An agricultural research station divides an experimental plot of land among three crops: W… — GRE General Test

An agricultural research station divides an experimental plot of land among three crops: Wheat, Rye, and Barley. Initially, Wheat occupies $\frac{4}{9}$ of the total plot area, Rye occupies $\frac{1}{3}$ of the total area, and Barley occupies the remaining area. First, $\frac{1}{3}$ of the Wheat area is converted to Rye. Next, $\frac{2}{5}$ of the new total Rye area is converted to Barley. Finally, $\frac{1}{4}$ of the new total Barley area is converted back to Wheat. If the final area occupied by Wheat is $126$ hectares, what was the total area, in hectares, of the experimental plot?

A
210
315Answer
C
270
D
378
E
540

Answer

315 hectares

By tracking the dynamic changes to each crop's area step-by-step, the final Wheat area is

\frac{54}{135} = \frac{2}{5}

of the total land area. Setting

\frac{2}{5} T = 126

yields

T = 315

hectares.

Step-by-Step Solution

Determine initial fraction allocations for each crop

Wheat:

\frac{4}{9}

, Rye:

\frac{1}{3} = \frac{3}{9}

, Barley:

1 - \left(\frac{4}{9} + \frac{3}{9}\right) = \frac{2}{9}

The sum of all three crop areas must equal the total area (

1

Apply the first conversion (

\frac{1}{3}

of Wheat area converted to Rye)

Wheat transferred:

\frac{1}{3} \times \frac{4}{9} = \frac{4}{27}

. New Wheat:

\frac{4}{9} - \frac{4}{27} = \frac{8}{27}

. New Rye:

\frac{1}{3} + \frac{4}{27} = \frac{13}{27}

. Barley:

\frac{2}{9} = \frac{6}{27}

Wheat loses

\frac{1}{3}

of its current area, and Rye gains that exact amount.

Apply the second conversion (

\frac{2}{5}

of new Rye area converted to Barley)

Rye transferred:

\frac{2}{5} \times \frac{13}{27} = \frac{26}{135}

. New Rye:

\frac{13}{27} - \frac{26}{135} = \frac{39}{135}

. New Barley:

\frac{6}{27} + \frac{26}{135} = \frac{56}{135}

. Wheat remains

\frac{8}{27} = \frac{40}{135}

Rye loses

\frac{2}{5}

of its updated area, and Barley gains that exact amount.

Apply the third conversion (

\frac{1}{4}

of new Barley area converted to Wheat)

Barley transferred:

\frac{1}{4} \times \frac{56}{135} = \frac{14}{135}

. New Wheat:

\frac{40}{135} + \frac{14}{135} = \frac{54}{135} = \frac{2}{5}

Barley loses

\frac{1}{4}

of its updated area, and Wheat gains that exact amount.

Solve for the total plot area

T

\frac{2}{5} T = 126 \implies T = \frac{126 \times 5}{2} = 315

hectares.

Set the final calculated fraction of Wheat equal to the given area of

126

hectares.

Key Concept

Sequential fractional updates of non-disjoint sub-amounts

Estimated Time:2m 30s

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