A baker has a flour mixture consisting only of wheat flour and rye flour. Currently, wheat… — GRE General Test

A baker has a flour mixture consisting only of wheat flour and rye flour. Currently, wheat flour accounts for $\frac{2}{5}$ of the total weight of the mixture. If the baker adds $9$ pounds of wheat flour to the mixture, wheat flour will account for $\frac{1}{2}$ of the new total weight of the mixture. What was the total weight, in pounds, of the original flour mixture?

Answer: 45 pounds

Answer

The total weight of the original flour mixture was 45 pounds.

The initial total weight of the mixture is 45 pounds. Initially, wheat flour makes up

\frac{2}{5} \times 45 = 18

pounds. Adding 9 pounds of wheat flour increases the wheat flour to

18 + 9 = 27

pounds and the total weight to

45 + 9 = 54

pounds. The new fraction of wheat flour is

\frac{27}{54} = \frac{1}{2}

, which satisfies the given conditions.

Step-by-Step Solution

Express the initial weight of wheat flour in terms of the initial total weight

W

Initial weight of wheat flour =

\frac{2}{5}W

Wheat flour represents

\frac{2}{5}

of the total mixture.

Formulate an equation reflecting the addition of 9 pounds of wheat flour.

\frac{\frac{2}{5}W + 9}{W + 9} = \frac{1}{2}

Adding 9 pounds of wheat flour increases both the amount of wheat flour and the total weight of the mixture by 9 pounds.

Solve the algebraic equation for

W

Cross-multiplying gives

2\left(\frac{2}{5}W + 9\right) = W + 9

, which simplifies to

\frac{4}{5}W + 18 = W + 9

. Subtracting

\frac{4}{5}W

and

9

from both sides gives

\frac{1}{5}W = 9

, so

W = 45

Isolating

W

gives the value of the original total weight.

Key Concept

Setting up and solving equations involving fractional parts when a quantity is added to both the part and the whole.

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