Problem 70 - Intermediate Algebra

Question

Insert either \(a<\) or \(a>\) symbol to make a true statement. $$ -\frac{7}{30} \quad-0.23 $$

Step-by-Step Solution

Answer

\(-\frac{7}{30} < -0.23\).

Step 1: Understand the Symbols

The symbols to use are \(a<\) (less than) and \(a>\) (greater than). The goal is to determine which symbol makes the given statement true.

Step 2: Convert Fractions to Decimal

Convert the fraction \(-\frac{7}{30}\) to a decimal. Divide \(-7\) by \(30\) to get approximately \(-0.2333\).

Step 3: Compare the Decimals

Now compare \(-0.2333\) and \(-0.23\). Since \(-0.2333\) is slightly less than \(-0.23\), we need to use the \(a<\) symbol.

Step 4: Insert the Correct Symbol

Insert the \(a<\) symbol between the numbers to make the true statement: \(-\frac{7}{30} < -0.23\).

Key Concepts

Decimal Conversion

Decimal conversion is a process of expressing numbers that are in one form, like fractions, into another form, namely decimals. It is particularly useful in comparing values that initially appear different due to their formats.

To convert a fraction, you simply divide the numerator by the denominator. Let's take the fraction \(-\frac{7}{30}\) as an example. Here, you divide -7 (the numerator) by 30 (the denominator) to obtain a decimal. Using a calculator or long division, dividing \(-7\) by \(30\) gives you approximately \(-0.2333\).

This decimal is an approximation of the exact fraction. It's a handy way to quickly grasp the size of the number in relation to other decimals. This is especially practical in tasks involving comparison, where decimals are much easier to work with visually.

Fractions

Fractions represent numbers that are not whole, showcasing a part of a whole. They consist of a numerator (top part) and a denominator (bottom part), separated by a line.

Understanding fractions is essential as they display values as a part of larger wholes. For example, the fraction \(-\frac{7}{30}\) indicates that -7 parts are of a division of 30 equal parts. It's a clear demonstration of how fractions cut numbers into pieces that can be analyzed more finely.

When dealing with fractions in equations or when comparing different values, converting fractions into decimals, as previously explained, can simplify the comparison and make complex concepts more approachable.

Comparing Decimals

Once numbers have been converted into a common format, such as decimals, comparing them becomes much more straightforward.

Let's look at comparing \(-0.2333\) and \(-0.23\). When comparing decimals, align them by their decimal point and assess their digits from left to right. Start with the highest place value and proceed to the right until a difference is found.

- Compare the tenths place: Both are \(-0.2\).- Compare the hundredths place: The first has a 3, and the second has a 3.- Compare the thousandths place: Here, \(-0.2333\) extends further (3 vs none); hence it is smaller than \(-0.23\).

Thus, \(-0.2333\) is less than \(-0.23\), confirming the placement of the "less than" symbol \(a<\) in the inequality. Mastering decimal comparison aids in logically arranging numbers in order, which is a vital mathematical skill.