A commercial bakery prepares a specialty grain blend using oats, wheat, and rye. Initially… — GRE General Test

A commercial bakery prepares a specialty grain blend using oats, wheat, and rye. Initially, the ratio of oats to wheat to rye by weight in the blend is $4 : 3 : 2$ . After $15\text{ kg}$ of oats and $15\text{ kg}$ of rye are added to the mixture while the amount of wheat remains unchanged, the ratio of oats to rye in the new blend becomes $3 : 2$ . Which of the following statements about the final grain blend must be true? Select all such statements.

The weight of wheat in the final blend is $22.5\text{ kg}$ .Answer
The total weight of the final grain blend is $97.5\text{ kg}$ .Answer
The ratio of wheat to rye in the final grain blend is $3 : 4$ .Answer
D
Oats account for exactly $50\%$ of the total weight of the final grain blend.
E
The total weight of the blend increased by $50\%$ from its initial weight.

Answer

The weight of wheat in the final blend is

22.5\text{ kg}

, the total weight of the final grain blend is

97.5\text{ kg}

, and the ratio of wheat to rye in the final grain blend is

3 : 4

Solving the ratio proportion equation yields a multiplier constant of

x = 7.5

. Substituting

x = 7.5

yields final weights of

45\text{ kg}

for oats,

22.5\text{ kg}

for wheat, and

30\text{ kg}

for rye. The weight of wheat is indeed

22.5\text{ kg}

, the total final weight is

45 + 22.5 + 30 = 97.5\text{ kg}

, and the ratio of wheat to rye is

22.5 : 30 = 3 : 4

. Thus, these three statements are correct.

Step-by-Step Solution

Define variables using the initial ratio.

Let the initial weight of oats be

4x

, wheat be

3x

, and rye be

2x

Expressing quantities in terms of a common ratio multiplier

x

ensures proportional relationships are preserved.

Set up an equation using the updated quantities and given new ratio.

\frac{4x + 15}{2x + 15} = \frac{3}{2}

15 kg was added to both oats and rye, establishing a new ratio of 3 to 2 between oats and rye.

Solve for the multiplier

x

2(4x + 15) = 3(2x + 15) \implies 8x + 30 = 6x + 45 \implies 2x = 15 \implies x = 7.5

Cross-multiplying yields the unique value for the ratio constant

x

Calculate the final weight of each component and the total weight.

Initial oats =

30\text{ kg}

, final oats =

45\text{ kg}

. Wheat (unchanged) =

22.5\text{ kg}

. Initial rye =

15\text{ kg}

, final rye =

30\text{ kg}

. Final total weight =

45 + 22.5 + 30 = 97.5\text{ kg}

Evaluating each component confirms all specific properties of the final mixture.

Verify each offered statement against calculated values.

Wheat weight is

22.5\text{ kg}

(True). Total weight is

97.5\text{ kg}

(True). Ratio of Wheat to Rye is

22.5 : 30 = 3 : 4

(True). Oats percentage is

\frac{45}{97.5} \approx 46.15\%

(False). Weight percent increase is

\frac{30}{67.5} \approx 44.44\%

(False).

Comparing calculated facts directly determines which statements must be true.

Key Concept

Ratio adjustment and algebraic formulation of multi-part mixture problems

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