Evaluate The Following Expression By Multiplying By The Reciprocal:$\[ \frac{6}{11} \div 3 \frac{2}{3} \\]

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Introduction

In mathematics, division can be rewritten as multiplication by the reciprocal. This concept is crucial in simplifying complex expressions and solving equations. In this article, we will evaluate the given expression by multiplying by the reciprocal.

Understanding the Expression

The given expression is 611÷323\frac{6}{11} \div 3 \frac{2}{3}. To evaluate this expression, we need to understand the concept of division as multiplication by the reciprocal. The reciprocal of a number is obtained by flipping its numerator and denominator.

Converting Division to Multiplication

To convert the given expression from division to multiplication, we need to find the reciprocal of the divisor, which is 3233 \frac{2}{3}. To find the reciprocal, we first need to convert the mixed number to an improper fraction.

Converting Mixed Number to Improper Fraction

A mixed number is a combination of a whole number and a fraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

323=(3×3)+23=9+23=1133 \frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}

Finding the Reciprocal

The reciprocal of 113\frac{11}{3} is 311\frac{3}{11}.

Evaluating the Expression

Now that we have the reciprocal of the divisor, we can rewrite the given expression as multiplication.

611÷323=611×311\frac{6}{11} \div 3 \frac{2}{3} = \frac{6}{11} \times \frac{3}{11}

Simplifying the Expression

To simplify the expression, we multiply the numerators and denominators separately.

611×311=6×311×11=18121\frac{6}{11} \times \frac{3}{11} = \frac{6 \times 3}{11 \times 11} = \frac{18}{121}

Conclusion

In this article, we evaluated the given expression by multiplying by the reciprocal. We first converted the mixed number to an improper fraction and then found the reciprocal of the divisor. Finally, we rewrote the expression as multiplication and simplified it to obtain the final result.

Frequently Asked Questions

Q: What is the reciprocal of a number?

A: The reciprocal of a number is obtained by flipping its numerator and denominator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.

Q: What is the final result of the given expression?

A: The final result of the given expression is 18121\frac{18}{121}.

Further Reading

References

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Introduction

In our previous article, we evaluated the expression 611÷323\frac{6}{11} \div 3 \frac{2}{3} by multiplying by the reciprocal. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the reciprocal of a number?

A: The reciprocal of a number is obtained by flipping its numerator and denominator. For example, the reciprocal of 611\frac{6}{11} is 116\frac{11}{6}.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. For example, to convert 3233 \frac{2}{3} to an improper fraction, we multiply 33 by 33 and add 22, which gives us 9+23=113\frac{9 + 2}{3} = \frac{11}{3}.

Q: What is the difference between division and multiplication by the reciprocal?

A: Division and multiplication by the reciprocal are equivalent operations. When we divide two numbers, we can rewrite it as multiplication by the reciprocal of the divisor. For example, 611÷323\frac{6}{11} \div 3 \frac{2}{3} can be rewritten as 611×311\frac{6}{11} \times \frac{3}{11}.

Q: How do I simplify an expression after multiplying by the reciprocal?

A: To simplify an expression after multiplying by the reciprocal, we multiply the numerators and denominators separately. For example, 611×311=6×311×11=18121\frac{6}{11} \times \frac{3}{11} = \frac{6 \times 3}{11 \times 11} = \frac{18}{121}.

Q: What are some common mistakes to avoid when evaluating expressions by multiplying by the reciprocal?

A: Some common mistakes to avoid when evaluating expressions by multiplying by the reciprocal include:

  • Not converting mixed numbers to improper fractions
  • Not finding the reciprocal of the divisor
  • Not multiplying the numerators and denominators separately
  • Not simplifying the expression after multiplying by the reciprocal

Q: How do I apply this concept to real-world problems?

A: This concept can be applied to real-world problems in various fields, such as finance, engineering, and science. For example, in finance, we may need to calculate the interest rate on a loan or investment, which involves dividing or multiplying by the reciprocal of a fraction.

Conclusion

In this article, we answered some frequently asked questions related to evaluating expressions by multiplying by the reciprocal. We hope that this article has provided you with a better understanding of this concept and how to apply it to real-world problems.

Further Reading

References