Simplify: ${ \frac{\frac{3}{7}+\frac{5}{21}}{\frac{2}{9}-\frac{7}{3}} }$Give Your Final Answer As A Reduced Fraction.

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Introduction

In mathematics, simplifying complex fractions is a crucial skill that helps in solving various mathematical problems. A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. Simplifying such fractions involves several steps, including finding a common denominator, adding or subtracting the fractions, and then simplifying the resulting fraction. In this article, we will simplify the given complex fraction 37+52129−73\frac{\frac{3}{7}+\frac{5}{21}}{\frac{2}{9}-\frac{7}{3}} and provide the final answer as a reduced fraction.

Step 1: Simplify the Numerator

To simplify the given complex fraction, we first need to simplify the numerator. The numerator is 37+521\frac{3}{7}+\frac{5}{21}. To add these two fractions, we need to find a common denominator. The least common multiple (LCM) of 7 and 21 is 21. Therefore, we can rewrite the fractions with a common denominator as follows:

37=3×37×3=921\frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}

Now, we can add the two fractions:

921+521=9+521=1421\frac{9}{21} + \frac{5}{21} = \frac{9 + 5}{21} = \frac{14}{21}

Step 2: Simplify the Denominator

Next, we need to simplify the denominator. The denominator is 29−73\frac{2}{9}-\frac{7}{3}. To subtract these two fractions, we need to find a common denominator. The least common multiple (LCM) of 9 and 3 is 9. Therefore, we can rewrite the fractions with a common denominator as follows:

73=7×33×3=219\frac{7}{3} = \frac{7 \times 3}{3 \times 3} = \frac{21}{9}

Now, we can subtract the two fractions:

29−219=2−219=−199\frac{2}{9} - \frac{21}{9} = \frac{2 - 21}{9} = \frac{-19}{9}

Step 3: Simplify the Complex Fraction

Now that we have simplified the numerator and the denominator, we can simplify the complex fraction. We can rewrite the complex fraction as follows:

1421−199\frac{\frac{14}{21}}{\frac{-19}{9}}

To simplify this fraction, we can multiply the numerator by the reciprocal of the denominator:

1421−199=1421×9−19\frac{\frac{14}{21}}{\frac{-19}{9}} = \frac{14}{21} \times \frac{9}{-19}

Step 4: Simplify the Resulting Fraction

Now, we can simplify the resulting fraction. We can multiply the numerators and the denominators:

14×921×−19=126−399\frac{14 \times 9}{21 \times -19} = \frac{126}{-399}

Step 5: Reduce the Fraction

Finally, we can reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 126 and 399 is 21. Therefore, we can reduce the fraction as follows:

126−399=126÷21−399÷21=6−19\frac{126}{-399} = \frac{126 \div 21}{-399 \div 21} = \frac{6}{-19}

Conclusion

In this article, we simplified the complex fraction 37+52129−73\frac{\frac{3}{7}+\frac{5}{21}}{\frac{2}{9}-\frac{7}{3}} and provided the final answer as a reduced fraction. We followed several steps, including finding a common denominator, adding or subtracting the fractions, and then simplifying the resulting fraction. The final answer is 6−19\frac{6}{-19}, which can be written as −619-\frac{6}{19}.

Final Answer

The final answer is: −619\boxed{-\frac{6}{19}}

Introduction

In our previous article, we simplified the complex fraction 37+52129−73\frac{\frac{3}{7}+\frac{5}{21}}{\frac{2}{9}-\frac{7}{3}} and provided the final answer as a reduced fraction. In this article, we will answer some frequently asked questions (FAQs) related to simplifying complex fractions.

Q&A

Q1: What is a complex fraction?

A1: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q2: How do I simplify a complex fraction?

A2: To simplify a complex fraction, you need to follow these steps:

  1. Simplify the numerator by finding a common denominator and adding or subtracting the fractions.
  2. Simplify the denominator by finding a common denominator and adding or subtracting the fractions.
  3. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.
  4. Simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q3: What is the least common multiple (LCM)?

A3: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.

Q4: How do I find the LCM of two numbers?

A4: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that appears in both lists.

Q5: What is the greatest common divisor (GCD)?

A5: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

Q6: How do I find the GCD of two numbers?

A6: To find the GCD of two numbers, you can list the factors of each number and find the largest factor that appears in both lists.

Q7: Can I simplify a complex fraction with a zero denominator?

A7: No, you cannot simplify a complex fraction with a zero denominator. If the denominator is zero, the fraction is undefined.

Q8: Can I simplify a complex fraction with a negative denominator?

A8: Yes, you can simplify a complex fraction with a negative denominator. When simplifying a complex fraction with a negative denominator, you need to multiply the numerator by the reciprocal of the denominator, which will change the sign of the denominator.

Q9: How do I simplify a complex fraction with a variable in the denominator?

A9: To simplify a complex fraction with a variable in the denominator, you need to follow the same steps as simplifying a complex fraction with a constant denominator. However, you may need to use algebraic manipulations to simplify the fraction.

Q10: Can I simplify a complex fraction with a fraction in the numerator and a fraction in the denominator?

A10: Yes, you can simplify a complex fraction with a fraction in the numerator and a fraction in the denominator. You need to follow the same steps as simplifying a complex fraction with a constant numerator and denominator.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying complex fractions. We covered topics such as the definition of a complex fraction, how to simplify a complex fraction, and how to find the least common multiple (LCM) and greatest common divisor (GCD). We also discussed how to simplify complex fractions with a zero denominator, a negative denominator, and a variable in the denominator.

Final Answer

The final answer is: −619\boxed{-\frac{6}{19}}