Question

Arrange the following decimals from smallest to largest: 0.73,0.307,0.703 ______

Short answer

The decimals in order from smallest to largest are 0.307, 0.703, 0.73.

Step-by-step solution

List the Decimals

Write down all the given decimals clearly on a piece of paper. The numbers are 0.73, 0.307, and 0.703.

Normalize the Decimals

To make it easier to compare, let's align the decimals by making sure they all have the same number of decimal places. Convert each number to have three decimal places: - 0.73 becomes 0.730 - 0.307 remains 0.307 - 0.703 becomes 0.703.

Compare the Decimals

Now compare the decimals by looking at each digit from left to right: - First compare the 0s: All numbers have the same digit. - Next compare the tenths digits: 7, 3, and 7. The smallest is 3. - The second decimal, 0.307, is the smallest. - Compare the remaining numbers: 0.730 and 0.703. The next smallest tenths digit is 7, look at hundredths: 3 < 7. - The third number, 0.703, comes next, followed by 0.730.

Order the Decimals

List the decimals in the order determined from the comparisons: 0.307, 0.703, 0.73. Write them down in this order.

Key concepts

Decimal Comparison

Comparing decimals may seem tricky, but it's as simple as comparing whole numbers. When you line up decimals, always look at each digit, starting from the left. Just like with whole numbers, the largest number is the one with the biggest left-most digit. In the decimal world, you compare tenths first, then hundredths, and so on, as you go rightward. This ensures that you correctly find which number is smaller or larger.
Always remember to:

• Align the decimals vertically.
• Start comparing from left to right.
• If digits are equal, move to the next to the right.

Understanding decimal comparison helps in properly sorting numbers and is essential in many real-life situations, such as dealing with money, measurements, or scientific data.

Number Normalization

Before comparing decimals, it's helpful to "normalize" them. This means making sure they all have the same number of decimal places. Why do we do this? It's simply to make the comparison easier and less prone to error.
Consider this:

• 0.73 becomes 0.730
• 0.307 stays as 0.307
• 0.703 becomes 0.703

By normalizing these numbers, you can see their real magnitude more clearly and make sure there is no confusion. This process doesn’t change the actual value but just the way it looks.

Mathematics Education

Decimal comparison and ordering are fundamental skills in mathematics education. They build a base for understanding more complex concepts in math and science. Teaching students these skills helps them see the power and precision of numbers.
The approach used in the exercise assigns students to work with real decimal numbers in a step-by-step manner:

• Listing numbers clearly.
• Normalizing for comparison.
• Comparing digits systematically.

This educational process enhances logical thinking, accuracy, and helps develop confidence in mathematical tasks that involve decimals.

Place Value Understanding

Place value is a key concept in understanding how decimals work. Each digit in a decimal has a place value, which determines its value in the number.
The place value for decimals works like this:

• The first digit after the decimal is the tenths place.
• The next digit is the hundredths place, followed by the thousandths place, and so on.

Understanding place value lets you read decimals correctly and perform operations like addition, subtraction, and comparison. It helps you grasp why a number like 0.73 is larger than 0.307 but smaller than 0.703 just by looking at the places of the digits.