A technology company allocates its quarterly research budget among three key projects: Pro… — GRE General Test

A technology company allocates its quarterly research budget among three key projects: Project Alpha, Project Beta, and Project Gamma. Initially, Project Alpha receives $\frac{2}{5}$ of the total budget. Of the remaining budget, Project Beta is allocated $\frac{4}{7}$ , and Project Gamma receives the rest. During a mid-quarter review, $\frac{1}{6}$ of Project Alpha's allocated budget and $\frac{1}{4}$ of Project Beta's allocated budget are transferred to Project Gamma. After these transfers, Project Gamma's final budget is what fraction of the total initial quarterly research budget?

A
$\frac{5}{21}$
B
$\frac{13}{50}$
$\frac{43}{105}$ Answer
D
$\frac{83}{165}$
E
$\frac{16}{105}$

Answer

\frac{43}{105}

Project Alpha receives

\frac{2}{5}

of the total budget, leaving

\frac{3}{5}

. Project Beta gets

\frac{4}{7}

\frac{3}{5}

, which is

\frac{12}{35}

. Project Gamma's initial share is

\frac{3}{5} - \frac{12}{35} = \frac{9}{35}

. Transferring

\frac{1}{6}

of Alpha's share contributes

\frac{1}{15}

, and transferring

\frac{1}{4}

of Beta's share contributes

\frac{3}{35}

. Adding these to Gamma's initial share yields

\frac{9}{35} + \frac{3}{35} + \frac{1}{15} = \frac{12}{35} + \frac{1}{15} = \frac{36 + 7}{105} = \frac{43}{105}

Step-by-Step Solution

Determine Project Alpha's initial allocation and the remaining budget fraction.

Project Alpha gets

\frac{2}{5}

of the total budget

B

. The remaining fraction is

1 - \frac{2}{5} = \frac{3}{5} B

Subtractions from the whole yield the remaining unallocated portion.

Calculate the initial allocations for Project Beta and Project Gamma.

Project Beta receives

\frac{4}{7} \times \frac{3}{5} B = \frac{12}{35} B

. Project Gamma receives the rest of the remaining budget:

\frac{3}{5} B - \frac{12}{35} B = \frac{21}{35} B - \frac{12}{35} B = \frac{9}{35} B

Project Beta's share is a fraction of the remaining budget, not the total budget.

Calculate the transferred amounts from Project Alpha and Project Beta to Project Gamma.

Transfer from Alpha =

\frac{1}{6} \times \frac{2}{5} B = \frac{1}{15} B

. Transfer from Beta =

\frac{1}{4} \times \frac{12}{35} B = \frac{3}{35} B

Transfers are fractional parts of each project's individual initial allocation.

Sum Project Gamma's initial share and the two transferred amounts.

Gamma's final fraction =

\frac{9}{35} + \frac{1}{15} + \frac{3}{35} = \frac{12}{35} + \frac{1}{15} = \frac{36}{105} + \frac{7}{105} = \frac{43}{105}

Finding the least common denominator (

105

) allows exact addition of the rational numbers.

Key Concept

Multi-step sequential operations with fractions and rational numbers

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic