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Zorluk: OrtaFractions and Rational Numbers

A logistics company received a shipment of identical freight containers. On Monday, the crew unloaded 27\frac{2}{7} of the total shipment. On Tuesday, they unloaded 35\frac{3}{5} of the remaining containers. On Wednesday, they unloaded 12\frac{1}{2} of the containers that remained after Tuesday's work. If 30 containers remained unloaded at the end of Wednesday, what was the total number of containers in the original shipment?

  1. A
    80
  2. B
    105
  3. C
    140
  4. 210Cevap
  5. E
    525

Cevap

The total number of containers in the original shipment was 210.
The correct answer is 210. Working forward, after Monday 57\frac{5}{7} of the initial shipment NN remains. On Tuesday, 25\frac{2}{5} of that remainder stays unloaded, which equals 25×57N=27N\frac{2}{5} \times \frac{5}{7}N = \frac{2}{7}N. On Wednesday, 12\frac{1}{2} of that remainder stays unloaded, yielding 12×27N=17N\frac{1}{2} \times \frac{2}{7}N = \frac{1}{7}N. Setting 17N=30\frac{1}{7}N = 30 gives N=210N = 210.

Adım Adım Çözüm

1
Determine the fraction of containers remaining after Monday.
Since 27\frac{2}{7} of the total shipment NN was unloaded on Monday, the fraction remaining is 1−27=571 - \frac{2}{7} = \frac{5}{7} of NN.
Subtracting the fraction unloaded on Monday from 1 gives the remaining fraction.
2
Calculate the fraction of containers remaining after Tuesday.
On Tuesday, 35\frac{3}{5} of the remaining 57N\frac{5}{7}N was unloaded, which is 35×57N=37N\frac{3}{5} \times \frac{5}{7}N = \frac{3}{7}N. The remaining fraction after Tuesday is 57N−37N=27N\frac{5}{7}N - \frac{3}{7}N = \frac{2}{7}N.
Alternatively, if 35\frac{3}{5} of the remainder was unloaded, then 1−35=251 - \frac{3}{5} = \frac{2}{5} of the remainder was left: 25×57N=27N\frac{2}{5} \times \frac{5}{7}N = \frac{2}{7}N.
3
Calculate the fraction of containers remaining after Wednesday.
On Wednesday, 12\frac{1}{2} of the remaining 27N\frac{2}{7}N was unloaded, leaving 1−12=121 - \frac{1}{2} = \frac{1}{2} of that remainder. Thus, the final fraction remaining is 12×27N=17N\frac{1}{2} \times \frac{2}{7}N = \frac{1}{7}N.
Multiplying the remaining fraction after Tuesday by the fraction left unhandled on Wednesday yields the overall fraction of the original shipment remaining.
4
Solve for the total initial number of containers NN.
Set 17N=30\frac{1}{7}N = 30, which gives N=30×7=210N = 30 \times 7 = 210.
Equating the calculated final remaining fraction to the given numerical count allows solving for the total original quantity.

Anahtar Kavram

Sequential Fraction of Remaining Quantities
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Konular

Fractions and Rational Numbers

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