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Difficulty: HardRatio and Proportion Word Problems

An agricultural processing plant produces a livestock feed blend consisting of corn, soybeans, and oats in an initial weight ratio of 5:3:25 : 3 : 2, respectively. To increase the protein content of a 1,200 kg1,200\text{ kg} batch of this blend, a technician removes a certain quantity of oats and replaces it with an equal weight of soybeans. If the resulting weight ratio of corn to soybeans to oats becomes 10:9:110 : 9 : 1, how many kilograms of oats were replaced by soybeans?

  1. A
    120
  2. 180Answer
  3. C
    240
  4. D
    540
  5. E
    720

Answer

180 kg of oats were replaced by soybeans.
The correct answer demonstrates that since corn is unchanged (600 kg600\text{ kg}), it corresponds to 1010 parts in the new 10:9:110 : 9 : 1 ratio. This yields 60 kg60\text{ kg} per part in the new ratio. Oats decrease from 240 kg240\text{ kg} (22 initial parts of 120 kg120\text{ kg}) to 60 kg60\text{ kg} (11 final part of 60 kg60\text{ kg}), indicating that 180 kg180\text{ kg} of oats were replaced by an equal weight of soybeans.

Step-by-Step Solution

1
Determine the initial weight of each component in the 1,200 kg batch.
The initial ratio is 5:3:25 : 3 : 2, giving a total of 5+3+2=105 + 3 + 2 = 10 parts. Each part corresponds to 1,200 kg10=120 kg\frac{1,200\text{ kg}}{10} = 120\text{ kg}. Therefore: Corn =5×120=600 kg= 5 \times 120 = 600\text{ kg}, Soybeans =3×120=360 kg= 3 \times 120 = 360\text{ kg}, Oats =2×120=240 kg= 2 \times 120 = 240\text{ kg}.
Finding the absolute initial quantities provides baseline values before the substitution occurs.
2
Analyze the invariant quantity after the substitution.
Since oats are replaced by an equal weight of soybeans, the total weight of the batch remains 1,200 kg1,200\text{ kg}, and the weight of corn remains unchanged at 600 kg600\text{ kg}.
Identifying unchanged quantities allows direct scaling of the new ratio.
3
Calculate the value of one ratio unit in the new ratio.
The new ratio of corn to soybeans to oats is 10:9:110 : 9 : 1. The 600 kg600\text{ kg} of corn represents 1010 parts of the new ratio. Thus, 1 part=600 kg10=60 kg1\text{ part} = \frac{600\text{ kg}}{10} = 60\text{ kg}. Alternatively, total parts =10+9+1=20= 10 + 9 + 1 = 20 parts, so 1 part=1,200 kg20=60 kg1\text{ part} = \frac{1,200\text{ kg}}{20} = 60\text{ kg}.
Determining the scale of the new ratio allows finding the new weight of each ingredient.
4
Compute the weight of oats replaced.
The new oats weight is 1×60 kg=60 kg1 \times 60\text{ kg} = 60\text{ kg}. The replaced quantity is Initial Oats −- Final Oats =240 kg−60 kg=180 kg= 240\text{ kg} - 60\text{ kg} = 180\text{ kg}.
The difference between initial and final oat weights equals the replaced quantity.

Key Concept

Solving multi-part ratio changes using invariant quantities and constant total mass scaling.
Estimated Time:2m 0s
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Topics

Ratio and Proportion Word Problems

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