Problem 67 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ \frac{3}{5} \quad 0.06 $$

Step-by-Step Solution

Answer

Insert \( > \): \( \frac{3}{5} > 0.06 \).

Step 1: Convert Fraction to Decimal

To compare the fraction \( \frac{3}{5} \) with the decimal 0.06, we first convert the fraction to a decimal. Dividing the numerator by the denominator: \( 3 \div 5 = 0.6 \). So, \( \frac{3}{5} = 0.6 \).

Step 2: Compare Decimals

Now, compare the two decimals: 0.6 and 0.06. Observe that 0.6 is greater than 0.06 because 0.6 represents six-tenths while 0.06 represents six-hundredths.

Step 3: Insert Correct Symbol

Since 0.6 is greater than 0.06, the correct symbol to insert between the fraction and the decimal is \( > \). Therefore, \( \frac{3}{5} > 0.06 \).

Key Concepts

Fraction to Decimal Conversion

Understanding how to convert fractions into decimals is an important mathematical skill. It allows you to compare and compute with different types of numbers more easily. To convert a fraction like \( \frac{3}{5} \) into a decimal, you divide the numerator by the denominator. This is because a fraction essentially represents a division operation. For example, dividing 3 by 5 yields 0.6. Thus, \( \frac{3}{5} \) as a decimal is 0.6.

• Numerator: The top number in a fraction.
• Denominator: The bottom number in a fraction, indicates into how many parts the whole is divided.
• Decimal: A way of expressing fractions in a base 10 system.

Comparing Numerical Values

Once you have both numbers in the same form, either as decimals or fractions, comparing them becomes a straightforward task. When comparing decimals, look at each digit starting from the left. The first non-matching digit will determine which number is larger.
For comparing 0.6 and 0.06:

• Since the 6 in 0.6 is in the tenths place, and the 6 in 0.06 is in the hundredths place, it's clear that 0.6 is greater because each tenth is larger than each hundredth.
• Just like comparing whole numbers, you compare from left to right.
• If one decimal number has more digits to the right, like 0.600 compared to 0.60, the extra zeros don't change the value.

Inequalities in Algebra

Inequalities are mathematical expressions that show the relative size of two values. The symbols used in inequalities are \(<\), \(>\), \(\leq\), or \(\geq\), which respectively mean less than, greater than, less than or equal to, and greater than or equal to.
In the given exercise, after converting \( \frac{3}{5} \) to 0.6 and upon comparing it to 0.06, you use the inequality symbol \(>\) to express that 0.6 is larger than 0.06.

• Inequality Symbols: Visual tools to compare the sizes or values.
• Purpose: Help depict scenarios where quantities are not equal.

Using inequalities not only helps in understanding size and magnitude differences but is also critical in solving algebraic equations that reflect real-world scenarios.