Problem 45 Replace each $\circ$ with \(&l... [FREE SOLUTION]

Chapter 5: Problem 45

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

Short Answer

Expert verified

Replace $\circ$ with $>$ for a true sentence.

Step by step solution

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 $-2 + -0.0833 = -2.0833$.

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.

Insert the Correct Symbol

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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

Short Answer

Step by step solution

Convert Mixed Number to Decimal

Compare the Numbers

Insert the Correct Symbol

Key Concepts

Mixed Numbers

Decimal Conversion

Negative Numbers

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