Printable worksheet for practicing significant figures and scientific notation in science.

Let's solve each part of this worksheet step by step with explanations.

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

Rules for Significant Figures:
1. All non-zero digits are significant.
2. Zeros between non-zero digits are significant.
3. Leading zeros (before non-zero digits) are not significant.
4. Trailing zeros (after non-zero digits) are significant only if there is a decimal point.
5. 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 are significant; last two zeros are not.
→ 3 significant digits

2. 0.0874000 cL
→ Leading zeros not significant.
→ 8, 7, 4, and the three trailing zeros after decimal are significant (because they follow a non-zero digit).
→ 6 significant digits

3. 170.5 km
→ All digits are significant (including zero between non-zeros and after decimal).
→ 4 significant digits

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

5. 36 kg
→ Both digits are non-zero → both significant.
→ 2 significant digits

6. 3.500 × 10⁻² s
→ In scientific notation: 3.500 has 4 digits, all significant.
→ 4 significant digits

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

8. 1.00 × 10³ mg
→ 1.00 has 3 digits → all significant.
→ 3 significant digits

9. 2 × 10² °C
→ Coefficient is just "2" → one significant digit.
→ 1 significant digit

10. 5.920 × 10⁻¹ A
→ 5.920 has four digits → all significant.
→ 4 significant digits

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✔ Answers for A:
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

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

1. 10100 mm
→ Move decimal to get 1.01 → moved 4 places right → exponent = +4
→ 1.01 × 10⁴

2. 0.0874000 cL
→ Move decimal to get 8.74 → moved 2 places left → exponent = -2
→ 8.74 × 10⁻²
(Note: We keep only significant digits — but since it’s conversion to scientific notation, we preserve all sig figs: 8.74000 → 8.74000 × 10⁻²)

But typically, we write the coefficient as 8.74000, so:
→ 8.74000 × 10⁻²

3. 170.5 km
→ 1.705 × 10² → moved decimal 2 places left
→ 1.705 × 10²

4. 5000 cm³
→ 5.000 × 10³ → assuming the zeros are not significant (since no decimal), but we’re converting the number, so write as is:
→ 5 × 10³ (only one sig fig, but value is 5000)
→ However, if we're writing exactly 5000 without decimal, it's ambiguous. But for scientific notation:
→ 5 × 10³

But note: If original was 5000 with no decimal, then only one sig fig → so 5 × 10³

5. 36 kg
→ 3.6 × 10¹
→ 3.6 × 10¹

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✔ Answers for B:
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
→ First two digits: 1 and 0 → round to 1.0 × 10⁴ mm
→ 1.0 × 10⁴ mm

2. 0.0874000 cL → 3 significant digits
→ First three significant digits: 8, 7, 4 → 0.0874 cL
→ 0.0874 cL (or 8.74 × 10⁻² cL)

3. 36 kg → 1 significant digit
→ Round to nearest ten → 40 kg
→ 40 kg (but only one sig fig → written as 4 × 10¹ kg)

4. 2.5689 × 10⁻² s → 4 significant digits
→ Already has 4 sig figs → no rounding needed
→ 2.569 × 10⁻² s? Wait!
→ 2.5689 has 5 sig figs → need to round to 4
→ Look at fifth digit: 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 → round to 3
→ 5.92 × 10⁻¹ A
→ 5.92 × 10⁻¹ A

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✔ Answers for C:
1. 1.0 × 10⁴ mm
2. 0.0874 cL or 8.74 × 10⁻² cL
3. 40 kg or 4 × 10¹ kg
4. 2.569 × 10⁻² s
5. 5.92 × 10⁻¹ A

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D. Perform the following operations

We must apply rules of 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.

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1. (0.17362 m)(31.22 m)(170.06 m)
→ Multiply all
→ First: 0.17362 × 31.22 = ?
→ 0.17362 × 31.22 ≈ 5.4225 → 5.4225
→ Then: 5.4225 × 170.06 ≈ 922.1 → let's compute:

Actually, better to do full multiplication:

$$
0.17362 \times 31.22 = 5.4225364 \\
5.4225364 \times 170.06 ≈ 922.14
$$

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

→ Limiting is 4 sig figs → round answer to 4 sig figs

→ 922.14 → 922.1 m³? Wait: 922.1 has 4 sig figs? Yes.

But 922.1 → 4 sig figs → correct.

→ 922.1 m³

✔ Answer: 922.1 m³

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2. (9.042 g) / [(5.24 cm)(9.5 cm)]

Denominator: 5.24 × 9.5
→ 5.24 × 9.5 = 50.08 → but 9.5 has 2 sig figs → so result should have 2 sig figs
→ 50.08 → rounds to 50 (2 sig figs)

Now: 9.042 / 50 = 0.18084 → but denominator has 2 sig figs → answer has 2 sig figs
→ 0.18 → but wait: 0.18 has 2 sig figs?

Yes: 1.8 × 10⁻¹ → 2 sig figs

So → 0.18 g/cm²

✔ Answer: 0.18 g/cm²

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3. 3.500 × 10⁻² s + 4.24 × 10⁻² s

Same power → add coefficients:
3.500 + 4.24 = 7.74 → but 4.24 has 3 sig figs, 3.500 has 4 → but for addition, look at decimal places.

Both have 4 decimal places? Let's see:

3.500 × 10⁻² = 0.03500
4.24 × 10⁻² = 0.0424

Add: 0.03500 + 0.0424 = 0.0774 → now check decimal places:

0.03500 → 5 decimal places
0.0424 → 4 decimal places → so answer should have 4 decimal places → 0.0774

But in terms of significant figures:
→ 0.0774 → 3 sig figs → because leading zeros don't count → 7,7,4 → 3 sig figs

But since both numbers have uncertainty in the last digit, we report based on precision.

→ Final answer: 7.74 × 10⁻² s

✔ Answer: 7.74 × 10⁻² s

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4. (36 kg – 12.5 kg) / (80.2 m² + 12 m²)

Numerator: 36 – 12.5 = 23.5 → but 36 has no decimal → uncertain in units place → so result should be to nearest whole number → 24 kg?
→ 36 has 2 sig figs, 12.5 has 3 → subtraction rule: look at decimal places.

36 → no decimal → ±0.5?
12.5 → ±0.05 → so difference: 23.5 → but 36 has precision to nearest 1 → so answer should be rounded to nearest 1 → 24 kg

Denominator: 80.2 + 12 = 92.2 → 80.2 has 1 decimal, 12 has none → so sum should have no decimals → 92 m²

Now divide: 24 kg / 92 m² = 0.26087... → now sig figs: numerator has 2 sig figs (24), denominator has 2 sig figs (92) → so answer has 2 sig figs

→ 0.26 g/cm²? Wait: units are kg/m² → but question asks for g/cm²?

→ We need unit conversion!

Wait! The problem says:
→ g/cm² → but inputs are in kg and m² → need to convert.

Let's re-express everything in grams and cm².

Numerator: 36 kg – 12.5 kg = 23.5 kg = 23,500 g
Denominator: 80.2 m² + 12 m² = 92.2 m²
→ 1 m² = 10,000 cm² → so 92.2 m² = 92.2 × 10⁴ cm² = 9.22 × 10⁵ cm²

Now:
$$
\frac{23,500\ \text{g}}{9.22 \times 10^5\ \text{cm}^2} = 0.02548... \approx 0.0255\ \text{g/cm}^2
$$

But we must consider sig figs from earlier steps:

Original numerator: 36 – 12.5 → 36 has 2 sig figs, 12.5 has 3 → difference: 24 kg (rounded to nearest 1) → 24 kg = 24,000 g → 2 sig figs

Denominator: 80.2 + 12 → 80.2 has 1 decimal, 12 has none → sum = 92 → 2 sig figs

So final division: 24,000 g / 92,000 cm² = 0.2608 → 2 sig figs → 0.26 g/cm²

But let's do exact:

23,500 g / 922,000 cm² = 0.02548 → but we used 24,000 / 92,000 = 0.2608 → but that's wrong.

Wait: we must use the actual values with proper rounding.

Step-by-step:

- 36 – 12.5 = 23.5 → but 36 has no decimal → so result must be rounded to nearest 1 → 24 kg → 24,000 g
- 80.2 + 12 = 92.2 → but 12 has no decimal → so round to nearest 1 → 92 m² → 92 × 10⁴ cm² = 9.2 × 10⁵ cm²

So:
$$
\frac{24,000\ \text{g}}{9.2 \times 10^5\ \text{cm}^2} = 0.02608... \approx 0.026\ \text{g/cm}^2
$$

Now sig figs: both numerator and denominator have 2 sig figs → answer has 2 sig figs → 0.026 → 2 sig figs

→ 0.026 g/cm²

✔ Answer: 0.026 g/cm²

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5. (7.50 × 10² mg / 3.0 × 10¹ cm) + (1.00 × 10³ mg / 4.5 × 10² cm)

First term:
$$
\frac{7.50 \times 10^2}{3.0 \times 10^1} = \frac{750}{30} = 25.0\ \text{mg/cm}
$$

Second term:
$$
\frac{1000}{450} = 2.222... \approx 2.22\ \text{mg/cm}
$$

Now add: 25.0 + 2.22 = 27.22 → but check decimal places:

- 25.0 → 1 decimal place
- 2.22 → 2 decimal places → so sum should have 1 decimal place

→ 27.22 → round to 27.2 mg/cm

Now sig figs: 25.0 has 3 sig figs, 2.22 has 3 → but addition rule applies → decimal places matter.

→ Final answer: 27.2 mg/cm

✔ Answer: 27.2 mg/cm

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✔ Final Answers Summary

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#### A. 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. Rounding
1. 1.0 × 10⁴ mm
2. 0.0874 cL
3. 40 kg
4. 2.569 × 10⁻² s
5. 5.92 × 10⁻¹ A

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#### D. Operations
1. 922.1 m³
2. 0.18 g/cm²
3. 7.74 × 10⁻² s
4. 0.026 g/cm²
5. 27.2 mg/cm

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