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Difficulty: MediumFractions and Rational Numbers

A charity foundation allocates its annual grant budget among three sectors: Healthcare, Education, and Environmental Conservation. First, 25\frac{2}{5} of the total annual budget is allocated to Healthcare. Next, 59\frac{5}{9} of the remaining budget is allocated to Education. The final remainder of $120,000\$120,000 is allocated to Environmental Conservation. What was the foundation's total annual grant budget?

  1. A
    $32,000\$32,000
  2. B
    $240,000\$240,000
  3. C
    $360,000\$360,000
  4. $450,000\$450,000Answer
  5. E
    $2,700,000\$2,700,000

Answer

$450,000\$450,000
The option specifying $450,000\$450,000 is correct because allocating 25\frac{2}{5} leaves 35\frac{3}{5} of the budget. Taking 59\frac{5}{9} of 35\frac{3}{5} gives 13\frac{1}{3} of the total. The combined allocations leave 1−(25+13)=4151 - (\frac{2}{5} + \frac{1}{3}) = \frac{4}{15} of the total budget, which equals $120,000\$120,000. Dividing $120,000\$120,000 by 415\frac{4}{15} yields $450,000\$450,000.

Step-by-Step Solution

1
Determine the fraction of the total budget remaining after the Healthcare allocation.
Since Healthcare receives 25\frac{2}{5} of the total budget, the remaining fraction is 1−25=351 - \frac{2}{5} = \frac{3}{5}.
Sequential fraction operations require updating the remaining whole after each step.
2
Calculate the fraction of the total budget allocated to Education.
Education receives 59\frac{5}{9} of the remaining 35\frac{3}{5}, which is 59×35=1545=13\frac{5}{9} \times \frac{3}{5} = \frac{15}{45} = \frac{1}{3} of the total budget.
To find a fraction of a remaining fraction, multiply the two fractions together.
3
Find the final remaining fraction allocated to Environmental Conservation.
Subtract both allocations from the whole: 1−(25+13)=1−1115=4151 - \left(\frac{2}{5} + \frac{1}{3}\right) = 1 - \frac{11}{15} = \frac{4}{15}.
The final dollar amount corresponds to the fraction of the total budget left over.
4
Solve for the total annual grant budget.
415×Total=$120,000  ⟹  Total=$120,000×154=$450,000\frac{4}{15} \times \text{Total} = \$120,000 \implies \text{Total} = \$120,000 \times \frac{15}{4} = \$450,000.
Dividing the part by its corresponding fractional proportion yields the total whole.

Key Concept

Sequential Fraction Operations and Finding the Whole from a Fractional Part
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Topics

Fractions and Rational Numbers

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