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Difficulty: HardFractions and Decimals

A municipal corporation is distributing its annual infrastructure development fund. Exactly 0.5‾0.\overline{5} of the total fund is allocated to urban housing projects. From the remaining fund, 37\frac{3}{7} is assigned to public transportation. The rest of the fund is distributed equally between two sectors: renewable energy and public parks. If the amount allocated to renewable energy is $3.2\$3.2 million, what is the total annual infrastructure development fund?

  1. A
    $12.6\$12.6 million
  2. B
    $22.4\$22.4 million
  3. $25.2\$25.2 millionAnswer
  4. D
    $403.2\$403.2 million

Answer

$25.2\$25.2 million
The correct sequence evaluates the recurring decimal 0.5‾0.\overline{5} as 59\frac{5}{9}. This leaves 49\frac{4}{9} of the fund. Transport takes 37\frac{3}{7} of this remainder, which is 1263\frac{12}{63}. The new remainder is 49−1263=1663\frac{4}{9} - \frac{12}{63} = \frac{16}{63}. This is split equally, meaning renewable energy gets 863\frac{8}{63} of the total. Solving 863x=3.2\frac{8}{63}x = 3.2 gives x=25.2x = 25.2.

Step-by-Step Solution

1
Convert the recurring decimal to a fraction.
0.5‾=590.\overline{5} = \frac{5}{9}
Fractional form is required to perform precise subsequent operations without rounding errors.
2
Calculate the remaining fraction of the fund after the housing allocation.
1−59=491 - \frac{5}{9} = \frac{4}{9}
The next allocation is taken from the remaining fund, not the total fund.
3
Determine the fraction of the total fund allocated to public transportation.
37×49=1263=421\frac{3}{7} \times \frac{4}{9} = \frac{12}{63} = \frac{4}{21}
Multiplying the proportion (37\frac{3}{7}) by the available remainder (49\frac{4}{9}) yields its share of the total.
4
Calculate the new remaining fund for the last two sectors.
49−421=2863−1263=1663\frac{4}{9} - \frac{4}{21} = \frac{28}{63} - \frac{12}{63} = \frac{16}{63}
Subtracting the transportation allocation from the previous remainder gives the final undistributed portion.
5
Determine the fraction of the total fund allocated to renewable energy.
12×1663=863\frac{1}{2} \times \frac{16}{63} = \frac{8}{63}
The problem states the final remainder is split equally between two sectors.
6
Equate the fraction to the given monetary value and solve for the total fund (xx).
863x=3.2  ⟹  x=3.2×638=0.4×63=25.2\frac{8}{63}x = 3.2 \implies x = 3.2 \times \frac{63}{8} = 0.4 \times 63 = 25.2
This establishes the algebraic relationship to find the initial total.

Key Concept

Successive Fractional Operations and Recurring Decimals

Alternative Method

Instead of calculating absolute remaining fractions through subtraction at each step, you can chain the remaining proportions through direct multiplication: Total × (1 - 5/9) × (1 - 3/7) × (1/2) = Total × (4/9) × (4/7) × (1/2) = Total × (8/63). Setting (8/63) equal to 3.2 million quickly yields 25.2 million.
Estimated Time:2m 30s
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Topics

Fractions and Decimals

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