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Chapter 1: Problem 51

Write a true sentence using either \(< or >\) $$ -\frac{5}{12} \square-\frac{11}{25} $$

Short Answer

Expert verified
-\frac{5}{12} > -\frac{11}{25}

Step by step solution

01

- Compare the Fractions

To determine whether \(-\frac{5}{12}\) is less than or greater than \(-\frac{11}{25}\), we need to compare these fractions. To do this, find a common denominator or directly compare by cross-multiplication.
02

- Cross-Multiply

Cross-multiplying the fractions gives: \(-5 \cdot 25\) and \(-11 \cdot 12\). This results in \(-125\) and \(-132\) respectively.
03

- Analyze the Results of Cross-Multiplication

Since \(-125 > -132\), \(-\frac{5}{12} > -\frac{11}{25}\). Therefore, \(-\frac{5}{12}\) is greater than \(-\frac{11}{25}\).
04

- Write the True Sentence

Since \(-\frac{5}{12}\) is greater than \(-\frac{11}{25}\), the true sentence using the inequality symbol is: \(-\frac{5}{12} > -\frac{11}{25}\).

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Key Concepts

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

Cross-Multiplication
Cross-multiplication is a powerful technique to compare two fractions. To use cross-multiplication, follow these steps: First, take the numerator of the first fraction and multiply it by the denominator of the second fraction. Then, take the numerator of the second fraction and multiply it by the denominator of the first fraction.
So for the fractions \(-\frac{5}{12}\) and \(-\frac{11}{25}\), we perform the following operations:
\(-5 \times 25 = -125\)
\(-11 \times 12 = -132\)
By comparing these products, we can easily see which fraction is greater or smaller. In this case, because \(-125\) is greater than \(-132\), we can conclude that \(-\frac{5}{12} > -\frac{11}{25}\).
Inequality Symbols
Understanding inequality symbols is crucial in math. The two common symbols we use are < and >.
  • The symbol > means 'greater than.' For example, 5 > 3 means 5 is greater than 3.
  • The symbol < means 'less than.' For example, 2 < 4 means 2 is less than 4.

Inequality symbols help compare two quantities. In the exercise, we used > to indicate that one fraction is greater than the other. By properly understanding and using these symbols, students can accurately represent the relationships between numbers or quantities.
Common Denominators
Another method to compare fractions is finding common denominators. This method requires expressing both fractions with the same denominator. This process involves two main steps:
  • Find the least common denominator (LCD) for the given fractions.
  • Convert each fraction to an equivalent fraction that has this common denominator.

For example, to compare \(-\frac{5}{12}\) and \(-\frac{11}{25}\) using common denominators:
The LCD of 12 and 25 is 300.
Convert \(-\frac{5}{12}\) to \(-\frac{125}{300}\) and \(-\frac{11}{25}\) to \(-\frac{132}{300}\).
Now, it鈥檚 easy to compare: \(-\frac{125}{300} > -\frac{132}{300}\)
Therefore, \(-\frac{5}{12} > -\frac{11}{25}\). This method often helps students who find cross-multiplication less intuitive.

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Most popular questions from this chapter

List in order from least to greatest. $$\frac{4}{5}, \frac{4}{3}, \frac{4}{8}, \frac{4}{6}, \frac{4}{9}, \frac{4}{2},-\frac{4}{3}$$

Write an equivalent expression without using grouping symbols. $$ -[5-6 x] $$

Evaluate \(3 x y\) for \(x=2\) and \(y=7\)

Simplify. $$ \frac{-3^{3}-2 \cdot 3^{2}}{8 \div 2^{2}-(6-|2-15|)} $$

Evaluate. $$ 45 \div 3^{2} x(x-1), \text { for } x=3 $$
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