Question

Perform each calculation and limit each answer to four significant figures. a) \(18,900 \times 76.33 \div 0.00336=?\) b) \(0.77604 \div 76,003 \times 8.888=?\)

Short answer

a) 4.300×10^8; b) 9.073×10^-5.

Step-by-step solution

Understanding the Problem

We need to perform two separate calculations and round each result to four significant figures.

Calculate Part (a) Numerator

First, calculate the numerator of part (a). Multiply the two numbers: \[ 18,900 \times 76.33 = 1,442,637 \]

Compute Part (a) Full Calculation

Divide the result from Step 2 by 0.00336 to complete the calculation for part (a): \[ \frac{1,442,637}{0.00336} = 429,958,035.7 \] Round this to four significant figures: 4.300 \times 10^8.

Calculate Part (b) Numerator

First, calculate the numerator of part (b). Multiply the two numbers: \[ 0.77604 \times 8.888 = 6.895 \]

Compute Part (b) Full Calculation

Divide the result from Step 4 by 76,003 to complete the calculation for part (b): \[ \frac{6.895}{76,003} = 0.000090725 \] Round this to four significant figures: 9.073 \times 10^{-5}.

Key concepts

Rounding

Rounding is an essential skill in mathematics that helps simplify numbers by reducing the number of digits while keeping the figure as close as possible to the original. This is particularly important when dealing with significant figures, where precision is maintained while unnecessary detail is minimized.

When rounding a number to a specific number of significant figures, you start counting from the first non-zero digit. For example, in the number 429,958,035.7, the first significant figure is 4. To round this to four significant figures:

• Identify the first four figures: 4, 2, 9, and 9.
• Check the digit after the fourth significant figure (in this case, 8): if it is 5 or greater, round up the last significant figure.
• Thus, you round 429,958,035.7 to 4.300 × 10^8.

This process ensures that the number is simplified without losing the necessary precision and remains practical for real-life applications.

Numerical Calculations

Numerical calculations are the backbone of solving mathematical problems accurately. They involve executing operations like addition, subtraction, multiplication, and division on numbers. When you perform these calculations, especially with large or small numbers, maintaining accuracy and order of operations is crucial.

In mathematical expressions, it's often helpful to approach them step-by-step:

• Start by solving operations inside parentheses or other grouping symbols.
• Next, address any exponents or roots.
• Then, perform multiplication and division from left to right.
• Finally, carry out addition and subtraction from left to right.

For example, in the given exercise, to find the numerical result of part (a), multiplication within the numerator is done first, followed by the division by 0.00336. Ensuring each step is executed accurately is vital for reaching the correct final result. Breaking down these calculations can make complex problems more manageable.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers conveniently by representing them as a product of a coefficient and a power of ten. This makes numbers easier to read and work with in scientific and engineering contexts.

In scientific notation, a number is written as: \[ \text{Coefficient} \times 10^{\text{exponent}} \]The coefficient is a number greater than or equal to 1 and less than 10, while the exponent indicates the number of places the decimal point has moved. For example, the number 429,958,035.7, rounded to four significant figures, is expressed as 4.300 × 10^8 in scientific notation.

Using scientific notation:

• Simplifies computations by reducing the complexity of numbers.
• Allows easy comparison of vastly different sized numbers.
• Keeps significant figures in focus, which is essential for precision in calculations.

Scientific notation is a powerful tool for handling both ends of the numerical spectrum, ensuring clarity and efficiency in representing and calculating with extensive ranges of numbers.