Problem 69 - Algebra 1

Question

Compare using \(<,=,\) or \(>.\) $$6.7 \underline{?} 6 .079$$

Step-by-Step Solution

Answer

6.7 > 6.079

Step 1: Identify the numbers

The numbers that need to be compared are 6.7 and 6.079.

Step 2: Compare the digits in each place value from left to right

First, compare the numbers in the ones place (the digits to the left of the decimal point). Both numbers have a 6 in the ones place. Next, compare the numbers in the tenths place (the first digit to the right of the decimal point). For 6.7, the tenths digit is 7. For 6.079, the tenths digit is 0.

Step 3: Determine which number is larger

Since 7 is larger than 0, it can be concluded that 6.7 is larger than 6.079.

Key Concepts

Decimal Numbers

Decimal numbers are a way of representing numbers that have a fractional component. They consist of two parts: the whole number part and the fractional part, separated by a decimal point. For example, in the number 6.7, the 6 is the whole number part, and the .7 is the fractional part. The digits after the decimal point denote values that are less than one. These values decrease in size from left to right.

• Understanding decimals is crucial in comparing numbers, performing arithmetic, and measuring quantities precisely.
• When comparing decimal numbers, focus on their digits from left to right, emphasizing place value.

Grasping the concept of decimals helps you understand various scenarios in real life, such as finance and measurements, which often require precision.

Place Value

Place value is a system that helps us understand the value of each digit in a number based on its position. Each digit represents a different power of ten, depending on its placement relative to the decimal point.

• The digit immediately to the left of the decimal is the ones place.
• The first digit to the right of the decimal is the tenths place, followed by the hundredths place, thousandths place, and so on.

For instance, in 6.7, the '7' is in the tenths place, meaning 7 tenths or 0.7. In 6.079, the '0' in the tenths place signifies 0 tenths, while '7' follows in the hundredths place, and '9' in the thousandths place.
Place value helps to clearly compare decimal numbers by highlighting which digits have the greater value, aiding in determining which number is larger or smaller.

Number Line

A number line is a visual tool used to represent numbers in a straight line. It shows numbers in order from smallest to largest, making it easier to understand the value of a number in relation to others.

• Decimal numbers can be placed on a number line, just like whole numbers.
• The number line helps visualize the difference and the position of numbers, including decimals.

By plotting points for 6.7 and 6.079 on a number line, you can clearly see that 6.7 is positioned further to the right, indicating it is the larger number.
This visual aspect of a number line helps reinforce the comparison process by showing how numbers are ordered and spaced.

Order of Magnitude

Order of magnitude gives us an idea of the scale or size of a number. It's a way to express the number's relative size by powers of ten. For example, the number 100 has an order of magnitude of 2, since it is 10 raised to the power of 2.

• While comparing decimal numbers, the first non-zero digit can help assess the order of magnitude.
• In the number 6.079, the significant digit after the whole number part will be in the tenths place, giving it a higher magnitude importance.

In the context of 6.7 and 6.079, while both numbers are close, 6.7 has a greater order of magnitude in the tenths place than 6.079, due to the 7 versus 0 comparison.
Understanding order of magnitude aids in quickly determining the relative size and importance of decimal numbers without needing to analyze each digit in detail.