Problem 40 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$6.18 \circ 6 \frac{1}{5}$$

Step-by-Step Solution

Answer

6.18 is less than 6.2, so the correct symbol is \(<\).

Step 1: Convert Mixed Number to Decimal

Begin by converting the mixed number \(6 \frac{1}{5}\) to a decimal. The fraction \(\frac{1}{5}\) is equivalent to 0.2, since 1 divided by 5 is 0.2. Therefore, \(6 \frac{1}{5}\) is equal to 6.2 in decimal form.

Step 2: Compare the Decimals

Now, compare the two decimal numbers: 6.18 and 6.2. Notice that 6.18 is less than 6.2, as 18 hundredths are smaller than 20 hundredths.

Step 3: Choose the Appropriate Symbol

Based on our comparison, since 6.18 is less than 6.2, use the symbol \(<\) to indicate this relationship. Thus, the correct statement is \(6.18 < 6 \frac{1}{5}\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, in the mixed number \(6 \frac{1}{5}\), the whole number is 6 and the fraction is \(\frac{1}{5}\). Mixed numbers are commonly used in everyday measurements, such as "7 and a half inches" or "2 and a quarter hours." This way of expressing numbers allows us to work comfortably with quantities that aren't whole numbers, which is particularly useful in practical situations.

Here are a few key points to remember about mixed numbers:

• The whole number part tells us how many complete units we have.
• The fraction part tells us the portion of a unit we have.
• Mixed numbers can always be converted into improper fractions or decimal numbers for easier computation.

Understanding mixed numbers is essential when comparing sizes or quantities, as it enables conversion for more straightforward calculations. When you're comfortable with changing mixed numbers into other forms, comparing and computing them becomes much more manageable.

Decimal Conversion

Decimal conversion involves turning numbers, such as fractions or mixed numbers, into decimal form. It's a simple yet powerful tool that can aid in solving math problems, especially when comparing and ordering different types of numbers.

When converting a mixed number to a decimal, you must first convert the fractional part of the mixed number into a decimal. For instance, to convert \(\frac{1}{5}\) into a decimal, you divide the numerator (1) by the denominator (5). This gives a decimal value of 0.2.

Then, add this decimal to the whole number part of the mixed number. So, for \(6 \frac{1}{5}\), you add 6 and 0.2 to get 6.2.

Key steps for decimal conversion:

• Convert the fractional part of the mixed number into a decimal by dividing.
• Add the resulting decimal to the whole number part to get a single decimal.
• Use the converted decimal number for further comparisons or calculations.

Turning numbers into their decimal form is beneficial because it streamlines comparison and arithmetic operations.

Inequalities

Inequalities are mathematical expressions used to show that one quantity is larger or smaller than another. They are represented using symbols like \(<, >, \) and \(=\). Understanding how to work with inequalities is crucial, as they help us compare numbers, whether in decimal or other forms.

In the exercise provided, we compared two decimal numbers to determine their relationship. The symbol \(<\) was used because 6.18 is smaller than 6.2. Here's how you can handle them:

Step-by-step approach when dealing with inequalities involves:

• Ensuring both numbers are in the same form (e.g., both as decimals).
• Examining each digit from left to right to see which number is larger or smaller.
• Using the correct inequality symbol to show the relationship.

With practice, working with inequalities becomes intuitive. They provide a means to solve practical problems, like determining if one value is higher, lower, or equal to another.