Question

Three quantities, the results of measurements, are to be added. They are \(2.0600,3.163,\) and \(1.12 .\) What is their sum to the correct number of significant figures?

Short answer

Answer: The sum of the three measurements is \(6.34\).

Step-by-step solution

Identify the number of decimal places in each measurement

In the given numbers: - \(2.0600\) has 4 decimal places - \(3.163\) has 3 decimal places - \(1.12\) has 2 decimal places In our final answer, it should have the least number of decimal places, which is 2 in this case.

Add the numbers

Add the three given numbers: \(2.0600 + 3.163 + 1.12 = 6.343\)

Round to the correct number of significant figures

Our final answer should have the least number of decimal places, which is 2. After rounding \(6.343\), we get: \(6.34\) Thus, the sum of the three measurements with the correct number of significant figures is \(6.34\).

Key concepts

Measurement

Understanding measurement is fundamental in science and daily life. It involves determining the size, quantity, or amount of something using various instruments or techniques. Measurements must be accurate and precise to be reliable.

Types of Measurements

Many kinds of measurements exist, each tailored to measure specific types of quantities:

• Length (meters, kilometers)
• Mass (grams, kilograms)
• Volume (liters, milliliters)
• Time (seconds, minutes)
• Temperature (Celsius, Kelvin)

When you combine measurements, like adding the quantities in the exercise, it's crucial to ensure that they are in compatible units.

Why Measurement Precision Matters

Each measurement in an experiment or calculation can affect the final result. Precise measurements help ensure that calculations, like sums or averages, reflect the true value as closely as possible. Inaccurate or imprecise measurements can lead to errors and significant deviations from expected outcomes.

Decimal Places

Decimal places are the digits that appear after the decimal point in a number. They are crucial for precision and accuracy in the result of calculations.

Understanding Decimal Places

For example, in the number 3.142, there are three decimal places. The more decimal places a measurement has, the more precise it is assumed to be. Here's what to keep in mind:

• Each decimal place represents a fraction of one whole. The first place is tenths, the second is hundredths, and so on.
• The precision of a calculation is determined by the least precise measurement (least number of decimal places) involved.

In the given exercise, although measurements like 2.0600 have more decimal places, the result must match the measurement with the fewest decimal places, which is 1.12 with two decimal places. This common practice ensures consistency and accuracy in results.

Rounding

Rounding is a mathematical process used to preserve the significant figures of a number while simplifying it. This technique is especially useful when numbers need to be expressed with a limited number of decimal places or digits.

How to Round Numbers

To round a number, you reduce the digits:

• Identify the decimal place you want to round to.
• Look at the digit immediately to the right of your desired decimal place.
• If this digit is 5 or more, increase the digit you're rounding up by one.
• If it is less than 5, keep your digit as it is.

In the exercise, the sum 6.343 must be rounded to two decimal places. Look at the third decimal place (3 in this case), since it is less than 5, the second decimal place remains the same and the answer is 6.34.

Significance of Rounding

Rounding helps express measurements and results in a manageable and understandable form, especially when dealing with very large or very small numbers. It ensures that the communicated results are clear and concise while maintaining a reasonable degree of accuracy.