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Difficulty: MediumRatio and Proportion Word Problems

A specialty tea manufacturer blends three varieties of tea leaves—Darjeeling, Assam, and Ceylon. Initially, the weights of Darjeeling, Assam, and Ceylon in a batch are in the ratio 3:4:53 : 4 : 5, respectively. After 2020 kilograms of Ceylon tea leaves are removed from the batch and replaced with 2020 kilograms of Darjeeling tea leaves, the new ratio of Darjeeling to Assam to Ceylon becomes 5:4:35 : 4 : 3. What was the total weight, in kilograms, of the initial tea batch?

Answer: 120 kg

Answer

The initial total weight of the tea batch was 120 kg.
By assigning xx as the multiplier for the initial ratio 3:4:53 : 4 : 5, the initial total weight is 12x12x. The swap of 2020 kg changes Darjeeling to 3x+203x + 20 and Ceylon to 5x−205x - 20, while Assam remains 4x4x. Setting 3x+204x=54\frac{3x + 20}{4x} = \frac{5}{4} gives x=10x = 10, making the initial total weight 12×10=12012 \times 10 = 120 kg.

Step-by-Step Solution

1
Define initial quantities using a common ratio multiplier xx.
Darjeeling = 3x3x, Assam = 4x4x, Ceylon = 5x5x. Total weight = 12x12x.
Expressing quantities in terms of a single variable xx allows setting up algebraic equations based on the changed quantities.
2
Adjust quantities after the exchange of tea leaves.
New Darjeeling = 3x+203x + 20, New Assam = 4x4x, New Ceylon = 5x−205x - 20.
Replacing 2020 kg of Ceylon with 2020 kg of Darjeeling increases Darjeeling by 2020 kg and decreases Ceylon by 2020 kg while leaving Assam unchanged.
3
Set up a proportion equation comparing the new ratio of Darjeeling to Assam.
3x+204x=54\frac{3x + 20}{4x} = \frac{5}{4}
The new ratio of Darjeeling to Assam is given as 5:45 : 4.
4
Solve the linear equation for xx.
x=10x = 10
Cross-multiplying yields 12x+80=20x12x + 80 = 20x, which simplifies to 8x=808x = 80.
5
Compute the total initial weight.
12×10=12012 \times 10 = 120 kg
The initial total weight is 12x12x, so substituting x=10x = 10 gives the final answer of 120120 kg.

Key Concept

Setting up algebraic equations for 3-part ratios under quantity addition/subtraction.
Estimated Time:1m 30s
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Topics

Ratio and Proportion Word Problems

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