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Difficulty: HardFractions and Rational Numbers

A university research lab receives a multi-year grant. In the first quarter, the lab spends 27\frac{2}{7} of the total grant on laboratory equipment. In the second quarter, the lab spends 35\frac{3}{5} of the remaining funds on research staff salaries. At the beginning of the third quarter, the lab receives a supplementary addition equal to 14\frac{1}{4} of the funds remaining at the end of the second quarter. If the total amount of money in the grant immediately following the third-quarter addition is $45,000\$45,000, what was the initial dollar amount of the research grant?

  1. A
    $84,000
  2. $126,000Answer
  3. C
    $157,500
  4. D
    $210,000
  5. E
    $315,000

Answer

$126,000
The correct answer of 126,000isfoundbytrackingthefractionoftheinitialgrant126,000 is found by tracking the fraction of the initial grant G remainingaftereachstep.Spending remaining after each step. Spending \frac{2}{7}leaves leaves \frac{5}{7}G .Spending. Spending \frac{3}{5}ofthatremainderleaves of that remainder leaves \frac{2}{5} \times \frac{5}{7}G = \frac{2}{7}G .Adding. Adding \frac{1}{4}of of \frac{2}{7}G increasesthebalanceby increases the balance by \frac{1}{14}G ,resultinginafinaltotalof, resulting in a final total of \frac{2}{7}G + \frac{1}{14}G = \frac{5}{14}G .Setting. Setting \frac{5}{14}G = 45,000 yields yields G = 126,000$.

Step-by-Step Solution

1
Determine the remaining fraction after the first quarter.
Fraction remaining = 1−27=571 - \frac{2}{7} = \frac{5}{7} of the initial grant GG.
The lab spent 27\frac{2}{7} of the total grant.
2
Calculate the remaining fraction after the second quarter.
Fraction remaining = (1−35)×57G=25×57G=27G\left(1 - \frac{3}{5}\right) \times \frac{5}{7}G = \frac{2}{5} \times \frac{5}{7}G = \frac{2}{7}G.
The lab spent 35\frac{3}{5} of the remaining funds, leaving 25\frac{2}{5} of those funds.
3
Account for the third-quarter supplementary addition.
Final fraction = 27G+14(27G)=27G+114G=414G+114G=514G\frac{2}{7}G + \frac{1}{4}\left(\frac{2}{7}G\right) = \frac{2}{7}G + \frac{1}{14}G = \frac{4}{14}G + \frac{1}{14}G = \frac{5}{14}G.
An additional 14\frac{1}{4} of the second-quarter remaining funds was added to the grant.
4
Solve for the initial grant amount GG.
514G=45,000  ⟹  G=45,000×145=9,000×14=126,000\frac{5}{14}G = 45,000 \implies G = 45,000 \times \frac{14}{5} = 9,000 \times 14 = 126,000.
The final remaining balance of $45,000\$45,000 corresponds to 514\frac{5}{14} of the original grant.

Key Concept

Multi-step sequential fraction operations and base tracking in word problems
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Topics

Fractions and Rational Numbers

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