Problem 83
Question
Complete the statement using \(<,>,\) or \(=.\) (Skills Review pp. \(763,770,771\) ) $$\frac{100}{9} ? 11 \frac{2}{9}$$
Step-by-Step Solution
Verified Answer
The correct answer is \(\frac{100}{9} < 11 \frac{2}{9}\).
1Step 1: Convert Mixed Numbers to Improper Fractions
The fraction \(100/9\) is already an improper fraction. To convert the number \(11 2/9\) to an improper fraction, multiply the whole part by the denominator of the fractional part and add the result to the numerator. Thus we have \(11*9+2=99+2=101\). So, \(11 2/9\) as an improper fraction is \(101/9\).
2Step 2: Compare the Fractions
Now we compare \(100/9\) and \(101/9\). When denominators are equal, the fraction with the larger numerator is greater. Therefore, \(101/9\) > \(100/9\). So the inequality sign that makes the statement true is \(>\).
3Step 3: Write the Final Answer
The final statement is \(\frac{100}{9} < 11 \frac{2}{9}\). This satisfies the inequality \(101/9>100/9\).
Key Concepts
Understanding Improper FractionsExplaining Mixed NumbersFundamentals of Fraction Comparison
Understanding Improper Fractions
An improper fraction is a type of fraction where the numerator, which is the top number, is larger than or equal to the denominator, the bottom number. This means that the fraction actually represents more than one whole unit.
For example, in the fraction \( \frac{100}{9} \), the numerator \(100\) is larger than the denominator \(9\), which qualifies it as an improper fraction.
Improper fractions are useful because they can make certain mathematical operations, like addition and subtraction, easier to manage, especially when dealing with mixed numbers or other improper fractions.
For example, in the fraction \( \frac{100}{9} \), the numerator \(100\) is larger than the denominator \(9\), which qualifies it as an improper fraction.
Improper fractions are useful because they can make certain mathematical operations, like addition and subtraction, easier to manage, especially when dealing with mixed numbers or other improper fractions.
- To identify an improper fraction, simply check if the numerator is larger than the denominator.
- Improper fractions can be converted to mixed numbers, providing a practical way to understand how many whole units are represented.
- They often arise naturally in situations involving division, where the result is not a whole number.
Explaining Mixed Numbers
A mixed number is a whole number combined with a fraction. This form is useful for expressing values that are greater than one but are not whole numbers, like \(11 \frac{2}{9}\).
To work with mixed numbers in equations, it's often necessary to convert them to improper fractions. This makes calculations straightforward.
For instance, to convert \(11 \frac{2}{9}\) into an improper fraction, multiply the whole number \(11\) by the denominator \(9\) and add the numerator \(2\). This provides the improper fraction \(\frac{101}{9}\).
To work with mixed numbers in equations, it's often necessary to convert them to improper fractions. This makes calculations straightforward.
For instance, to convert \(11 \frac{2}{9}\) into an improper fraction, multiply the whole number \(11\) by the denominator \(9\) and add the numerator \(2\). This provides the improper fraction \(\frac{101}{9}\).
- Mixed numbers make it easier to understand and visualize quantities in real-world terms.
- They're handy when needing to express or talk about quantities in a more relatable manner than improper fractions.
- To convert them to improper fractions, always multiply the whole number by the denominator, then add the numerator.
Fundamentals of Fraction Comparison
Comparing fractions is an essential skill in mathematics. Whether fractions are proper or improper, the process remains consistent. The key to compare fractions efficiently is to consider their numerators and denominators.
When fractions have the same denominator, like \(\frac{100}{9}\) and \(\frac{101}{9}\), it's simple to compare them. Just look at the numerators. The fraction with the larger numerator is greater.
In this exercise, comparing \(\frac{100}{9}\) with \(\frac{101}{9}\) shows that \(\frac{101}{9}\) is larger because \(101\) is more than \(100\).
When fractions have the same denominator, like \(\frac{100}{9}\) and \(\frac{101}{9}\), it's simple to compare them. Just look at the numerators. The fraction with the larger numerator is greater.
In this exercise, comparing \(\frac{100}{9}\) with \(\frac{101}{9}\) shows that \(\frac{101}{9}\) is larger because \(101\) is more than \(100\).
- If fractions have the same denominator, compare the numerators directly.
- If denominators differ, consider finding a common denominator or convert them to decimals.
- Practicing fraction comparison helps in solving inequalities and understanding relative sizes in problems.
Other exercises in this chapter
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