Which Expression Is Equivalent To $\frac{-2 \frac{1}{4}}{-\frac{2}{3}}$?A. $\frac{9}{4}+\frac{3}{2}$ B. $-\frac{9}{4}+\left(-\frac{2}{3}\right$\] C. $-\frac{9}{4}+\frac{2}{3}$ D.

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Introduction

When dealing with fractions, it's essential to understand how to simplify and manipulate them to find equivalent expressions. In this article, we will explore the concept of equivalent expressions and how to apply it to a given problem involving fractions. We will examine the expression −214−23\frac{-2 \frac{1}{4}}{-\frac{2}{3}} and determine which of the provided options is equivalent to it.

Understanding Equivalent Expressions

Equivalent expressions are mathematical expressions that have the same value, but may be written differently. In the context of fractions, equivalent expressions can be obtained by multiplying or dividing both the numerator and denominator by the same non-zero value. This concept is crucial in simplifying complex fractions and making them easier to work with.

Simplifying the Given Expression

To simplify the expression −214−23\frac{-2 \frac{1}{4}}{-\frac{2}{3}}, we need to first convert the mixed number −214-2 \frac{1}{4} to an improper fraction. This can be done by multiplying the whole number part by the denominator and then adding the numerator.

−214=−2×4+14=−8+14=−74-2 \frac{1}{4} = \frac{-2 \times 4 + 1}{4} = \frac{-8 + 1}{4} = \frac{-7}{4}

Now that we have converted the mixed number to an improper fraction, we can rewrite the original expression as:

−74−23\frac{\frac{-7}{4}}{-\frac{2}{3}}

Dividing by a Fraction

When dividing by a fraction, we can multiply by its reciprocal instead. The reciprocal of a fraction is obtained by swapping its numerator and denominator. In this case, the reciprocal of −23-\frac{2}{3} is −32-\frac{3}{2}.

−74−23=−74×−32\frac{\frac{-7}{4}}{-\frac{2}{3}} = \frac{-7}{4} \times -\frac{3}{2}

Multiplying Fractions

To multiply fractions, we simply multiply the numerators together and the denominators together.

−74×−32=−7×−34×2=218\frac{-7}{4} \times -\frac{3}{2} = \frac{-7 \times -3}{4 \times 2} = \frac{21}{8}

Comparing with the Options

Now that we have simplified the expression to 218\frac{21}{8}, we can compare it with the provided options to determine which one is equivalent.

Option A: 94+32\frac{9}{4}+\frac{3}{2}

To compare this option with our simplified expression, we need to find a common denominator for the two fractions. The least common multiple of 4 and 2 is 4, so we can rewrite the second fraction as:

32=3×22×2=64\frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}

Now we can add the two fractions:

94+64=154\frac{9}{4} + \frac{6}{4} = \frac{15}{4}

This is not equivalent to our simplified expression, so we can rule out option A.

Option B: −94+(−23)-\frac{9}{4}+\left(-\frac{2}{3}\right)

To compare this option with our simplified expression, we need to find a common denominator for the two fractions. The least common multiple of 4 and 3 is 12, so we can rewrite the fractions as:

−94=−9×34×3=−2712-\frac{9}{4} = -\frac{9 \times 3}{4 \times 3} = -\frac{27}{12}

−23=−2×43×4=−812-\frac{2}{3} = -\frac{2 \times 4}{3 \times 4} = -\frac{8}{12}

Now we can add the two fractions:

−2712−812=−3512-\frac{27}{12} - \frac{8}{12} = -\frac{35}{12}

This is not equivalent to our simplified expression, so we can rule out option B.

Option C: −94+23-\frac{9}{4}+\frac{2}{3}

To compare this option with our simplified expression, we need to find a common denominator for the two fractions. The least common multiple of 4 and 3 is 12, so we can rewrite the fractions as:

−94=−9×34×3=−2712-\frac{9}{4} = -\frac{9 \times 3}{4 \times 3} = -\frac{27}{12}

23=2×43×4=812\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

Now we can add the two fractions:

−2712+812=−1912-\frac{27}{12} + \frac{8}{12} = -\frac{19}{12}

This is not equivalent to our simplified expression, so we can rule out option C.

Conclusion

After comparing the simplified expression 218\frac{21}{8} with the provided options, we can conclude that none of the options A, B, or C are equivalent to the given expression. However, we can see that option D is not provided in the given options.

Introduction

In the previous article, we explored the concept of equivalent expressions and how to apply it to a given problem involving fractions. We also simplified the expression −214−23\frac{-2 \frac{1}{4}}{-\frac{2}{3}} and compared it with the provided options. In this article, we will address some frequently asked questions (FAQs) about equivalent expressions.

Q: What is an equivalent expression?

A: An equivalent expression is a mathematical expression that has the same value as another expression, but may be written differently. Equivalent expressions can be obtained by multiplying or dividing both the numerator and denominator by the same non-zero value.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to first convert any mixed numbers to improper fractions. Then, you can multiply the numerator and denominator by the reciprocal of the denominator to eliminate the fraction in the denominator.

Q: What is the difference between equivalent expressions and equivalent fractions?

A: Equivalent expressions and equivalent fractions are related concepts. Equivalent fractions are fractions that have the same value, but may be written differently. Equivalent expressions, on the other hand, are mathematical expressions that have the same value, but may be written differently.

Q: Can I use equivalent expressions to solve equations?

A: Yes, you can use equivalent expressions to solve equations. By simplifying an equation using equivalent expressions, you can make it easier to solve.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you need to simplify both expressions and compare their values. If the values are the same, then the expressions are equivalent.

Q: Can I use equivalent expressions to simplify algebraic expressions?

A: Yes, you can use equivalent expressions to simplify algebraic expressions. By simplifying an algebraic expression using equivalent expressions, you can make it easier to work with.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Some common mistakes to avoid when working with equivalent expressions include:

  • Not simplifying the expression enough
  • Not using the correct method to simplify the expression
  • Not checking if the simplified expression is equivalent to the original expression

Q: How do I know if an expression is equivalent to another expression?

A: To determine if an expression is equivalent to another expression, you need to simplify both expressions and compare their values. If the values are the same, then the expressions are equivalent.

Q: Can I use equivalent expressions to solve word problems?

A: Yes, you can use equivalent expressions to solve word problems. By simplifying a word problem using equivalent expressions, you can make it easier to solve.

Conclusion

In this article, we addressed some frequently asked questions (FAQs) about equivalent expressions. We hope that this article has provided you with a better understanding of equivalent expressions and how to apply them to solve problems.

Additional Resources

If you are looking for additional resources to learn more about equivalent expressions, we recommend the following:

  • Khan Academy: Equivalent Expressions
  • Mathway: Equivalent Expressions
  • Wolfram Alpha: Equivalent Expressions

We hope that this article has been helpful in your understanding of equivalent expressions. If you have any further questions or need additional help, please don't hesitate to ask.