Question

The masses of four sugar cubes are measured to be \(25.3 \mathrm{~g}, 24.7 \mathrm{~g}, 26.0 \mathrm{~g},\) and \(25.8 \mathrm{~g} .\) Express the answers to the following questions in scientific notation, with standard SI units and an appropriate number of significant figures. a) If the four sugar cubes were crushed and all the sugar collected, what would be the total mass, in kilograms, of the sugar? b) What is the average mass, in kilograms, of these four sugar cubes?

Short answer

Step 1: Convert the masses of the sugar cubes to kilograms. Step 2: Calculate the total mass by adding up the masses of all sugar cubes. Step 3: Express the total mass in scientific notation with appropriate significant figures. Step 4: Calculate the average mass by dividing the total mass by the number of sugar cubes. Step 5: Express the average mass in scientific notation with appropriate significant figures.

Step-by-step solution

Convert masses to kilograms

First, we need to convert the masses of the sugar cubes from grams to kilograms. To do this, we will divide the masses by 1000 since 1 kilogram is equal to 1000 grams.

Calculate the total mass

Add up the masses of all the sugar cubes to find the total mass in kilograms.

Express the total mass in scientific notation

Rewrite the total mass in scientific notation and with an appropriate number of significant figures.

Calculate the average mass

Divide the total mass by the number of sugar cubes (4 in this case) to find the average mass in kilograms.

Express the average mass in scientific notation

Rewrite the average mass in scientific notation and with an appropriate number of significant figures.

Key concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. It's essential to understand which numbers are significant to ensure accuracy in calculations. For example:

• All non-zero digits are significant.
• Zeros between significant figures are significant.
• Leading zeros are not significant.
• Trailing zeros are significant if they are after a decimal point.

In our exercise, the measured masses such as 25.3 g have three significant figures. When expressing results like total and average masses, we maintain these figures to preserve measurement precision. This practice is crucial when converting these masses into kilograms and presenting them in scientific notation. This ensures consistency and precision in scientific calculations.

SI Units

SI units, or the International System of Units, provide a standard for scientific measurements. The base unit for mass in the SI system is the kilogram (kg). Using a common standard like SI units helps scientists worldwide understand and compare measurements easily. Each mass in the original exercise was measured in grams (g), which is a subunit of the kilogram. Converting from grams to kilograms involves dividing by 1000, because 1 kilogram equals 1000 grams. Proper use of SI units is fundamental in scientific communication, ensuring that calculations with values in different forms are correct and easily interpreted.

Mass Conversion

Mass conversion is the process of changing mass from one unit to another, often required in scientific contexts to align measurements with standard units. In the exercise, converting grams to kilograms is necessary. Here's how to perform the conversion: 1. Take each mass value in grams. 2. Divide by 1000 to convert to kilograms. For example, 25.3 g becomes 0.0253 kg. This step is crucial for obtaining the total and average masses in consistent units. Using correct units simplifies the process of further calculations like summation and averaging, and setting these values into scientific notation.

Average Mass

The average mass provides insight into the typical mass of each sugar cube, smoothening out variations. To calculate it: 1. Sum all individual masses (already converted to kilograms). 2. Divide by the number of items, which is 4 sugar cubes in this example. This gives the average mass in kilograms. Expressing this average in scientific notation makes it easier to read and compare, especially when dealing with very small numbers. Maintaining significant figures throughout is also important to reflect the precision of initial measurements.