Significant Figures & Scientific Notation worksheet - Free Printable

Let's solve each part of Activity 1: Significant Figures and Scientific Notation step by step.

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A. Find the number of significant digits in the following measurements.

Rules for significant figures:
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros (before the first non-zero digit) are not significant.
- Trailing zeros (after the last non-zero digit) are significant only if there is a decimal point.
- In scientific notation, all digits in the coefficient are significant.

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1. 10100 mm
→ No decimal point. Trailing zeros are not significant.
→ 1, 0, 1 → only the two 1s and the zero between them count.
→ 3 significant digits

2. 0.0874000 cL
→ Leading zeros not significant.
→ 8, 7, 4, and trailing zeros after decimal are significant.
→ 874000 → 6 digits
→ 6 significant digits

3. 170.5 km
→ All digits are significant. Decimal present, so trailing zero (between 7 and 5) is significant.
→ 1, 7, 0, 5 → 4 significant digits

4. 5000 cm³
→ No decimal point. Trailing zeros not significant.
→ Only '5' is significant.
→ 1 significant digit

5. 36 kg
→ Two non-zero digits.
→ 2 significant digits

6. 3.500 × 10⁻² s
→ Coefficient: 3.500 → all digits significant.
→ 4 significant digits

7. 6.19 × 10⁻¹ mi
→ 6.19 → three digits
→ 3 significant digits

8. 1.00 × 10³ mg
→ 1.00 → three digits (zeros after decimal are significant)
→ 3 significant digits

9. 2 × 10² °C
→ 2 → one digit only
→ 1 significant digit

10. 5.920 × 10⁻¹ A
→ 5.920 → four digits (trailing zero after decimal is significant)
→ 4 significant digits

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✔ A Answers:
1. 3
2. 6
3. 4
4. 1
5. 2
6. 4
7. 3
8. 3
9. 1
10. 4

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B. Write the following measurements in scientific notation format.

Scientific notation: $ a \times 10^n $, where $ 1 \leq a < 10 $

1. 10100 mm
→ Move decimal to make it 1.01 × 10⁴
→ 1.01 × 10⁴

2. 0.0874000 cL
→ Move decimal right: 8.74000 × 10⁻²
→ 8.74000 × 10⁻²

3. 170.5 km
→ 1.705 × 10²
→ 1.705 × 10²

4. 5000 cm³
→ 5 × 10³
→ 5 × 10³

5. 36 kg
→ 3.6 × 10¹
→ 3.6 × 10¹

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✔ B Answers:
1. 1.01 × 10⁴
2. 8.74000 × 10⁻²
3. 1.705 × 10²
4. 5 × 10³
5. 3.6 × 10¹

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C. Round off the following measurements to the indicated number of significant digits.

1. 10100 mm → 2 significant digits
→ 1.0 × 10⁴ mm (since 10100 rounded to 2 sig fig → 10000 = 1.0 × 10⁴)
→ 1.0 × 10⁴ mm

2. 0.0874000 cL → 3 significant digits
→ First 3 significant digits: 8, 7, 4 → 0.0874
→ But we need to express it with 3 sig fig: 8.74 × 10⁻²
→ 8.74 × 10⁻² cL

3. 36 kg → 1 significant digit
→ Round to nearest ten: 40 kg
→ 40 kg

4. 2.5689 × 10⁻² s → 4 significant digits
→ Already has 4 sig figs: 2, 5, 6, 8 → next digit is 9 → round up
→ 2.569 × 10⁻² s
→ 2.569 × 10⁻² s

5. 5.920 × 10⁻¹ A → 3 significant digits
→ Already has 4 sig figs (5,9,2,0). Round to 3:
→ Look at fourth digit: 0 → leave as is
→ 5.92 × 10⁻¹ A
→ 5.92 × 10⁻¹ A

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✔ C Answers:
1. 1.0 × 10⁴ mm
2. 8.74 × 10⁻² cL
3. 40 kg
4. 2.569 × 10⁻² s
5. 5.92 × 10⁻¹ A

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D. Perform the following operations.

We must follow rules for significant figures in calculations:

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

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#### 1. (0.17362 m)(31.22 m)(170.06 m) = ? m³

First, multiply:
→ 0.17362 × 31.22 = 5.4202664
→ 5.4202664 × 170.06 ≈ 921.747... m³

Now check sig figs:
- 0.17362 → 5 sig figs
- 31.22 → 4 sig figs
- 170.06 → 5 sig figs

Least is 4, so answer should have 4 sig figs.

→ 921.7 m³? Wait: 921.7 has 4 sig figs, but let's write properly.

921.747 → round to 4 sig figs → 922 m³? Let's compute carefully:

Actually:
0.17362 × 31.22 = 5.4202664 → 5.420 (4 sig figs)
5.420 × 170.06 = ?

5.420 × 170.06 = 921.747 → round to 4 sig figs → 922 m³

But wait: 921.747 → to 4 sig figs: look at 5th digit → 7 → round up 1 to 2 → 922 → 922 m³

Wait: 921.747 → 4 sig figs: 922 → yes.

But actually, 921.747 → 922 → 3 sig figs? No: 922 has 3 sig figs?

No: 922 → 3 sig figs, but we need 4.

So better: 921.7 → that’s 4 sig figs.

Wait: 921.7 → digits: 9,2,1,7 → 4 sig figs.

But 921.747 → round to 4 sig figs: look at fifth digit → 4 → less than 5 → keep 921.7

So 921.7 m³

But let’s do full calculation:

0.17362 × 31.22 = 5.4202664
5.4202664 × 170.06 = 921.747... → ≈ 921.7 m³ (4 sig figs)

✔ 921.7 m³

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#### 2. $ \frac{9.042\ \text{g}}{(5.24\ \text{cm})(9.5\ \text{cm})} $ = ? g/cm²

First denominator: 5.24 × 9.5 = ?

5.24 × 9.5 = 5.24 × (9 + 0.5) = 47.16 + 2.62 = 49.78 cm²

Now: 9.042 / 49.78 ≈ 0.1816 g/cm²

Now check sig figs:

- Numerator: 9.042 → 4 sig figs
- Denominator: 5.24 → 3 sig figs; 9.5 → 2 sig figs → multiplication: least is 2 sig figs

So final answer must have 2 sig figs

0.1816 → round to 2 sig figs → 0.18 g/cm²

✔ 0.18 g/cm²

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#### 3. $ 3.500 \times 10^{-2}\ \text{s} + 4.24 \times 10^{-2}\ \text{s} $

Same power of 10 → add coefficients:

3.500 + 4.24 = 7.74

So: $ 7.74 \times 10^{-2} $ s

Now check sig figs: Addition → look at decimal places

- 3.500 → 3 decimal places
- 4.24 → 2 decimal places → least is 2

So result should have 2 decimal places → 7.74 → already has 2 → okay

→ 7.74 × 10⁻² s

✔ 7.74 × 10⁻² s

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#### 4. $ \frac{(36\ \text{kg} - 12.5\ \text{kg})}{(80.2\ \text{m}^2 + 12\ \text{m}^2)} $ = ? g/cm²

Wait — units don’t match. Numerator is kg, denominator is m² → result is kg/m², but question says g/cm² → need conversion.

But let's proceed step by step.

Numerator: 36 kg – 12.5 kg = 23.5 kg
→ 36 has no decimal → uncertainty ±0.5? But 12.5 has 1 decimal → subtraction rule: least precise decimal place

36 → no decimal → assume ±0.5 → so result → 23.5 → but 36 is ambiguous.

Standard rule: 36 has 2 sig figs, 12.5 has 3 → subtraction: align decimals:

36.0
–12.5
= 23.5 → but 36 has no decimal → so result should be reported to nearest unit → 24 kg? Or 23.5?

Better: 36 → uncertain in units place → so difference uncertain in units place → 24 kg (rounded to nearest 1)

But let's keep 23.5 for now.

Denominator: 80.2 + 12 = 92.2 m² → 12 has no decimal → so sum to nearest unit → 92 m²

So numerator: 23.5 kg → but from subtraction: 36 – 12.5 = 23.5 → but 36 has precision to nearest 1 → so result should be 24 kg (rounded to nearest 1)

So numerator: 24 kg

Denominator: 80.2 + 12 = 92.2 → but 12 has no decimal → so sum is 92 m²

Now: 24 kg / 92 m² = 0.26087... kg/m²

Convert to g/cm²:

1 kg = 1000 g
1 m² = 10,000 cm²

So:
0.26087 kg/m² = 0.26087 × 1000 g / 10000 cm² = 260.87 / 10000 = 0.026087 g/cm²

Now sig figs:

- Numerator: 24 kg → 2 sig figs
- Denominator: 92 m² → 2 sig figs

So division → 2 sig figs

0.026087 → 2 sig figs → 2.6 × 10⁻² g/cm²

✔ 2.6 × 10⁻² g/cm²

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#### 5. $ \frac{7.50 \times 10^2\ \text{mg}}{3.0 \times 10^1\ \text{cm}} + \frac{1.00 \times 10^3\ \text{mg}}{4.5 \times 10^2\ \text{cm}} $ = ? mg/cm

Compute each term:

First: $ \frac{750\ \text{mg}}{30\ \text{cm}} = 25\ \text{mg/cm} $

Second: $ \frac{1000\ \text{mg}}{450\ \text{cm}} = 2.222...\ \text{mg/cm} $

Now add: 25 + 2.222 = 27.222 mg/cm

Now check sig figs:

- First term: 7.50 × 10² / 3.0 × 10¹ → 7.50 (3 sig) / 3.0 (2 sig) → result has 2 sig figs → 25 → but 25 has 2 sig figs → okay
- Second: 1.00 × 10³ / 4.5 × 10² → 1000 / 450 = 2.222 → 1.00 (3 sig), 4.5 (2 sig) → result has 2 sig figs → 2.2 mg/cm

Now addition: 25 + 2.2 = 27.2 → but 25 has no decimal → precision to nearest 1 → 2.2 has 1 decimal → so sum should be to nearest 1 → 27 mg/cm

But 25 is exact? No: 25 comes from 750 / 30 → both have 2 sig figs → 25 → 2 sig figs

But in addition: 25 → uncertainty ±0.5, 2.2 → ±0.05 → total uncertainty dominated by 25

So sum: 27.2 → but should report to nearest 1 → 27 mg/cm

Alternatively: 25 + 2.2 = 27.2 → but 25 has precision to ones place → so answer should be 27 mg/cm

✔ 27 mg/cm

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✔ Final Answers:

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A. Number of significant digits:

1. 3
2. 6
3. 4
4. 1
5. 2
6. 4
7. 3
8. 3
9. 1
10. 4

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B. Scientific notation:

1. 1.01 × 10⁴
2. 8.74000 × 10⁻²
3. 1.705 × 10²
4. 5 × 10³
5. 3.6 × 10¹

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C. Rounded values:

1. 1.0 × 10⁴ mm
2. 8.74 × 10⁻² cL
3. 40 kg
4. 2.569 × 10⁻² s
5. 5.92 × 10⁻¹ A

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D. Operations:

1. 921.7 m³
2. 0.18 g/cm²
3. 7.74 × 10⁻² s
4. 2.6 × 10⁻² g/cm²
5. 27 mg/cm

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