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Zorluk: OrtaRatio and Proportion Word Problems

A craft brewery blends three types of hops—Cascade, Centennial, and Mosaic—in a master batch. Initially, the weights of Cascade, Centennial, and Mosaic hops are in the ratio 3:4:53 : 4 : 5, respectively. After 1212 kilograms of Cascade hops and 1212 kilograms of Centennial hops are added to the batch, the new ratio of Cascade to Centennial to Mosaic becomes 5:6:55 : 6 : 5. What was the initial weight, in kilograms, of Centennial hops in the master batch?

  1. A
    66
  2. B
    1818
  3. 2424Cevap
  4. D
    3030
  5. E
    3636

Cevap

The initial weight of Centennial hops in the master batch was 2424 kilograms.
Let the initial weights of Cascade, Centennial, and Mosaic hops be 3x3x, 4x4x, and 5x5x kilograms, respectively. Adding 1212 kg to Cascade and Centennial gives new weights of 3x+123x + 12 kg and 4x+124x + 12 kg, while Mosaic stays at 5x5x kg. Equating the ratio of New Cascade to New Mosaic (3x+125x\frac{3x + 12}{5x}) to the new ratio term (55=1\frac{5}{5} = 1) yields 3x+12=5x3x + 12 = 5x, so 2x=122x = 12 and x=6x = 6. Therefore, the initial weight of Centennial hops was 4x=4(6)=244x = 4(6) = 24 kg.

Adım Adım Çözüm

1
Define initial quantities using a common ratio multiplier xx.
Cascade = 3x3x kg, Centennial = 4x4x kg, Mosaic = 5x5x kg.
An initial ratio of 3:4:53 : 4 : 5 means the actual weights are multiples of xx.
2
Express the new quantities after adding the specified weights.
New Cascade = 3x+123x + 12, New Centennial = 4x+124x + 12, New Mosaic = 5x5x.
1212 kg of Cascade and 1212 kg of Centennial were added, while Mosaic remained unchanged.
3
Set up a proportion using the unchanged quantity and the new ratio.
Since Mosaic quantity remains 5x5x and its component in the new ratio 5:6:55 : 6 : 5 is 55, the multiplier for the new ratio is also xx.
Comparing New Cascade to New Mosaic: 3x+125x=55=1  ⟹  3x+12=5x  ⟹  2x=12  ⟹  x=6\frac{3x + 12}{5x} = \frac{5}{5} = 1 \implies 3x + 12 = 5x \implies 2x = 12 \implies x = 6.
4
Calculate the initial weight of Centennial hops.
Initial Centennial weight = 4x=4×6=244x = 4 \times 6 = 24 kilograms.
Multiplying the ratio part for Centennial (44) by the multiplier (x=6x = 6) gives the initial weight.

Anahtar Kavram

Ratio scaling and setting up algebraic equations for altered multi-part ratios.
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Ratio and Proportion Word Problems

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