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Difficulty: HardFractions and Decimals

A large agricultural cooperative is dividing a massive tract of land for different crops. They allocate 0.428571‾0.\overline{428571} of the total land to cultivate sunflowers, 0.16‾0.1\overline{6} of the total land to cultivate maize, and 0.05‾0.0\overline{5} of the total land to cultivate organic vegetables. The remaining land, which measures exactly 110110 hectares, is preserved as a wildlife reserve. What is the total area of the tract of land, in hectares?

Answer: 315 hectares

Answer

315
The total area is found by properly converting all recurring decimals into exact fractions (3/73/7, 1/61/6, and 1/181/18), summing them to find the total allocated land (41/6341/63), determining the remaining land fraction (22/6322/63), and setting it equal to the given 110110 hectares. Solving for the whole yields exactly 315315 hectares.

Step-by-Step Solution

1
Convert the pure recurring decimal 0.428571‾0.\overline{428571} into a simplified fraction.
The fraction is 37\frac{3}{7}.
Recognizing that 17=0.142857‾\frac{1}{7} = 0.\overline{142857}, we can multiply by 33 to get 0.428571‾0.\overline{428571}. Alternatively, using the algebraic method: 428571999999=37\frac{428571}{999999} = \frac{3}{7}.
2
Convert the mixed recurring decimals 0.16‾0.1\overline{6} and 0.05‾0.0\overline{5} into fractions.
0.16‾=16−190=1590=160.1\overline{6} = \frac{16-1}{90} = \frac{15}{90} = \frac{1}{6} and 0.05‾=5−090=590=1180.0\overline{5} = \frac{5-0}{90} = \frac{5}{90} = \frac{1}{18}.
To operate with mixed recurring decimals, subtract the non-repeating part from the entire number, and place it over a denominator consisting of 9s (for repeating digits) followed by 0s (for non-repeating digits after the decimal point).
3
Calculate the total fraction of land allocated to the three crops.
37+16+118=37+3+118=37+418=37+29=27+1463=4163\frac{3}{7} + \frac{1}{6} + \frac{1}{18} = \frac{3}{7} + \frac{3+1}{18} = \frac{3}{7} + \frac{4}{18} = \frac{3}{7} + \frac{2}{9} = \frac{27+14}{63} = \frac{41}{63}.
Finding a common denominator (6363) allows us to sum the individual crop fractions to determine the total proportion of cultivated land.
4
Determine the fraction representing the wildlife reserve and calculate the total land area.
Reserve fraction = 1−4163=22631 - \frac{41}{63} = \frac{22}{63}. Total Area = 110×6322=315110 \times \frac{63}{22} = 315 hectares.
The unallocated fraction represents the reserve area. Setting this fraction of the total area (TT) equal to 110110 hectares (2263×T=110\frac{22}{63} \times T = 110) gives the final answer.

Key Concept

Fractions and Decimals
Estimated Time:2m 30s
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Topics

Fractions and Decimals

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