The question states that we are looking for the reciprocal of 8/19.
Recall what a reciprocal is. A reciprocal, when looking at a number, is what is multiplied with it for it to equal 0. It is commonly regarded as just the fraction flipped. (For example, the reciprocal of 1/2 would be 2, or 2/1. 2 * 1/2 = 1)
This goes back to the simplifying, when multiplying two fractions together.
If you were to multiply two fractions, say [tex]\frac{5}{4}[/tex] and [tex] \frac{4}{5}[/tex], you could look at the numerator of one and the denominator of another, and cancel them out. In this case, the numerator of one is 5, and the other is also 5. Cancelling those out makes both one. If you cancel out 4 and 4 too, you get 1/1 * 1/1, or just 1.
We could go on asking why these methods work forever. Why does cancelling out a numerator and a denominator of separate fractions work? (A simple explanation is when the separate fractions are multiplied together, you could divide both sides by the factor which was originally used to cancel out with, which is what was stated above but simpler.)
Since we know the reciprocal is just the flipped form of the fraction, we can simply flip 8/19 to get our answer.
8/19 flipped is 19/8, which is our answer.
