Question

Round off each number in the following calculation to one fewer significant figure, and find the answer: $$\frac{10.8 \times 6.18 \times 2.381}{24.3 \times 1.8 \times 19.5}$$

Short answer

0.19

Step-by-step solution

- Identify Significant Figures

Identify the number of significant figures in each number in the calculation.\(10.8\) (3 significant figures), \(6.18\) (3 significant figures), \(2.381\) (4 significant figures), \(24.3\) (3 significant figures), \(1.8\) (2 significant figures), \(19.5\) (3 significant figures).

- Round to One Fewer Significant Figure

Round each number to one fewer significant figure: \(10.8\) becomes \(11\) (2 significant figures), \(6.18\) becomes \(6.2\) (2 significant figures), \(2.381\) becomes \(2.38\) (3 significant figures), \(24.3\) becomes \(24\) (2 significant figures), \(1.8\) becomes \(1.8\) (still 2 significant figures), and \(19.5\) becomes \(20\) (2 significant figures).

- Perform Calculation with Rounded Numbers

Substitute the rounded numbers into the original expression and perform the calculation: \( \frac{11 \times 6.2 \times 2.38}{24 \times 1.8 \times 20} \).

- Simplify the Numerator

Calculate the product of the numbers in the numerator: \(11 \times 6.2 \times 2.38 = 162.876\).

- Simplify the Denominator

Calculate the product of the numbers in the denominator: \(24 \times 1.8 \times 20 = 864\).

- Divide the Simplified Numerator by the Simplified Denominator

Perform the division \( \frac{162.876}{864} = 0.1885\).

- Round the Final Result to Appropriate Significant Figures

Round the final result to the significant figures equal to the smallest number of significant figures among the original rounded numbers which is 2: \(0.1885\) rounds to \(0.19\).

Key concepts

numerical rounding

Numerical rounding helps to simplify calculations by reducing the number of digits in a number. When rounding off numbers, we look at the digit immediately after the last significant figure we want to keep.
If this digit is 5 or higher, we round up. If it is lower than 5, we round down.
For example, when rounding 6.18 to two significant figures, the number becomes 6.2.
This is because the third digit, 8, causes us to round the second digit, 1, up to 2. By rounding off numbers, our calculations become easier, but we must be careful as it can also slightly affect the precision of the results.

significant figures

Significant figures are digits in a number that contribute to its precision. They include all the non-zero digits, any zeros between them, and any trailing zeros in the decimal part.
For instance, 2.381 has four significant figures: 2, 3, 8, and 1.
When performing calculations, it's essential to keep track of significant figures to maintain accuracy. In the provided exercise, reducing each number's significant figures was needed to simplify the problem. This involved reducing each number by one significant figure while maintaining as much accuracy as possible.
So, 2.381 is rounded to 2.38, retaining three significant figures.

step-by-step calculation

Solving complex problems step by step helps break down the task into manageable parts.
In our exercise, we first identified the significant figures in each number before rounding them off.
By simplifying each number, we restructured the initial complex expression to an approachable form.
We then calculated the product of numbers in the numerator and denominator separately before dividing them.
Lastly, we rounded our final result to match the smallest number of significant figures among our rounded numbers, providing us with the final answer.
This systematic approach ensures clarity and accuracy in solving mathematical problems.