Final answer:
A fraction over a fraction, also known as a complex fraction, simplifies by multiplying the numerators and denominators of both fractions respectively. The division by a fraction is equivalent to multiplication by its reciprocal. Simplification of the result may involve canceling out common factors.
Step-by-step explanation:
Understanding Fractions over Fractions
To represent a fraction over a fraction, you are dealing with a complex fraction. A complex fraction is simply one fraction divided by another. To understand this, it's essential to know that a fraction like a/b, where a is the numerator and b is the denominator, signifies the division of a by b.
Now, take for example 1/2 over 3/4. This can also be seen as (1/2) / (3/4), which means you are dividing half by three quarters. To simplify, you multiply by the reciprocal of the denominator fraction. This turns division into multiplication, obeying the rule that dividing by a number is the same as multiplying by its reciprocal.
The general rule for multiplying fractions is to multiply the numerators together and multiply the denominators together. For addition and subtraction, we often need a common denominator, which we can find by the intuitive understanding that denominators can interact through multiplication to find a shared base. Units in complex fractions can cancel each other out, leading to a simpler form of the result.
When multiplicative relationships are understood, simplifying complex fractions becomes straightforward. For instance, if you have 1/6 of 1/5, you would multiply the numerators (1*1) and the denominators (6*5), which initially gives you 1/30. If there were any common factors in the numerator and denominator, they could be canceled out to simplify the fraction further.
In conclusion, a fraction over a fraction simplifies by multiplying the top fraction's numerator by the bottom fraction's numerator, and the top fraction's denominator by the bottom fraction's denominator, simplifying the result as necessary.