Question

Express the results of the following calculations with the correct number of significant figures: (a) \(\frac{3.41-0.23}{5.233} \times 0.205\) (b) \(\frac{5.556 \times 2.3}{4.223-0.08}\)

Short answer

(a) 0.125, (b) 3.1

Step-by-step solution

Perform Calculation for Part (a)

Calculate the result of the expression \( \frac{3.41 - 0.23}{5.233} \times 0.205 \). Start by performing the subtraction in the numerator: \( 3.41 - 0.23 = 3.18 \).

Divide the Subtracted Result by Denominator for Part (a)

Next, divide \( 3.18 \) by \( 5.233 \) to get \( \frac{3.18}{5.233} \approx 0.6077 \).

Multiply by the Remaining Factor for Part (a)

Multiply \( 0.6077 \) by \( 0.205 \) to obtain \( 0.6077 \times 0.205 \approx 0.12458 \).

Determine Significant Figures for Part (a)

The result of part (a) must reflect the number of significant figures from the data used. The number with the least significant figures in our calculation is \( 0.205 \), which has 3 significant figures. Thus, round \( 0.12458 \) to 3 significant figures: \( 0.125 \).

Perform Calculation for Part (b)

Calculate the result of the expression \( \frac{5.556 \times 2.3}{4.223-0.08} \). Start by multiplying in the numerator: \( 5.556 \times 2.3 = 12.7788 \).

Perform Subtraction in Denominator for Part (b)

Subtract \( 0.08 \) from \( 4.223 \) to find \( 4.223 - 0.08 = 4.143 \).

Divide the Multiplied Result by Denominator for Part (b)

Then, divide \( 12.7788 \) by \( 4.143 \) to get \( \frac{12.7788}{4.143} \approx 3.084 \).

Determine Significant Figures for Part (b)

The final result should reflect the least number of significant figures. Here, the term with the least significant figures is \( 2.3 \), having 2 significant figures. Round \( 3.084 \) to 2 significant figures: \( 3.1 \).

Key concepts

Understanding Numerical Calculations

Numerical calculations involve processing numbers to acquire a result. It might include various mathematical operations like addition, subtraction, multiplication, and division. These calculations are a foundation of mathematics and require precision and accuracy. For instance, in our exercise, the calculations involved operations like subtraction, multiplication, and division. It’s crucial to follow these operations step by step carefully because each operation contributes to the final result.

While performing numerical calculations, always remember:

• Start by identifying the operations required.
• Process the operations in the correct mathematical order, following the rules of precedence.
• Ensure accuracy at each step to maintain the correctness of the ultimate result.

Taking these considerations into account helps achieve accurate and reliable outcomes in any problem-solving scenario.

Following Rounding Rules

Rounding is a critical step in numerical calculations and is essential for maintaining numerical precision. It allows numbers to be expressed in forms that are more manageable, while still conveying the necessary information as accurately as possible.

When rounding a number, follow these general rules:

• If the digit after the rounding digit is less than 5, leave the rounding digit unchanged.
• If the digit after the rounding digit is 5 or greater, increase the rounding digit by 1.

This process ensures that numbers conform to the predefined level of significance—hence improving the precision.

Rounding helps when dealing with significant figures, which convey the precision level of a measurement or calculation outcome. Applying the correct rounding rules ensures the reliability of the result, just as noted in our example where outcomes were rounded to match the significant figures of the data that was used.

Enhancing Numerical Precision

Numerical precision is about how closely a calculated number can approximate a true value or reference number. Precision is dependent on significant figures—the number of meaningful digits including non-zero digits and any zeros between them or following them in a decimal.

To enhance numerical precision:

• Use numbers with more digits and significant figures when necessary to keep results as close to accurate as possible.
• Be mindful of the rules governing significant figures during calculations, especially during estimation and rounding.
• Keep track of every step in a multi-operation procedure because errors early in calculations can affect the final result.

For example, when multiple calculations or transformations occur, the final answer should reflect the precision tied to initial numbers used. In our exercise, this meant rounding off to fewer significant figures to respect and reflect their precision in inputs.

Mastering Arithmetic Operations

Arithmetic operations are the calculations involving basic mathematical procedures: addition, subtraction, multiplication, and division. Understanding how to carry out each operation efficiently and accurately is vital for successful problem-solving. These operations form the backbone of nearly all numeric manipulation.

For optimal outcomes when performing arithmetic operations, remember:

• Change the order of operations can affect the result; always follow the precedence (PEMDAS/BODMAS rule).
• Ensure numbers are correctly aligned, especially with decimals, so that each has its proper place value.
• Verify calculations step-by-step to identify and correct any errors as they arise.

By mastering these operations, you can effectively engage a wide array of mathematical challenges, just as seen in how we methodically approached each section of our initial example problem.