Following up on tkhunny, let p and q be non-zero rational numbers.What is the inverses or reciprocals of the following complex fractions?
(3/4)/5, (9/10)/1, and (5/7)/(7/8)
Are these correct? 5/(4/3), 1/(10/9), and (8/7)/(7/5)
The reciprocal of a/b is b/a, it is NOT (1/a)/(1/b). Why not give it another go.
More explicit for your collection, the reciprocal of (a/c)/(b/d) is (b/d)/(a/c) and NOT (d/b)/(c/a). In fact, that last expression is EXACTLY the same as where you started.
What is the inverses or reciprocals of the following complex fractions?
(3/4)/5, (9/10)/1, and (5/7)/(7/8)
Are these correct? 5/(4/3), 1/(10/9), and (8/7)/(7/5)
There are two main ways to think about reciprocals.
First, a reciprocal means flipping a fraction upside-down, simply swapping the numerator and denominator. The reciprocal of 3/4 is 4/3. Note that we just interchange the 3 and the 4, without changing what each of them is. The reciprocal of (3/4)/(5), in exactly the same way, is just (5)/(3/4), not (5)/(4/3), where I have flipped (3/4) itself. Do you see the difference? Only move, don't change.
But why is the reciprocal important? It's because the reciprocal of a fraction is also the multiplicative inverse -- that is, the number you can multiply it by to get 1. For example, (3/4)*(4/3) = 12/12 = 1. The same should be true for the reciprocal of a complex fraction like your examples. If you multiply (3/4)/(5) by your (5)/(4/3), you won't get 1. (Try it and see.) But if you multiply by the actual reciprocal, (5)/(3/4), you will get 1. Whenever you aren't sure whether two numbers are reciprocals, you can do this multiplication to find out.
In fact, if you have learned to simplify complex fractions, you will find that what you called the reciprocal of (3/4)/(5), namely (5)/(4/3), is in fact just another name for the same number; both are equal to 15/4. The only numbers that are equal to their reciprocal are 1 and -1. You actually took the reciprocal of the reciprocal, which took you back to where you started. The same is true of all your examples.
What is the inverses or reciprocals of the following complex fractions?
(3/4)/5, (9/10)/1, and (5/7)/(7/8)
Are these correct? 5/(4/3), 1/(10/9), and (8/7)/(7/5)
I have tried to correct my examples. Are theses now correct? 5/(3/4) 1/(9/10), and (8/7)/(5/7)
The last one is still wrong. I imagine it was just a slip.
But are you sure you aren't supposed to give simplified answers, as is standard for questions about fractions? What was the actual wording of the exercise you are answering?
BobFredStevePete Reciprocal ==> StevePeteBobFred
Notice how Pete and Steve have the same relationship. They did NOT turn into Steve/Pete.
Notice how Fred and Bob have the same relationship. They did NOT turn into Bob/Fred.
I believe that my reciprocal examples are incorrect. They should be 5/(3/4), 1/(9/10), and (7/5)/(5/7). tkhunny provided an excellent example that helped my see how to make the reciprocal of these complex fractions.
I can prove that these are correct by multiplying them as follows:
(3/4)/ 5 * 5/(3/4) = 1
(9/10)/1 * 1/(9/10) = 1
(5/7)/(7/8) * (7/8)/(5/7) = 1
A fraction multiplied by its reciprocal is one (1).
Correct -- except that you had another typo when you wrote (7/5)/(5/7), meaning (7/8)/(5/7) as you wrote later.