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To understand the concept of dividing by a fraction, we need to first grasp what a reciprocal is. A reciprocal of a number is what you multiply that number by to get 1.
For fractions, finding the reciprocal means flipping the numerator and the denominator.
For example, the reciprocal of \(\frac{1}{4}\) is \(\frac{4}{1}\) or simply 4.
This means that instead of dividing by \(\frac{1}{4}\), you can multiply by its reciprocal.
Understanding this basic concept is crucial because it simplifies the process of division into something more familiar: multiplication.

Once we have understood the reciprocal, the next step in our problem involves multiplying integers.
Multiplying integers is straightforward; you multiply the numbers and then determine the sign of the product.
Let's use our example \(-3 \div \frac{1}{4}\). We found the reciprocal of \(\frac{1}{4}\) to be 4. Now, we need to multiply \(-3\) by 4: \(-3 \times 4 = -12\).
When you multiply a negative number by a positive number, the result is negative, which gives us \-12\. This concept is consistent no matter what integers you are working with.
Practicing these steps reinforces your understanding of how to approach similar problems.

The final step is to ensure our answer is in its simplest form. Simplification means reducing the number or expression to its most basic level.
In this context, \(-12\) is already in its simplest form because there are no factors, other than 1, that can divide into both the numerator and denominator.
If our division result had a fraction, like \(\frac{6}{9}\), we would simplify it to \(\frac{2}{3}\) by dividing both the numerator and the denominator by their greatest common divisor.
Simplifying your results makes them more understandable and easier to use in further calculations. By following these steps, you can confidently solve division problems involving fractions and ensure your answers are clear and concise.