Question

Round off or add zeros to the following calculated answers to give a final answer with three significant figures: a. \(0.00001258 \mathrm{~L}\) b. \(3.528 \times 10^{2} \mathrm{~kg}\) c. \(125111 \mathrm{~m}\) d. \(58.703 \mathrm{~g}\) e. \(3 \times 10^{-3} \mathrm{~s}\) f. \(0.010826 \mathrm{~g}\)

Short answer

a. 0.0000126 L, b. 3.53 × 10² kg, c. 125000 m, d. 58.7 g, e. 3.00 × 10⁻³ s, f. 0.0108 g

Step-by-step solution

Title - Round off 0.00001258 L

First, identify the first three significant figures in the number 0.00001258. These are 1, 2, and 5. The fourth significant figure is 8, which is greater than 5. Hence, round the third figure up by 1. Thus, the number rounded to three significant figures is 0.0000126.

Title - Round off 3.528 × 10² kg

Identify the first three significant figures, which are 3, 5, and 2. The fourth figure is 8, which means the third figure (2) should be rounded up by 1. Thus, 3.528 × 10² rounded to three significant figures is 3.53 × 10².

Title - Round off 125111 m

Identify the first three significant figures: 1, 2, and 5. The fourth figure is 1, which means no rounding is needed. Thus the number rounded to three significant figures is 125000 m.

Title - Round off 58.703 g

Identify the first three significant figures: 5, 8, and 7. The fourth figure is 0, which means no rounding is needed. Thus the number rounded to three significant figures is 58.7 g.

Title - Adjust zeros for 3 × 10⁻³ s

The number 3 × 10⁻³ has only one significant figure. To express it with three significant figures, add zeros at required places, thus it becomes 3.00 × 10⁻³ s.

Title - Round off 0.010826 g

Identify the first three significant figures: 1, 0, and 8. The fourth figure is 2, which means no rounding is needed. Thus the number rounded to three significant figures is 0.0108 g.

Key concepts

rounding significant figures

Rounding significant figures is a crucial skill in mathematics and science.
It helps ensure that the precision of your measurements and calculations is represented accurately.
When you round a number to a certain number of significant figures, you are including numbers that meaningfully contribute to its precision.

For example, let's consider 0.00001258 L.
The first three significant figures here are 1, 2, and 5.
The fourth digit, 8, tells us how to round the last significant digit.
Since 8 is greater than 5, you round 5 up by 1, making it 6.
Thus, 0.00001258 L rounded to three significant figures becomes 0.0000126 L.

Another example is 58.703 g.
Here, the first three significant figures are 5, 8, and 7.
The fourth digit is 0, which is less than 5.
Therefore, you do not need to round up, so the number remains 58.7 g when rounded to three significant figures.

significant figures in measurements

In measurements, significant figures tell you how precise a measurement is.
They include all the digits you are certain about plus the first uncertain or estimated digit.
This helps convey the accuracy of your instruments and methods.

Let's take 125111 m as an example.
The first three significant figures are 1, 2, and 5.
The fourth digit is 1, which means no rounding is needed.
So you write it as 125000 m to indicate that only the first three digits are significant.

In another example, consider 0.010826 g.
The first three significant figures in this number are 1, 0, and 8.
The fourth digit is 2, so you don't need to round up.
This means the number rounded to three significant figures is 0.0108 g.
Properly noting significant figures ensures you maintain correct precision in scientific calculations.

scientific notation

Scientific notation is a way to express very large or very small numbers in a simpler form.
It is helpful in making calculations easier and more readable.
Scientific notation consists of two parts: a coefficient and an exponent (power of 10).

For instance, the number 3.528 × 10² kg can be rounded to three significant figures.
Identify the first three significant figures: 3, 5, and 2.
The fourth digit is 8, so you round 2 up by 1, making it 3.53 × 10² kg.

Similarly, a tiny number like 3 × 10⁻³ s has only one significant figure.
To write it with three significant figures, you add zeros, resulting in 3.00 × 10⁻³ s.
This makes the precision and accuracy clear, making scientific notation invaluable for handling measurements across various magnitudes.