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Difficulty: Very hardRatio and Proportion Word Problems

A commercial bakery prepares a specialty grain blend using Organic Wheat, Spelt, and Rye flour. Initially, the ratio of Organic Wheat to Spelt to Rye flour in the batch is 4:3:24 : 3 : 2 by weight. To meet specific recipe requirements, 3030 kilograms of Organic Wheat and 1515 kilograms of Spelt are added to the batch, while 1010 kilograms of Rye are removed. After these adjustments, the ratio of Organic Wheat to Rye flour in the batch becomes 3:13 : 1. What was the total weight, in kilograms, of the initial batch of flour before any adjustments were made?

  1. A
    120
  2. B
    135
  3. C
    210
  4. 270Answer
  5. E
    305

Answer

270 kilograms
By representing the initial amounts of Organic Wheat, Spelt, and Rye as 4x4x, 3x3x, and 2x2x kilograms respectively, the initial total weight is 9x9x kilograms. Following the adjustments, the new amount of Wheat is 4x+304x + 30 kg and the new amount of Rye is 2x−102x - 10 kg. Equating their ratio to 3:13 : 1 gives 4x+302x−10=31\frac{4x + 30}{2x - 10} = \frac{3}{1}. Solving for xx gives 4x+30=6x−304x + 30 = 6x - 30, leading to 2x=602x = 60 or x=30x = 30. Therefore, the initial total weight of the flour batch was 9×30=2709 \times 30 = 270 kilograms.

Step-by-Step Solution

1
Define initial quantities using a common multiplier xx.
Organic Wheat =4x= 4x, Spelt =3x= 3x, Rye =2x= 2x. Initial total weight =4x+3x+2x=9x= 4x + 3x + 2x = 9x.
The initial ratio 4:3:24 : 3 : 2 specifies the relative proportions of the three components in the batch.
2
Express the updated amounts after additions and removals.
New Organic Wheat =4x+30= 4x + 30, New Spelt =3x+15= 3x + 15, New Rye =2x−10= 2x - 10.
Incorporating the specified additions (+30+30 kg Wheat, +15+15 kg Spelt) and removal (−10-10 kg Rye).
3
Set up an equation using the new ratio of Organic Wheat to Rye (3:13 : 1).
4x+302x−10=31\frac{4x + 30}{2x - 10} = \frac{3}{1}
The problem states that the updated ratio of Wheat to Rye is 3:13 : 1.
4
Solve for the multiplier xx.
4x+30=3(2x−10)  ⟹  4x+30=6x−30  ⟹  2x=60  ⟹  x=304x + 30 = 3(2x - 10) \implies 4x + 30 = 6x - 30 \implies 2x = 60 \implies x = 30.
Cross-multiplying and solving the linear equation yields x=30x = 30.
5
Calculate the total weight of the initial batch.
Initial total =9x=9×30=270= 9x = 9 \times 30 = 270 kilograms.
Substituting x=30x = 30 back into the initial total expression 9x9x gives the required initial batch mass.

Key Concept

Ratio Modification and Multi-Part Proportions
Estimated Time:2m 0s
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Topics

Ratio and Proportion Word Problems

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