At a logistics distribution hub, a shipment of incoming packages was processed over two sh… — GMAT

At a logistics distribution hub, a shipment of incoming packages was processed over two shifts. During the morning shift, $0.35$ of the total shipment was processed and dispatched. During the evening shift, $\frac{4}{13}$ of the remaining packages were processed. If $450$ packages remained unprocessed at the end of both shifts, what was the total number of packages in the initial shipment?

Answer: 1000 packages

Answer

1000 packages

To find the initial shipment size, first express

0.35

as the fraction

\frac{7}{20}

. Subtracting this from

1

leaves

\frac{13}{20}

of the initial total after the morning shift. Next, processing

\frac{4}{13}

of these remaining packages leaves

1 - \frac{4}{13} = \frac{9}{13}

of that remaining amount. Multiplying

\frac{9}{13} \times \frac{13}{20}

shows that

\frac{9}{20}

of the original total remains unprocessed. Finally, setting

\frac{9}{20}

of the total equal to

450

packages yields an initial total of

1000

packages.

Step-by-Step Solution

Convert decimal portion to fraction and find remaining fraction after morning shift

Remaining fraction after morning shift is

\frac{13}{20}

Since

0.35 = \frac{7}{20}

of the total shipment was processed,

1 - \frac{7}{20} = \frac{13}{20}

of the initial shipment remained.

Determine the remaining fraction of the shipment after the evening shift

Final remaining fraction of the initial shipment is

\frac{9}{20}

The evening shift processed

\frac{4}{13}

of the remaining packages, leaving

1 - \frac{4}{13} = \frac{9}{13}

of the remaining packages. Thus,

\frac{9}{13} \times \frac{13}{20} = \frac{9}{20}

of the initial shipment remained.

Solve for the total initial number of packages

Total initial packages =

1000

Setting

\frac{9}{20} N = 450

gives

N = 450 \times \frac{20}{9} = 1000

Key Concept

Combining decimal-to-fraction conversions with successive remaining fraction calculations

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