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Difficulty: HardFractions and Rational Numbers

At a financial trading firm, incoming trade orders pass through a three-stage automated verification pipeline:

1. In Stage 1, 16\frac{1}{6} of all incoming orders are rejected as invalid, and 15\frac{1}{5} of the remaining valid orders are flagged for manual review.
2. In Stage 2, of the orders that were cleared directly without manual review, 38\frac{3}{8} are executed immediately, while the rest are placed in a waiting queue.
3. In Stage 3, 13\frac{1}{3} of the queued orders are processed, leaving the remaining queued orders unprocessed.

If exactly 250250 queued orders remain unprocessed at the end of Stage 3, how many incoming orders were flagged for manual review in Stage 1?

  1. A
    100
  2. 150Answer
  3. C
    180
  4. D
    200
  5. E
    225

Answer

150 orders were flagged for manual review in Stage 1.
Let NN represent the total number of incoming trade orders. In Stage 1, after 16N\frac{1}{6}N orders are rejected, 56N\frac{5}{6}N remain. Flagging 15\frac{1}{5} of these remaining valid orders means 15×56N=16N\frac{1}{5} \times \frac{5}{6}N = \frac{1}{6}N orders are flagged for manual review, leaving 56N−16N=23N\frac{5}{6}N - \frac{1}{6}N = \frac{2}{3}N orders cleared directly. In Stage 2, 58\frac{5}{8} of the cleared orders enter the queue, which equals 58×23N=512N\frac{5}{8} \times \frac{2}{3}N = \frac{5}{12}N. In Stage 3, 23\frac{2}{3} of the queued orders remain unprocessed, giving 23×512N=518N\frac{2}{3} \times \frac{5}{12}N = \frac{5}{18}N. Setting 518N=250\frac{5}{18}N = 250 yields N=900N = 900. The number of orders flagged for manual review is 16×900=150\frac{1}{6} \times 900 = 150.

Step-by-Step Solution

1
Determine the fraction of orders cleared directly in Stage 1.
The fraction of valid orders remaining after rejection is 1−16=561 - \frac{1}{6} = \frac{5}{6}. The fraction flagged for manual review is 15×56=16\frac{1}{5} \times \frac{5}{6} = \frac{1}{6} of the total. Thus, the fraction cleared directly is 56−16=46=23\frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} of total incoming orders.
Sequential fractional deductions must be applied to the remaining amounts at each sub-stage.
2
Determine the fraction of orders placed in the queue during Stage 2.
Of the 23\frac{2}{3} directly cleared orders, 38\frac{3}{8} are executed immediately, so 1−38=581 - \frac{3}{8} = \frac{5}{8} are placed in the queue. The fraction of total incoming orders queued is 58×23=512\frac{5}{8} \times \frac{2}{3} = \frac{5}{12}.
Fraction of a fraction requires multiplying the operational fraction by the current subtotal.
3
Determine the fraction of total incoming orders that remain unprocessed at the end of Stage 3.
Since 13\frac{1}{3} of queued orders are processed, 1−13=231 - \frac{1}{3} = \frac{2}{3} of the queued orders remain unprocessed. The fraction of total orders unprocessed is 23×512=518\frac{2}{3} \times \frac{5}{12} = \frac{5}{18}.
Multiply the remaining fraction in Stage 3 by the total queued fraction from Stage 2.
4
Calculate the total number of incoming orders (NN) and the target flagged orders.
Setting 518N=250\frac{5}{18}N = 250 gives N=250×185=900N = 250 \times \frac{18}{5} = 900 total incoming orders. The number of orders flagged for manual review is 16×900=150\frac{1}{6} \times 900 = 150.
Equate the given numerical quantity to its fractional equivalent to find the total, then compute the required part.

Key Concept

Multi-step fraction calculations involving fractions of remaining quantities and working backward from a known part to the whole.
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Topics

Fractions and Rational Numbers

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