Apply a decimal scaling transformation that adjusts the dataset [150, 300, 450] so that the largest number is less than 1. What is the result?

Question

BlackTom AI · Accepted Answer

Decimal scaling divides each value x by 10^j, where j is the smallest integer such that max(|x|) / 10^j < 1.

1) Relevant knowledge points and concepts
- Decimal scaling transformation: x' = x / 10^j.
- Determine j as the smallest integer satisfying max(|x|) < 10^j (equivalently max(|x|)/10^j < 1).
- We must ensure the largest transformed value is strictly less than 1.

2) Step-by-step reasoning
- Identify the maximum absolute value in the dataset: max(|150|, |300|, |450|) = 450.
- Find the smallest integer j with 450 < 10^j.
  - 10^1 = 10 (450 < 10? No)
  - 10^2 = 100 (450 < 100? No)
  - 10^3 = 1000 (450 < 1000? Yes)
- Therefore j = 3.

3) Calculations with intermediate steps
- Apply x' = x / 10^3 = x / 1000 to each element:
  - 150 / 1000 = 0.150 → written as 0.15
  - 300 / 1000 = 0.300 → written as 0.30
  - 450 / 1000 = 0.450 → written as 0.45

4) Why this answer is correct
- Using j = 3 ensures the largest transformed value is 450/1000 = 0.45, which is less than 1, meeting the requirement. This is the minimal j that achieves max < 1, so the transformation follows the decimal scaling rule correctly.

5) Why the other options are incorrect
- a. [15, 30, 45] corresponds to dividing by 10 (j = 1); the largest value 45 is not < 1.
- c. [1.5, 3.0, 4.5] corresponds to dividing by 100 (j = 2); the largest value 4.5 is not < 1.
- d. [0.025, 0.030, 0.045] would correspond to dividing by 6000 or inconsistent scaling (300/6000 = 0.05, not 0.03); in any case it is not the correct decimal-scaling result using the minimal j.

6) Conclusion
- The correctly transformed dataset is [0.15, 0.30, 0.45], which matches option b.

Apply a decimal scaling transformation that adjusts the dataset [150, 300, 450] so that the largest number is less than 1. What is the result?单项选择题

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