I see you uploaded a worksheet on significant figures. Let’s go through the different parts and solutions:

1. Identifying Significant Figures:

For the numbers listed, we count the significant figures (you’ve already provided some answers which seem correct). Here’s the confirmation:

• 246.32 → 5 significant figures.
• 107.854 → 6 significant figures.
• 100.3 → 4 significant figures.
• 0.678 → 3 significant figures.
• 1008 → 4 significant figures.
• 0.00340 → 3 significant figures (zeros before are placeholders, but the zero after the 4 counts).
• 14.600 → 5 significant figures (trailing zeros in a number with a decimal are significant).
• 0.0001 → 1 significant figure (only the 1 counts).
• 700000 → 1 significant figure (trailing zeros in a number without a decimal are not significant).
• 350.670 → 6 significant figures.
• 1.000 → 4 significant figures (the trailing zeros after the decimal are significant).
• 320001 → 6 significant figures.

2. Addition/Subtraction with Significant Figures:

You need to round your final answer to the least number of decimal places found in the original numbers.

• 32.567 + 135.0 + 1.4567:

  • Adding them gives $169.0237$.
  • The number with the fewest decimal places is 135.0 (1 decimal place).
  • Round to 1 decimal place: 169.0.

• 246.24 + 238.278 + 98.3:

  • Adding them gives $582.818$.
  • The number with the fewest decimal places is 98.3 (1 decimal place).
  • Round to 1 decimal place: 582.8.

• 658.0 + 23.5478 + 1345.29:

  • Adding them gives $2026.8378$.
  • The number with the fewest decimal places is 658.0 (1 decimal place).
  • Round to 1 decimal place: 2026.8.

3. Multiplication/Division with Significant Figures:

For multiplication and division, round to the smallest number of significant figures.

• 23.7 × 3.8:

  • The answer is $89.06$.
  • Both numbers have 2 significant figures, so round to 89.

• 45.76 × 0.25:

  • The answer is $11.44$.
  • The number with the fewest significant figures is 0.25 (2 significant figures), so round to 11.

• 810.4 × 0.001:

  • The answer is $0.8104$.
  • The number with the fewest significant figures is 0.001 (1 significant figure), so round to 0.8.

• 6.47 × 6.45:

  • The answer is $41.7615$.
  • Both numbers have 3 significant figures, so round to 41.8.

• 43.678 × 64.1:

  • The answer is $2799.0818$.
  • The number with the fewest significant figures is 64.1 (3 significant figures), so round to 2790.

• 1.678 ÷ 0.42:

  • The answer is $3.9952$.
  • The number with the fewest significant figures is 0.42 (2 significant figures), so round to 4.0.

• 28.367 ÷ 3.74:

  • The answer is $7.5852$.
  • The number with the fewest significant figures is 3.74 (3 significant figures), so round to 7.59.

• 4278 ÷ 1.006:

  • The answer is $4251.493$.
  • The number with the fewest significant figures is 1.006 (4 significant figures), so round to 4251.

Let me know if you'd like more details on any of the steps.

Questions:

1. Why do trailing zeros in a decimal number count as significant?
2. What are placeholder zeros, and how do they affect significant figures?
3. How do significant figures differ between addition and multiplication?
4. How would rounding change if no significant figures rules applied?
5. What would happen if you had exact numbers in these problems?

Tip:

When multiplying or dividing, focus on the total number of significant figures, not decimal places.