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Understanding the reciprocal of a fraction is essential for solving division problems involving fractions. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
For a fraction \(\frac{a}{b}\), its reciprocal will be \(\frac{b}{a}\).
This simple flip of numbers allows us to turn a division problem into a multiplication problem, which is often easier to solve.
For example: the reciprocal of \(\frac{8}{15}\) is \(\frac{15}{8}\).
When dividing fractions, you multiply by the reciprocal of the divisor.
So, \(\frac{6}{125} \div \ \frac{8}{15}\becomes\ \frac{6}{125} \times\ \frac{15}{8}\).
Notice how it changes from division to multiplication after the reciprocal is found.

Once you've multiplied the fractions, it's important to simplify the result.
To simplify a fraction means to reduce it to its lowest terms.
This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For instance, when you multiply \(\frac{6}{125}\) and \(\frac{15}{8}\), you get \(\frac{90}{1000}\).
This fraction can be simplified by dividing both the numerator and the denominator by their GCD.
But first, let's explain how to find the GCD.

The greatest common divisor (GCD) is the largest number that evenly divides two given numbers.
Finding the GCD helps in simplifying fractions.
To determine the GCD of 90 and 1000, you can list the factors of each number:

• The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
• The factors of 1000 are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000.

You can see that the largest common factor is 10.
Therefore, the GCD of 90 and 1000 is 10.
To simplify the fraction \(\frac{90}{1000}\), divide both the numerator and the denominator by 10:
\(\frac{90 \div \ 10}{1000 \div \ 10}\backslash\ = \ \frac{9}{100}\).
Now the fraction is in its simplest form.