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Difficulty: HardRatio and Proportion Word Problems

At a textile weaving mill, high-grade cotton, synthetic polyester, and natural silk threads are blended to produce a specialized fabric. Initially, the weights of cotton, polyester, and silk in the blending vat are in the ratio of 5:3:25 : 3 : 2, respectively. To meet a custom order requirement, 1212 kilograms of silk are added to the vat, and 44 kilograms of polyester are removed, while the amount of cotton remains unchanged. If the new ratio of cotton weight to silk weight is 5:45 : 4, how many kilograms of synthetic polyester are in the blending vat after these adjustments?

Answer: 14 kg

Answer

14 kg
Representing the initial weights as 5x5x, 3x3x, and 2x2x, the addition of 1212 kg of silk updates the silk quantity to 2x+122x + 12. Setting the unchanged cotton quantity (5x5x) over the new silk quantity equal to 54\frac{5}{4} yields x=6x = 6. The initial weight of polyester was 3(6)=183(6) = 18 kg, and removing 44 kg gives a final polyester weight of 1414 kg.

Step-by-Step Solution

1
Express initial component weights using a common ratio multiplier xx
Cotton = 5x5x, Polyester = 3x3x, Silk = 2x2x
The given initial ratio of cotton to polyester to silk is 5:3:25 : 3 : 2.
2
Set up a proportion for the updated Cotton-to-Silk ratio
5x2x+12=54\frac{5x}{2x + 12} = \frac{5}{4}
Adding 1212 kg of silk increases its weight to (2x+12)(2x + 12) kg, while cotton remains at 5x5x kg.
3
Solve the algebraic proportion for xx
x=6x = 6
Cross-multiplying gives 20x=5(2x+12)=10x+6020x = 5(2x + 12) = 10x + 60, which simplifies to 10x=6010x = 60, so x=6x = 6.
4
Calculate the final weight of synthetic polyester
14 kg
Initial polyester was 3(6)=183(6) = 18 kg. Subtracting the 44 kg removed leaves 18−4=1418 - 4 = 14 kg.

Key Concept

Multi-part ratios and algebraic setting of updated proportional quantities
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Topics

Ratio and Proportion Word Problems

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