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Question

Difficulty: HardFractions and Decimals

During a system stress test, four data processing modules record their average latency per transaction (in seconds) as given below. Arrange the modules in ascending order based on their latency times (from fastest to slowest).

  1. 1Module Y: 0.256‾0.2\overline{56} seconds
  2. 2Module X: 935\frac{9}{35} seconds
  3. 3Module W: The HCF (Highest Common Factor) of 3433\frac{34}{33} and 8522\frac{85}{22} seconds
  4. 4Module Z: The product of 1.28‾1.2\overline{8} and 0.20.2 seconds

Answer

The correct ascending sequence from fastest to slowest latency is Module Y, followed by Module X, then Module W, and finally Module Z.
By converting all latency values into extended decimals, we find Module Y (0.25656...0.25656...) is the smallest, followed by Module X (0.25714...0.25714...), then Module W (0.25757...0.25757...), and finally Module Z (0.25777...0.25777...). Therefore, the correct ascending order is Module Y, Module X, Module W, Module Z.

Step-by-Step Solution

1
Calculate the exact value for Module W using the HCF of fractions formula.
HCF = HCF(34,85)LCM(33,22)=1766\frac{\text{HCF}(34, 85)}{\text{LCM}(33, 22)} = \frac{17}{66}. Converted to a decimal, this is 0.25757...0.25757... (0.257‾0.2\overline{57}).
To standardize the mathematical expression into a comparable decimal format.
2
Convert the value for Module X into a decimal.
935≈0.2571428...\frac{9}{35} \approx 0.2571428...
Decimals allow for direct digit-by-digit comparison.
3
Expand the recurring decimal for Module Y to several decimal places.
0.256‾=0.2565656...0.2\overline{56} = 0.2565656...
To provide enough precision to compare digits past the third decimal place.
4
Convert the recurring decimal in Module Z to a fraction, multiply, and convert back to a decimal.
1.28‾=128−1290=11690=58451.2\overline{8} = \frac{128-12}{90} = \frac{116}{90} = \frac{58}{45}. Then, 5845×15=58225\frac{58}{45} \times \frac{1}{5} = \frac{58}{225}. Converting to decimal gives 0.25777...0.25777... (0.257‾0.25\overline{7}).
To determine its exact decimal value for accurate comparison against the other modules.
5
Compare the decimal values digit by digit from left to right to establish the ascending order.
0.25656...0.25656... (Y) < 0.25714...0.25714... (X) < 0.25757...0.25757... (W) < 0.25777...0.25777... (Z).
The question asks for the fastest to slowest modules, which corresponds to the shortest to longest latency times (ascending numerical order).

Key Concept

Advanced comparison of fractions, recurring decimals, and the application of HCF/LCM rules for fractions.

Alternative Method

Find a common denominator for all fractions: 66 for W (17/66), 35 for X (9/35), 990 for Y (254/990), and 225 for Z (58/225). The LCM of 66, 35, 990, and 225 is 34650. Convert all fractions to this denominator to compare the exact numerators, though decimal conversion to the fourth digit is faster.
Estimated Time:2m 30s
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Topics

Fractions and Decimals

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