Worksheet on Scientific Notation and Significant Figures.

Absolutely! Let’s solve both problems step by step, explaining the rules of significant figures and how to apply them correctly.

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## ✔ PART 1: Determine the number of significant figures

Rules for Significant Figures:

1. Non-zero digits are always significant.
2. Leading zeros (before first non-zero digit) are NEVER significant.
3. Captive zeros (between non-zero digits) are ALWAYS significant.
4. Trailing zeros in a number WITH a decimal point ARE significant.
5. Trailing zeros in a number WITHOUT a decimal point are ambiguous — but often considered NOT significant unless specified (e.g., with scientific notation or a bar).
6. In scientific notation, all digits in the coefficient are significant.

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a) 0.02

→ Leading zeros don’t count. Only the “2” is significant.
✔ Answer: 1

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b) 0.020

→ Leading zeros don’t count. The trailing zero *after* the decimal and after a non-zero digit IS significant.
Digits: 2 and 0 → 2 sig figs.
✔ Answer: 2

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c) 501

→ All non-zero digits are significant. Zero is captive → significant.
Digits: 5, 0, 1 → all 3 are significant.
✔ Answer: 3

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d) 6051.00

→ Decimal present → trailing zeros are significant. All digits including zeros between and at end are significant.
Digits: 6, 0, 5, 1, 0, 0 → 6 sig figs.
✔ Answer: 6

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e) 501.0

→ Decimal present → trailing zero is significant. Captive zero also significant.
Digits: 5, 0, 1, 0 → 4 sig figs.
✔ Answer: 4

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f) 2.000 × 10²

→ In scientific notation, all digits in the coefficient (2.000) are significant.
Digits: 2, 0, 0, 0 → 4 sig figs.
✔ Answer: 4

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g) 0.982 × 10⁻³

→ Same as writing 9.82 × 10⁻⁴ (but we count sig figs in the coefficient as written).
Coefficient: 0.982 → leading zero doesn't count, but 9,8,2 are significant → 3 sig figs.
✔ Answer: 3

> 💡 Note: You could rewrite it as 9.82 × 10⁻⁴ to make it clearer, but even as 0.982 × 10⁻³, the coefficient has 3 sig figs.

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h) 0.0114 × 10⁴

→ Coefficient: 0.0114 → leading zeros don’t count. Digits 1,1,4 are significant → 3 sig figs.
✔ Answer: 3

> 💡 Again, better to write as 1.14 × 10² for clarity, but original form still has 3 sig figs.

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## ✔ PART 2: Calculate using correct number of significant digits

General Rules for Calculations:

- Multiplication/Division: Answer has same number of sig figs as the factor with the least sig figs.
- Addition/Subtraction: Answer has same number of decimal places as the term with the least decimal places.

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a) 1.35 × 2.467

→ 1.35 has 3 sig figs; 2.467 has 4 → answer must have 3 sig figs.
Calculation: 1.35 × 2.467 = 3.33045 → round to 3 sig figs → 3.33
✔ Answer: 3.33

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b) 1.035 + 4.2

→ Addition: look at decimal places.
1.035 → 3 decimal places
4.2 → 1 decimal place → answer must have 1 decimal place.
Calculation: 1.035 + 4.2 = 5.235 → round to 1 decimal → 5.2
✔ Answer: 5.2

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c) 12.01 + 35.2 + 6.9

→ All are addition → look at decimal places.
12.01 → 2 decimals
35.2 → 1 decimal
6.9 → 1 decimal → least is 1 decimal place.
Sum: 12.01 + 35.2 + 6.9 = 54.11 → round to 1 decimal → 54.1
✔ Answer: 54.1

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d) 55.46 − 28.9

→ Subtraction → decimal places:
55.46 → 2 decimals
28.9 → 1 decimal → answer must have 1 decimal place.
Calculation: 55.46 − 28.9 = 26.56 → round to 1 decimal → 26.6
✔ Answer: 26.6

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e) 0.21 × 3.2 × 100.1

→ Multiplication → find least sig figs:
0.21 → 2 sig figs
3.2 → 2 sig figs
100.1 → 4 sig figs → so answer must have 2 sig figs.
Calculation: 0.21 × 3.2 = 0.672 → × 100.1 ≈ 67.2672 → round to 2 sig figs → 67
✔ Answer: 67

> 💡 Note: 67 has 2 sig figs. If you want scientific notation to avoid ambiguity: 6.7 × 10¹

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f) 505.0 − 450.25

→ Subtraction → decimal places:
505.0 → 1 decimal
450.25 → 2 decimals → answer must have 1 decimal place.
Calculation: 505.0 − 450.25 = 54.75 → round to 1 decimal → 54.8
✔ Answer: 54.8

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g) 1.5 × 10⁶ / 3.76 × 10²

→ Division → sig figs:
1.5 → 2 sig figs
3.76 → 3 sig figs → answer must have 2 sig figs.
Calculation: (1.5 / 3.76) × 10⁶⁻² = 0.398936... × 10⁴ = 3.98936 × 10³ → round to 2 sig figs → 4.0 × 10³
✔ Answer: 4.0 × 10³

> 💡 Why 4.0? Because 2 sig figs require us to keep the zero — 4.0 has 2 sig figs, whereas 4 has only 1.

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h) 1.28 × 10³ / 1.4267 × 10²

→ Division → sig figs:
1.28 → 3 sig figs
1.4267 → 5 sig figs → answer must have 3 sig figs.
Calculation: (1.28 / 1.4267) × 10³⁻² ≈ 0.8971 × 10¹ = 8.971 → round to 3 sig figs → 8.97
✔ Answer: 8.97

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## 📝 FINAL ANSWERS:

1. Significant Figures:

a) 1
b) 2
c) 3
d) 6
e) 4
f) 4
g) 3
h) 3

2. Calculations:

a) 3.33
b) 5.2
c) 54.1
d) 26.6
e) 67 (or 6.7 × 10¹)
f) 54.8
g) 4.0 × 10³
h) 8.97

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Let me know if you’d like these explained visually or need help with rounding rules again! 😊