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Let's evaluate the numbers in the list provided in the question and then find their approximate values to determine the correct order from least to greatest. Here are the numbers:

1. \(\frac{63}{20} = 3.15\)
2. \(\sqrt{10} \approx 3.162\)
3. \(\frac{1}{3} \approx 0.333\)
4. \(\pi \approx 3.1416\)
5. \(\log_{10} 1000 = 3\)

### Step-by-step comparison of these values:

- \(\frac{1}{3} \approx 0.333\) (smallest)
- \(\log_{10} 1000 = 3\) (next smallest)
- \(\pi \approx 3.1416\)
- \(\frac{63}{20} = 3.15\)
- \(\sqrt{10} \approx 3.162\) (largest)

### Now, let's analyze the answer choices:

(A) \( \log_{10} 1000, \frac{63}{20}, \sqrt{10}, \frac{1}{3}, \pi \)
- Order: \( 3, 3.15, 3.162, 0.333, 3.1416 \) → Incorrect (not in order)

(B) \( \frac{63}{20}, \log_{10} 1000, \frac{1}{3}, \sqrt{10}, \pi \)
- Order: \( 3.15, 3, 0.333, 3.162, 3.1416 \) → Incorrect

(C) \( \sqrt{10}, \log_{10} 1000, \pi, \frac{63}{20}, \frac{1}{3} \)
- Order: \( 3.162, 3, 3.1416, 3.15, 0.333 \) → Incorrect

(D) \( \log_{10} 1000, \pi, \frac{63}{20}, \sqrt{10}, \frac{1}{3} \)
- Order: \( 3, 3.1416, 3.15, 3.162, 0.333 \) → **Correct**

Thus, the correct answer is **(D)**.

Would you like more details on this or have any further questions?

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Here are 5 related questions to explore further:
1. What is the significance of logarithms in ordering numbers?
2. How can you estimate the value of square roots without a calculator?
3. Why is \(\pi\) used so often in mathematics?
4. What are other logarithmic properties that simplify problems like this?
5. How does comparing fractions help in ordering numbers?

**Tip:** When ordering numbers, it helps to approximate them to a few decimal places if you're not given exact values.

In which of the following are the numbers shown ordered from least to greatest? Use the list of numbers: 63/20, sqrt(10), 1/3, pi, log_10 1000.

Learn how to order real numbers, including logarithms, square roots, pi, and fractions. This problem demonstrates key mathematical concepts like approximation and comparison, using numbers like log_10 1000, sqrt(10), pi, 63/20, and 1/3.

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