Problem 45 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$-2 \frac{1}{12} \circ-2.09$$

Step-by-Step Solution

Answer

Replace \(\circ\) with \(>\) for a true sentence.

Step 1: Convert Mixed Number to Decimal

The first number is a mixed number, \(-2 \frac{1}{12}\). To convert it to a decimal, we first convert the fraction \(\frac{1}{12}\) into a decimal. Divide 1 by 12 to get approximately 0.0833. So, \(-2 \frac{1}{12}\) becomes \(-2 + -0.0833 = -2.0833\).

Step 2: Compare the Numbers

Now compare the decimal equivalents of both numbers: \(-2.0833\) and \(-2.09\). Since both are negative, the number with the lesser absolute value is greater. Therefore, since \(2.0833 < 2.09\), it follows that \(-2.0833 > -2.09\) because it is closer to zero.

Step 3: Insert the Correct Symbol

Given the comparison in the previous step, replace the \(\circ\) with \(>\). The correct inequality is \(-2 \frac{1}{12} > -2.09\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that combine a whole number and a fraction. They offer an easy way to express quantities that are not complete integers. For example, in the number \(-2 \frac{1}{12}\), \(-2\) is the whole number and \(\frac{1}{12}\) is the fractional part.

When working with mixed numbers, it can be helpful to convert them into improper fractions or decimals to make calculations easier.

• The whole number part and the fractional part are combined by adding the two parts together.
• In calculations like subtraction or comparison, it's often necessary to convert mixed numbers to a single form - either improper fractions or decimals.

In our exercise, we first needed to address the fraction \(\frac{1}{12}\) by converting it into decimal form, which simplifies further operations.

Decimal Conversion

Converting between fractions and decimals is a fundamental skill in handling various mathematical problems. To convert a fraction to a decimal, you divide the numerator by the denominator.

In this exercise, \(\frac{1}{12}\) was converted to a decimal approximately equal to \(0.0833\).

• This conversion is necessary to uniformly compare numbers in one form.
• Converting helps in performing precise calculations and in visualizing number sizes more easily.

For instance, once \(-2 \frac{1}{12}\) was converted to \(-2.0833\), it's easier to directly compare it with another decimal \(-2.09\) without dealing with fractions.

Negative Numbers

Negative numbers can sometimes be tricky, especially when comparing them. Remember, negative numbers represent values less than zero, and the more negative a number is, the smaller its value.

When comparing negative numbers:

• A negative number with a smaller absolute value is actually greater.
• Negative numbers closer to zero are larger.

In our exercise, the decimals \(-2.0833\) and \(-2.09\) appear close, but because \(2.0833 < 2.09\), we find that \(-2.0833 > -2.09\). This result occurs as \(-2.0833\) is closer to zero than \(-2.09\), making it larger on the number line.