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Difficulty: HardRatios, Rates, and Proportions

An artisan bakery produces a signature flour blend by combining wheat, rye, and oat flour. Initially, the ratio of wheat flour to rye flour by weight is 5:35:3, and the ratio of rye flour to oat flour by weight is 4:14:1. The baker prepares an initial batch of this mixture weighing exactly 140140 pounds. To adjust the recipe for a special order, pure oat flour is added to the batch until oat flour accounts for exactly 20%20\% of the total weight of the new mixture. How many pounds of pure oat flour must the baker add?

  1. A
    12
  2. B
    16
  3. 20Answer
  4. D
    28
  5. E
    32

Answer

20 pounds
To solve this problem, first express the ratios of wheat, rye, and oat flour with a common term for rye. Since Wheat : Rye = 5 : 3 (or 20 : 12) and Rye : Oat = 4 : 1 (or 12 : 3), the combined ratio Wheat : Rye : Oat is 20 : 12 : 3, giving 35 total parts. In a 140-pound batch, each part equals 4 pounds, meaning the batch contains 12 pounds of oat flour and 128 pounds of non-oat flour. When additional oat flour is added, the non-oat weight stays fixed at 128 pounds. For oat flour to be 20% of the new mixture, non-oat flour must be 80%. Dividing 128 by 0.80 gives a new total weight of 160 pounds. The new oat weight is 20% of 160, which is 32 pounds. Subtracting the initial 12 pounds yields 20 pounds of added oat flour.

Step-by-Step Solution

1
Combine the two given ratios into a single three-part ratio.
Wheat : Rye = 5:3=20:125 : 3 = 20 : 12, and Rye : Oat = 4:1=12:34 : 1 = 12 : 3. Thus, Wheat : Rye : Oat = 20:12:320 : 12 : 3.
Scaling the ratios so that the common element (Rye) has equal parts allows unification into a single ratio.
2
Calculate the initial weight of each component in the 140-pound batch.
Total ratio parts = 20+12+3=3520 + 12 + 3 = 35 parts. Each part is 140/35=4140 / 35 = 4 pounds. Initial Oat flour = 3×4=123 \times 4 = 12 pounds. Non-oat flour (Wheat + Rye) = (20+12)×4=128(20 + 12) \times 4 = 128 pounds.
Determining the weight of the constant non-oat portion is key to solving mixture adjustment problems.
3
Set up an equation for the new mixture where oat flour makes up 20% of the total weight.
Since non-oat flour remains unchanged at 128128 pounds, it represents 80%80\% of the new total weight. New Total Weight = 128/0.80=160128 / 0.80 = 160 pounds.
If oat flour is 20% of the new total, the non-oat components must constitute the remaining 80%.
4
Calculate the weight of oat flour added.
New Oat Weight = 20%20\% of 160=32160 = 32 pounds. Added Oat Flour = 32−12=2032 - 12 = 20 pounds.
Subtracting the initial oat weight from the final required oat weight yields the amount added.

Key Concept

Combining multi-part ratios and solving ratio adjustment problems by holding unchanged components constant.
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Topics

Ratios, Rates, and Proportions

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