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To divide one fraction by another, we use the concept of the reciprocal. A reciprocal of a fraction is simply the fraction flipped upside down. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). This switching of the numerator and the denominator is essential when dealing with fraction division.

The logic behind it is simple. When we divide by a fraction, what we are essentially doing is multiplying by its reciprocal. This transforms our problem from division to multiplication, making it easier to solve.

So, when given a division problem like \(\frac{3}{4} \div \frac{2}{3}\), the first step is to rewrite it in terms of multiplication by using the reciprocal of the divisor. Thus, it becomes \(\frac{3}{4} \times \frac{3}{2}\).

Once you have rewritten the division problem as a multiplication problem, the next step is to multiply the fractions. This involves two simple steps:

• Multiply the numerators (the top parts) of the two fractions.
• Multiply the denominators (the bottom parts) of the two fractions.

For our example \(\frac{3}{4} \times \frac{3}{2}\), multiplying the numerators gives us:

\(3 \times 3 = 9\)

And multiplying the denominators gives us:

\(4 \times 2 = 8\).

So, after performing the multiplication, we obtain the new fraction \(\frac{9}{8}\).

The final step is to simplify our resulting fraction if necessary. Simplification involves reducing the fraction to its simplest form by removing any common factors from the numerator and the denominator. For our fraction \(\frac{9}{8}\), both numbers share no common factors other than 1, which means \(\frac{9}{8}\) is already in its simplest form.

However, it's also helpful to recognize that \(\frac{9}{8}\) is an improper fraction (where the numerator is larger than the denominator). It can be converted into a mixed number for easier interpretation. In this case, \(\frac{9}{8}\) can be converted to \(1 \frac{1}{8}\), which tells us we have one whole and an additional eighth.

Understanding these steps ensures you can handle fraction division easily and accurately, leading to a deep comprehension of the underlying mathematical processes.