Choose The Correct Symbol To Compare The Expressions. Do Not Multiply. 2 1 8 ? 2 1 8 × 22 10 2 \frac{1}{8} \quad ? \quad 2 \frac{1}{8} \times \frac{22}{10} 2 8 1 ​ ? 2 8 1 ​ × 10 22 ​ A. □ \square □ B. $\ \textless \ $ C. = = =

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Introduction

When comparing expressions, it's essential to choose the correct symbol to represent the relationship between them. In this article, we'll explore the correct symbol to use when comparing the expressions 2182 \frac{1}{8} and 218×22102 \frac{1}{8} \times \frac{22}{10}. We'll examine the different options and provide a step-by-step guide to help you make an informed decision.

Understanding the Expressions

The first expression, 2182 \frac{1}{8}, represents a mixed number, which is a combination of a whole number and a fraction. The second expression, 218×22102 \frac{1}{8} \times \frac{22}{10}, represents the product of a mixed number and a fraction.

Option A: □\square

The first option is to use the □\square symbol, which represents "not equal to." However, this symbol is not suitable for comparing the expressions in this case. The reason is that the second expression is a product of a mixed number and a fraction, which can be simplified to a single fraction. Therefore, the □\square symbol is not the correct choice.

Option B: $\ \textless \ $

The second option is to use the $\ \textless \ $ symbol, which represents "less than." However, this symbol is also not suitable for comparing the expressions in this case. The reason is that the second expression is a product of a mixed number and a fraction, which can be simplified to a single fraction. Therefore, the $\ \textless \ $ symbol is not the correct choice.

Option C: ==

The third option is to use the == symbol, which represents "equal to." This symbol is the correct choice for comparing the expressions in this case. The reason is that the second expression can be simplified to a single fraction, which is equal to the first expression.

Step-by-Step Guide

To choose the correct symbol, follow these steps:

  1. Simplify the second expression: The second expression is a product of a mixed number and a fraction. To simplify it, multiply the numerator and denominator of the fraction by the whole number.
  2. Convert the mixed number to an improper fraction: The first expression is a mixed number. To convert it to an improper fraction, multiply the whole number by the denominator and add the numerator.
  3. Compare the expressions: Once you have simplified the second expression and converted the first expression to an improper fraction, compare the two expressions.
  4. Choose the correct symbol: Based on the comparison, choose the correct symbol to represent the relationship between the expressions.

Conclusion

In conclusion, the correct symbol to compare the expressions 2182 \frac{1}{8} and 218×22102 \frac{1}{8} \times \frac{22}{10} is the == symbol. This symbol represents "equal to," which is the correct relationship between the two expressions. By following the step-by-step guide, you can choose the correct symbol and make an informed decision.

Example

To illustrate the concept, let's consider an example. Suppose we want to compare the expressions 3143 \frac{1}{4} and 314×25123 \frac{1}{4} \times \frac{25}{12}. To simplify the second expression, multiply the numerator and denominator of the fraction by the whole number:

314×2512=3×254×12=75483 \frac{1}{4} \times \frac{25}{12} = \frac{3 \times 25}{4 \times 12} = \frac{75}{48}

To convert the first expression to an improper fraction, multiply the whole number by the denominator and add the numerator:

314=3×4+14=12+14=1343 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}

Now, compare the two expressions:

134=7548\frac{13}{4} = \frac{75}{48}

Since the two expressions are equal, the correct symbol to use is the == symbol.

Tips and Variations

  • When comparing expressions, always simplify the second expression and convert the first expression to an improper fraction.
  • Use the == symbol to represent "equal to" and the $\ \textless \ $ symbol to represent "less than."
  • Be careful when comparing expressions with different denominators. In such cases, find the least common multiple (LCM) of the denominators and convert both expressions to have the same denominator.

Conclusion

Q: What is the correct symbol to use when comparing the expressions 2182 \frac{1}{8} and 218×22102 \frac{1}{8} \times \frac{22}{10}?

A: The correct symbol to use is the == symbol, which represents "equal to." This is because the second expression can be simplified to a single fraction, which is equal to the first expression.

Q: Why is the □\square symbol not suitable for comparing the expressions in this case?

A: The □\square symbol represents "not equal to," but in this case, the second expression can be simplified to a single fraction, which is equal to the first expression. Therefore, the □\square symbol is not the correct choice.

Q: Why is the $\ \textless \ $ symbol not suitable for comparing the expressions in this case?

A: The $\ \textless \ $ symbol represents "less than," but in this case, the second expression can be simplified to a single fraction, which is equal to the first expression. Therefore, the $\ \textless \ $ symbol is not the correct choice.

Q: How do I simplify the second expression in the example 218×22102 \frac{1}{8} \times \frac{22}{10}?

A: To simplify the second expression, multiply the numerator and denominator of the fraction by the whole number:

218×2210=2×228×10=44802 \frac{1}{8} \times \frac{22}{10} = \frac{2 \times 22}{8 \times 10} = \frac{44}{80}

Q: How do I convert the first expression in the example 2182 \frac{1}{8} to an improper fraction?

A: To convert the first expression to an improper fraction, multiply the whole number by the denominator and add the numerator:

218=2×8+18=16+18=1782 \frac{1}{8} = \frac{2 \times 8 + 1}{8} = \frac{16 + 1}{8} = \frac{17}{8}

Q: What is the least common multiple (LCM) of the denominators in the example 2182 \frac{1}{8} and 218×22102 \frac{1}{8} \times \frac{22}{10}?

A: The least common multiple (LCM) of the denominators is 80.

Q: How do I compare the expressions 2182 \frac{1}{8} and 218×22102 \frac{1}{8} \times \frac{22}{10} using the LCM?

A: To compare the expressions, convert both expressions to have the same denominator, which is 80:

218=178=17×108×10=170802 \frac{1}{8} = \frac{17}{8} = \frac{17 \times 10}{8 \times 10} = \frac{170}{80}

218×2210=44802 \frac{1}{8} \times \frac{22}{10} = \frac{44}{80}

Since the two expressions are equal, the correct symbol to use is the == symbol.

Q: What are some common mistakes to avoid when comparing expressions?

A: Some common mistakes to avoid when comparing expressions include:

  • Not simplifying the second expression
  • Not converting the first expression to an improper fraction
  • Not using the correct symbol to represent the relationship between the expressions
  • Not finding the least common multiple (LCM) of the denominators when comparing expressions with different denominators.

Q: How can I practice choosing the correct symbol to compare expressions?

A: You can practice choosing the correct symbol to compare expressions by working through examples and exercises. Start with simple examples and gradually move on to more complex ones. Make sure to simplify the second expression and convert the first expression to an improper fraction before comparing them.