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Difficulty: MediumRatio and Proportion Word Problems

An artisanal coffee roaster creates a custom blend using Arabica, Robusta, and Liberica beans in the initial ratio of 5:3:25 : 3 : 2 by weight, respectively. To adjust the flavor profile, the roaster adds 6060 kilograms of Arabica beans and 2020 kilograms of Robusta beans to the blend, while leaving the amount of Liberica beans unchanged. If the new ratio of Arabica to Robusta to Liberica beans is 4:2:14 : 2 : 1, respectively, what was the initial total weight, in kilograms, of the blend?

  1. A
    100
  2. B
    120
  3. C
    160
  4. 200Answer
  5. E
    280

Answer

200 kilograms
Let the initial weights of Arabica, Robusta, and Liberica beans be 5x5x, 3x3x, and 2x2x kilograms, respectively. The initial total weight is 5x+3x+2x=10x5x + 3x + 2x = 10x kg. After adding 6060 kg of Arabica and 2020 kg of Robusta, the new weights are 5x+605x + 60, 3x+203x + 20, and 2x2x. Using the new ratio of Robusta to Liberica (2:12 : 1), we set up the equation 3x+202x=2\frac{3x + 20}{2x} = 2, which simplifies to 3x+20=4x3x + 20 = 4x, giving x=20x = 20. Substituting x=20x = 20 into the initial total weight expression yields 10(20)=20010(20) = 200 kg.

Step-by-Step Solution

1
Define initial quantities using a common multiplier xx.
Arabica =5x= 5x, Robusta =3x= 3x, Liberica =2x= 2x. Initial total weight =5x+3x+2x=10x= 5x + 3x + 2x = 10x.
Expressing ratio terms in terms of a single multiplier allows algebraic modeling of changes to each component.
2
Express the new quantities after adding the specified weights.
New Arabica =5x+60= 5x + 60, New Robusta =3x+20= 3x + 20, New Liberica =2x= 2x.
Arabica and Robusta amounts increase by 60 kg and 20 kg respectively, while Liberica remains constant.
3
Set up a proportion using the new ratio of Robusta to Liberica (2:12 : 1).
3x+202x=21  ⟹  3x+20=4x  ⟹  x=20\frac{3x + 20}{2x} = \frac{2}{1} \implies 3x + 20 = 4x \implies x = 20.
Equating the ratio of the updated Robusta and Liberica quantities to 2:12:1 yields the multiplier xx.
4
Verify consistency with the Arabica to Liberica ratio (4:14 : 1).
5(20)+602(20)=16040=41\frac{5(20) + 60}{2(20)} = \frac{160}{40} = \frac{4}{1}, which matches the given final ratio.
Ensures that x=20x = 20 satisfies the complete 3-part ratio of 4:2:14 : 2 : 1.
5
Calculate the initial total weight.
Initial total =10x=10×20=200= 10x = 10 \times 20 = 200 kg.
Multiplying the initial sum of ratio parts (10) by x=20x = 20 gives the original total blend weight.

Key Concept

Ratio and Proportion Word Problems
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Topics

Ratio and Proportion Word Problems

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