Problem 40 - Prealgebra

Question

For each pair of numbers, place the correct symbol, \(<\) or \(>,\) between the numbers. Write the following numbers in order from smallest to largest. \(0.4 \quad 0.04 \quad 0.42 \quad 0.24\) \(0.2 \quad 0.02\)

Step-by-Step Solution

Answer

The order from smallest to largest is: 0.02, 0.04, 0.2, 0.24, 0.4, 0.42.

Step 1: Compare the first pair

Let's compare the numbers in the first pair, 0.4 and 0.04. To do this, look at the tenths place since 0.4 has a digit 4 in the tenths place while 0.04 has a 0. Therefore, 0.4 is greater than 0.04. Thus, the symbol should be \(>\), i.e., \(0.4 > 0.04\).

Step 2: Compare the second pair

Next, compare the numbers in the second pair, 0.42 and 0.24. Similarly, look at the tenths place: 0.42 has a 4, while 0.24 has a 2. Therefore, 0.42 is greater than 0.24. Thus, the symbol should be \(>\), i.e., \(0.42 > 0.24\).

Step 3: Compare the third pair

For the numbers 0.2 and 0.02, look at the tenths place: 0.2 has a digit 2 in the tenths place, whereas 0.02 has a 0. Therefore, 0.2 is greater than 0.02. Hence, the correct symbol is \(>\), i.e., \(0.2 > 0.02\).

Step 4: Order numbers from smallest to largest

Now, list all given numbers from smallest to largest. The smallest is 0.02, followed by 0.04, then 0.2, followed by 0.24, then 0.4, and the largest is 0.42.

Key Concepts

Comparing Decimal Numbers

When comparing decimal numbers, it is important to focus on each number's digit starting from the left. This often means looking at the digits in the tenths, hundredths, or thousandths places. To start:

• Identify the highest order digit, usually the tenths place for two-digit decimals.
• If the tenths digit differs, you can decide immediately which number is larger based on this single digit.
• If the tenths digit is the same, move to the next place value, the hundredths, to determine the larger number.

For example, in comparing 0.4 and 0.04, 0.4 has 4 tenths and 0.04 has 0 tenths, meaning 0.4 is greater because 4 is larger than 0.

Decimal Place Value

Understanding decimal place value is crucial when working with numbers that have decimal points. Each position to the right of the decimal point represents a fraction of a base 10 value:

• The first digit after the decimal point is the tenths place, representing 1/10.
• The second digit is in the hundredths place, representing 1/100.
• The third digit is the thousandths place, representing 1/1000.

When determining size based on place value, always start from the leftmost (highest value) digit. For example, in 0.24, the digit 2 signifies "two-tenths," which is more substantial than “four-hundredths" in the number 0.04. This system helps in placing numbers in order and comparing them effectively.

Numerical Order

Ordering numbers from smallest to largest involves careful assessment of their place values. With decimals, the tenths place is crucial. Begin by:

• Listing the numbers to see the sequence clearly.
• Comparing the leftmost non-zero digit until one number is larger.
• Arranging the decimals based on their places, starting with the smallest.

As in the step-by-step solution, you determine that 0.02 is the smallest and 0.42 the largest. The order would become 0.02, 0.04, 0.2, 0.24, 0.4, and then 0.42. By following a systematic approach, you can efficiently determine and list decimals in their numerical order.