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Difficulty: Very hardDecimals and Scientific Notation

A supercomputer processor operates at a standard speed of 1.25×1091.25 \times 10^9 calculations per second. During a maintenance diagnostic, the processor's operational speed is reduced by 96%96\%. Operating exclusively at this reduced speed, how many seconds will it take the processor to complete a workload of 1.8×10131.8 \times 10^{13} calculations?

  1. A
    3.6×1043.6 \times 10^4
  2. B
    1.5×1041.5 \times 10^4
  3. 3.6×1053.6 \times 10^5Answer
  4. D
    1.44×1051.44 \times 10^5
  5. E
    3.6×1063.6 \times 10^6

Answer

3.6×1053.6 \times 10^5 seconds
The correct answer is derived by first finding the reduced processing speed: 4%4\% of 1.25×1091.25 \times 10^9 equals 0.04×1.25×109=5.0×1070.04 \times 1.25 \times 10^9 = 5.0 \times 10^7 calculations per second. Dividing the target workload of 1.8×10131.8 \times 10^{13} by 5.0×1075.0 \times 10^7 yields 1.85.0×1013−7=0.36×106=3.6×105\frac{1.8}{5.0} \times 10^{13-7} = 0.36 \times 10^6 = 3.6 \times 10^5 seconds.

Step-by-Step Solution

1
Determine the reduced operational speed percentage.
Remaining speed percentage is 100%−96%=4%=0.04100\% - 96\% = 4\% = 0.04.
A 96%96\% reduction means the processor operates at 4%4\% of its original capacity.
2
Calculate the reduced operational speed in scientific notation.
Reduced speed =0.04×(1.25×109)=0.05×109=5.0×107= 0.04 \times (1.25 \times 10^9) = 0.05 \times 10^9 = 5.0 \times 10^7 calculations per second.
Multiplying the decimal coefficient 0.040.04 by 1.251.25 gives 0.050.05, which adjusts to 5.0×1075.0 \times 10^7 in scientific notation.
3
Divide total workload by reduced speed to compute time in seconds.
Time =1.8×10135.0×107=(1.85.0)×1013−7=0.36×106= \frac{1.8 \times 10^{13}}{5.0 \times 10^7} = \left(\frac{1.8}{5.0}\right) \times 10^{13 - 7} = 0.36 \times 10^6 seconds.
Workload divided by rate gives total duration. Exponents are subtracted when dividing powers of ten.
4
Convert the final result to standard scientific notation.
0.36×106=3.6×1050.36 \times 10^6 = 3.6 \times 10^5 seconds.
Moving the decimal point one place to the right requires decreasing the exponent of 10 by 1.

Key Concept

Decimals and Scientific Notation Operations
Estimated Time:2m 0s
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Topics

Decimals and Scientific Notation

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