Significant Figures Worksheet Answer Key showing examples and solutions for determining significant figures and performing calculations.

Problem Explanation and Solution

The worksheet focuses on significant figures, which are the digits in a number that carry meaning contributing to its precision. This includes all digits except:

1. Leading zeros (e.g., in 0.0023, the leading zeros are not significant).
2. Trailing zeros when they are merely placeholders to indicate the scale of the number (e.g., in 700000, the trailing zeros may not be significant unless specified).

#### Task 1: Indicate how many significant figures there are in each of the following measured values.

We need to count the significant figures for each number based on the rules of significant figures.

- 246.32: All non-zero digits are significant. → 5 sig figs
- 1.008: Zeros between non-zero digits are significant. → 4 sig figs
- 700000: Only the "7" is significant unless otherwise specified (trailing zeros are placeholders). → 1 sig fig
- 107.854: All non-zero digits are significant. → 6 sig figs
- 0.00340: Leading zeros are not significant, but the trailing zero after the decimal point is significant. → 3 sig figs
- 350.670: Zeros between non-zero digits and trailing zeros after the decimal point are significant. → 6 sig figs
- 100.3: All non-zero digits and the zero between non-zero digits are significant. → 4 sig figs
- 14.600: Zeros at the end of a number after the decimal point are significant. → 5 sig figs
- 1.0000: All digits are significant. → 5 sig figs
- 0.678: All non-zero digits are significant. → 3 sig figs
- 0.0001: Only the "1" is significant; leading zeros are not. → 1 sig fig
- 320001: All non-zero digits are significant. → 6 sig figs

#### Task 2: Calculate the answers to the appropriate number of significant figures.

For addition and subtraction, the result should have the same number of decimal places as the least precise measurement.

1. 32.567 + 135.0 + 1.4567
- Least precise measurement: 135.0 (1 decimal place)
- Sum: \( 32.567 + 135.0 + 1.4567 = 169.0237 \)
- Rounded to 1 decimal place: 169.0

2. 246.24 + 238.278 + 98.3
- Least precise measurement: 98.3 (1 decimal place)
- Sum: \( 246.24 + 238.278 + 98.3 = 582.818 \)
- Rounded to 1 decimal place: 582.8

3. 658.0 + 23.5478 + 1345.29
- Least precise measurement: 658.0 (1 decimal place)
- Sum: \( 658.0 + 23.5478 + 1345.29 = 2026.8378 \)
- Rounded to 1 decimal place: 2026.8

#### Task 3: Calculate the answers to the appropriate number of significant figures.

For multiplication and division, the result should have the same number of significant figures as the least precise measurement.

1. 23.7 × 3.8
- Significant figures: 23.7 (3 sig figs), 3.8 (2 sig figs)
- Product: \( 23.7 \times 3.8 = 89.86 \)
- Rounded to 2 sig figs: 90.

2. 45.76 × 0.25
- Significant figures: 45.76 (4 sig figs), 0.25 (2 sig figs)
- Product: \( 45.76 \times 0.25 = 11.44 \)
- Rounded to 2 sig figs: 11

3. 81.04 g × 0.010
- Significant figures: 81.04 (4 sig figs), 0.010 (2 sig figs)
- Product: \( 81.04 \times 0.010 = 0.8104 \)
- Rounded to 2 sig figs: 0.81

4. 6.47 × 64.5
- Significant figures: 6.47 (3 sig figs), 64.5 (3 sig figs)
- Product: \( 6.47 \times 64.5 = 417.615 \)
- Rounded to 3 sig figs: 417

5. 43.678 × 64.1
- Significant figures: 43.678 (5 sig figs), 64.1 (3 sig figs)
- Product: \( 43.678 \times 64.1 = 2800.0658 \)
- Rounded to 3 sig figs: 2.80 × 10³

6. 1.678 / 0.42
- Significant figures: 1.678 (4 sig figs), 0.42 (2 sig figs)
- Quotient: \( 1.678 / 0.42 = 3.995238 \)
- Rounded to 2 sig figs: 4.0

7. 28.367 / 3.74
- Significant figures: 28.367 (5 sig figs), 3.74 (3 sig figs)
- Quotient: \( 28.367 / 3.74 = 7.58476 \)
- Rounded to 3 sig figs: 7.58

8. 4278 / 1.006
- Significant figures: 4278 (4 sig figs), 1.006 (4 sig figs)
- Quotient: \( 4278 / 1.006 = 4252.4851 \)
- Rounded to 4 sig figs: 4252

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Task 1:} \\
246.32: 5 \text{ sig figs}, \quad 1.008: 4 \text{ sig figs}, \quad 700000: 1 \text{ sig fig}, \quad 107.854: 6 \text{ sig figs}, \quad 0.00340: 3 \text{ sig figs}, \\
350.670: 6 \text{ sig figs}, \quad 100.3: 4 \text{ sig figs}, \quad 14.600: 5 \text{ sig figs}, \quad 1.0000: 5 \text{ sig figs}, \quad 0.678: 3 \text{ sig figs}, \\
0.0001: 1 \text{ sig fig}, \quad 320001: 6 \text{ sig figs}. \\
\\
\text{Task 2:} \\
32.567 + 135.0 + 1.4567 = 169.0, \quad 246.24 + 238.278 + 98.3 = 582.8, \quad 658.0 + 23.5478 + 1345.29 = 2026.8. \\
\\
\text{Task 3:} \\
a) 23.7 \times 3.8 = 90., \quad b) 45.76 \times 0.25 = 11, \quad c) 81.04 \times 0.010 = 0.81, \quad d) 6.47 \times 64.5 = 417, \\
e) 43.678 \times 64.1 = 2.80 \times 10^3, \quad f) 1.678 / 0.42 = 4.0, \quad g) 28.367 / 3.74 = 7.58, \quad h) 4278 / 1.006 = 4252.
\end{array}
}
\]