Question

Solve the following problems, stating answers to the proper number of significant figures: (a) \(12.62+1.5+0.25=\) ? (b) \(\left(2.25 \times 10^{3}\right)\left(4.80 \times 10^{4}\right)=\) ? (c) \(\frac{(452)(6.2)}{14.3}=\) ? (d) \((0.0394)(12.8)=\) ? (e) \(\frac{0.4278}{59.6}=\) ? (f) \(10.4+3.75\left(1.5 \times 10^{4}\right)=\) ?

Short answer

(a) 14.4, (b) 1.08x10^8, (c) 1.9x10^2, (d) 0.504, (e) 0.00718, (f) 5.63x10^4

Step-by-step solution

- Solve part (a)

First, perform the addition: \(12.62 + 1.5 + 0.25 = 14.37\) Determine the number of significant figures: The least precise measurement is 1.5 (2 significant figures), so the result should have the same number of decimal places: Final answer: \(14.4\)

- Solve part (b)

First, perform the multiplication: \((2.25 \times 10^{3})(4.80 \times 10^{4}) = 10.8 \times 10^{7}\) Combine the exponents: \(10.8 \times 10^{7} = 1.08 \times 10^{8}\) Determine the number of significant figures: Both measurements have 3 significant figures, so the result should also have 3 significant figures: Final answer: \(1.08 \times 10^{8}\)

- Solve part (c)

First, perform the multiplication and division: \(\frac{(452)(6.2)}{14.3} = \frac{2802.4}{14.3} = 195.2356643\) Determine the number of significant figures: The least number of significant figures in the measurements is 2 (6.2), so the result should have 2 significant figures: Final answer: \(1.9 \times 10^{2}\)

- Solve part (d)

First, perform the multiplication: \((0.0394)(12.8) = 0.50432\) Determine the number of significant figures: The least number of significant figures in the measurements is 3 (0.0394), so the result should have 3 significant figures: Final answer: \(0.504\)

- Solve part (e)

First, perform the division: \(\frac{0.4278}{59.6} = 0.0071795302\) Determine the number of significant figures: The least number of significant figures in the measurements is 3 (0.4278), so the result should have 3 significant figures: Final answer: \(0.00718\)

- Solve part (f)

First, solve the multiplication inside the expression: \(3.75(1.5 \times 10^{4}) = 56250\) Next, perform the addition: \(10.4 + 56250 = 56260.4\) Determine the number of significant figures: The least precise measurement in the addition term is 10.4 (2 decimal places), so the result should be rounded accordingly: Final answer: \(5.63 \times 10^{4}\)

Key concepts

scientific notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. This form uses a base number between 1 and 10, multiplied by a power of 10. It’s especially useful in scientific calculations to manage extreme values easily. For example, instead of writing out 45,000,000, we can write it as \(4.5 \times 10^{7}\). This makes calculations easier and helps avoid errors in counting zeros.
When performing operations like multiplication or division with scientific notation, you handle the base numbers and exponents separately. For instance, when multiplying \(2.25 \times 10^3\) by \(4.80 \times 10^4\), you first multiply the base numbers (2.25 and 4.80) and then add the exponents (3 and 4) to get the answer \(1.08 \times 10^8\).

multiplication and division

Multiplication and division with significant figures require careful consideration of the precision in each measurement. When multiplying or dividing numbers, the final result should have the same number of significant figures as the factor with the least significant figures.
For example, when multiplying 0.0394 by 12.8, you first perform the multiplication to get 0.50432. The number with the fewest significant figures here is 0.0394 (3 significant figures), so you round your result to 0.504.
Similarly, when performing division like \(\frac{0.4278}{59.6} = 0.0071795302\), the number with the fewest significant figures is 0.4278 (4 significant figures), so you round the final answer to 0.00718.

addition and subtraction

Addition and subtraction with significant figures follow different rules compared to multiplication and division. The key is to look at the decimal places. Your result should have the same number of decimal places as the number with the least decimal precision.
Consider the problem \(12.62 + 1.5 + 0.25\). Before you sum up these numbers, note that 1.5 has the fewest decimal places (1 decimal place). So, your final answer should also have one decimal place, making it 14.4.
Another example from the exercise is \(10.4 + 56250\). Here, 10.4 has one decimal place, so your final sum, once rounded, should also be represented with one decimal place resulting in 56260.4, and in scientific notation, it will be \(5.63 \times 10^4\).

precision in measurements

Precision in measurements is crucial for ensuring that calculated results are reliable and meaningful. Any measurement has a degree of uncertainty, and significant figures are a way to express this precision.
When handling operations, it's important to maintain the consistency of significant figures to reflect the inherent precision of the measurements.

• For multiplication and division, the number of significant figures in the result should be the same as in the measurement with the least significant figures.
• For addition and subtraction, the number of decimal places in the result should match the number with the least decimal places in the measurement.

Understanding these rules helps prevent errors and ensures that your results correctly reflect the precision of the initial data. Whether you're solving problems related to physical measurements or just working on science homework, keeping track of significant figures is essential for accurate and credible results.