Problem 128 - Elementary and Intermediate Algebra

Question

Fill in the blank with \(>\) or \(<: 0.3 \quad \frac{1}{3}\)

Step-by-Step Solution

Answer

0.3 is less than \(\frac{1}{3}\), so 0.3 < \(\frac{1}{3}\).

Step 1: Convert Fraction to Decimal

First, convert the fraction \( \frac{1}{3} \) into a decimal to easily compare it with 0.3. Divide 1 by 3 to get the decimal representation, which is 0.3333... (repeating).

Step 2: Compare Decimals

Now that both numbers are in decimal form, compare 0.3 with 0.3333.... The number 0.3 is less than 0.3333....

Step 3: Fill in the Blank

Based on the comparison, 0.3 is less than \( \frac{1}{3} \). Therefore, fill in the blank with '<' to show the relationship: 0.3 \(<\) \(\frac{1}{3}\).

Key Concepts

Comparison of Decimals

Decimals are numerical values that lie between integers. They allow us to express numbers that are smaller than whole units. Understanding how to compare these values is an essential skill in mathematics.

When comparing decimals, consider each place value from left to right. Begin with the tenths place, then the hundredths, and so on.

For example:

• Compare 0.3 and 0.333. Focus on the tenths place first. In 0.3, there is a 3 in the tenths place, while 0.333 also has a 3 in the tenths place.
• Move to the next place value, the hundredths. Here, 0.3 does not have a digit, which implies a zero, while 0.333 has a 3. Clearly, 0.300 is less than 0.333.

Comparing decimal values can also involve using symbols such as greater than (>), less than (<), or equal to (=). Familiarizing oneself with these comparisons is fundamental to dealing with inequalities in mathematics.

Fraction to Decimal Conversion

Converting fractions to decimals gives us another way to express the same quantity, often making it easier to perform operations like comparison and addition.

In the provided exercise, the fraction \(\frac{1}{3}\) was converted to a decimal. To do this, the numerator (1) was divided by the denominator (3) to yield 0.3333..., a repeating decimal.

Generally, to convert a fraction:

• Divide the top number (numerator) by the bottom number (denominator).
• If the division isn't exact, you'll end up with a repeating decimal, indicated by a line over the repeating digits or simply by writing those digits several times, like 0.3333.

Understanding this conversion is crucial because it provides insights into the nature of numbers, such as seeing rational numbers expressed as endless repeating decimals. This is essential in comparing sizes and calculating precise values.

Inequalities in Mathematics

Inequalities are statements showing that two values are unequal in size. They are vital in understanding relationships between numbers and solving algebraic expressions.

Inequality symbols are used to denote such relationships:

• "\(<\)" means "less than";
• "\(>\)" means "greater than";
• "\(=\)" means "equal to."

In the exercise, the inequality needed was between 0.3 and \(\frac{1}{3}\), thus resulting in 0.3 \(<\) \(\frac{1}{3}\). Expressing these differences in value is helpful in various mathematical contexts, from simple comparisons to solving equations.

Using inequalities helps in problem-solving, determining feasible solutions in algebraic problems, and understanding constraints in real-world scenarios, such as calculating budgets or understanding speed limits.