How do you solve a fraction with a fraction denominator?

To solve a fraction with a fraction in the denominator (also known as a complex fraction), you typically simplify it by multiplying both the numerator and the denominator of the main fraction by the reciprocal of the fraction in the denominator. This eliminates the fraction in the denominator and simplifies the entire expression.

Here's a breakdown of the process:

1. Identify the Complex Fraction: Recognize the main fraction and the fraction within its denominator. For example: (a/b) / (c/d). Here, (a/b) is the main fraction's numerator and (c/d) is the denominator.

2. Find the Reciprocal of the Denominator Fraction: The reciprocal of a fraction is obtained by swapping its numerator and denominator. So, the reciprocal of (c/d) is (d/c).

3. Multiply Numerator and Denominator by the Reciprocal: Multiply both the main fraction's numerator and its denominator by the reciprocal you found in step 2:

   [(a/b) * (d/c)] / [(c/d) * (d/c)]

4. Simplify: The denominator simplifies to 1 because (c/d) * (d/c) = 1. The numerator becomes (a*d) / (b*c). Therefore, the complex fraction simplifies to (ad)/(bc).

Example:

Let's say you have the complex fraction: (1/2) / (3/4)

1. The fraction in the denominator is 3/4.

2. The reciprocal of 3/4 is 4/3.

3. Multiply the numerator and denominator by 4/3:

   [(1/2) * (4/3)] / [(3/4) * (4/3)]

4. Simplify:

   (4/6) / 1 = 4/6

5. Further simplify: 4/6 = 2/3

In summary, dividing by a fraction is the same as multiplying by its reciprocal. This principle is the core of simplifying complex fractions.