Question

Perform the following mathematical operations and express the result to the correct number of significant figures. a. \(\frac{2.526}{3.1}+\frac{0.470}{0.623}+\frac{80.705}{0.4326}\) b. \((6.404 \times 2.91) /(18.7-17.1)\) c. \(6.071 \times 10^{-5}-8.2 \times 10^{-6}-0.521 \times 10^{-4}\) d. \(\left(3.8 \times 10^{-12}+4.0 \times 10^{-13}\right) /\left(4 \times 10^{12}+6.3 \times 10^{13}\right)\) e. \(\frac{9.5+4.1+2.8+3.175}{4}\) (Assume that this operation is taking the average of four numbers. Thus 4 in the denominator is exact.) f. \(\frac{8.925-8.905}{8.925} \times 100\) (This type of calculation is done many times in calculating a percentage error. Assume that this example is such a calculation; thus 100 can be considered to be an exact number.)

Short answer

The short answers for the given problems are as follows: a. \(188.3\) b. \(11.6\) c. \(4.768 \times 10^{-5}\) d. \(5.78 \times 10^{-26}\) e. \(4.894\) f. \(0.224\%\)

Step-by-step solution

a. Division and Addition

Perform the divisions first and round the obtained values according to the lowest significant figures in the divisor and dividend: \[\frac{2.526}{3.1} = 0.815\] \[\frac{0.470}{0.623} = 0.754\] \[\frac{80.705}{0.4326} = 186.7\] Now, sum these values and round the result according to the lowest decimal place: \[0.815 + 0.754 + 186.7 = 188.268 \approx 188.3\]

b. Multiplication, Subtraction, and Division

Perform the multiplication first and round the obtained value according to the lowest significant figures in the multiplicands: \[6.404 \times 2.91 = 18.63464 \approx 18.6\] Next, subtract the two numbers and round the result according to the lowest decimal place: \[18.7 - 17.1 = 1.6\] Finally, divide the two values and round the result according to the lowest significant figures in the divisor and dividend: \[\frac{18.6}{1.6}= 11.625 \approx 11.6\]

c. Subtraction

Perform the subtractions by considering the least amount of decimal places in the subtraction and rounding the results accordingly: \[6.071 \times 10^{-5} - 8.2 \times 10^{-6} = 5.289 \times 10^{-5}\] \[5.289 \times 10^{-5} - 0.521 \times 10^{-4} = 4.768 \times 10^{-5}\]

d. Addition and Division

Add the numbers in the numerator and denominator. Keep the lowest amount of significant figures while adding: \[(3.8 \times 10^{-12} + 4.0 \times 10^{-13}) = 3.84 \times 10^{-12}\] \[(4 \times 10^{12}+6.3 \times 10^{13}) = 6.64 \times 10^{13}\] Now, divide the two values and round the result according to the lowest significant figures in the divisor and dividend: \[\frac{3.84 \times 10^{-12}}{6.64 \times 10^{13}} = 5.78 \times 10^{-26}\]

e. Addition and Division

Perform the addition first as it is an average of four numbers, the four in the denominator is an exact number: \[9.5 + 4.1 + 2.8 + 3.175 = 19.575\] Now divide the sum by 4 and round to the lowest decimal place: \[\frac{19.575}{4} = 4.89375 \approx 4.894\]

f. Subtraction, Division, and Multiplication

Perform the subtraction first and keep the lowest amount of decimal places: \[8.925 - 8.905 = 0.020\] Now divide the result by the given value and round according to the lowest significant figures in the divisor and dividend: \[\frac{0.020}{8.925} = 0.0022402\] Finally, multiply the result by 100 (since it's an exact number): \[0.0022402 \times 100 = 0.22402 \approx 0.224\% \]

Key concepts

Mathematical Operations

Mathematical operations involve a series of processes such as addition, subtraction, multiplication, and division. Each operation has its specific rules that dictate how calculations are performed. For instance, when multiplying or dividing numbers, we need to consider significant figures. The number of significant figures in the final result should match the number with the fewest significant figures used in the operation. Likewise, when adding or subtracting, the result should be rounded to the least precise decimal place. These standard rules ensure that calculations are accurate and consistent across different scenarios.

Rounding Rules

Rounding is essential when dealing with significant figures, as it affects the reliability and accuracy of numerical results. The basic rule is that if the digit following your desired precision is 5 or greater, you round up the last retained digit by one. For example, if you need to round 2.375 to two decimal places, it becomes 2.38. However, if the next digit is less than 5, you simply drop it, such as rounding 2.374 to 2.37. These rules are straightforward but crucial for maintaining precision in reporting and communicating scientific data.

Percentage Error

Percentage error is a way to express how close an experimental or calculated value is to the actual or accepted value. It is calculated using the formula: \[ \text{Percentage Error} = \left( \frac{\text{Experimental Value} - \text{Accepted Value}}{\text{Accepted Value}} \right) \times 100 \% \]This formula helps highlight the accuracy and reliability of a given set of data. For instance, in the exercise, the calculation of percentage error involved determining the accuracy of a measurement by comparing it to a known standard. Accurate percentage error calculations are critical in fields requiring high precision, like chemistry and engineering.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers conveniently. It involves using powers of ten to simplify numeric representation. For example, instead of writing 0.0000075, we can express it as \(7.5 \times 10^{-6}\). This format is beneficial for making arithmetic operations easier, particularly multiplication and division, which involve adjusting the exponents as necessary. Furthermore, scientific notation ensures that significant figures are easier to count, as they appear clearly in the prefixed decimal. This notation is invaluable in scientific calculations, allowing for streamlined computation and improved clarity in representing data.