9x4x + 3x + 2x = 9x).

Key Concept

Altering linear ratios by solving for a common multiplier across multi-part ratio components.
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Topics

Ratio and Proportion Word Problems

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