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)
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.
"Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.
Following up on tkhunny, let p and q be non-zero rational numbers.
What is the reciprocal of [tex]\dfrac{p}{q}?[/tex]
If and only if y is the reciprocal of x, then and only then x * y = 1. That's the definition.
And obviously [tex]\dfrac{p}{q} * \dfrac{q}{p} = 1.[/tex]
Every rational number can be expressed as the ratio of two integers. Again that is a definition. So,
[tex]p = \dfrac{a}{b} \text { and } q = \dfrac{c}{d} \implies [/tex]
[tex]\text {The reciprocal of } \dfrac{p}{q} = \dfrac{\dfrac{a}{b}}{\dfrac{c}{d}} = \dfrac{ad}{bc} \text { is } \dfrac{q}{p} = \dfrac{\dfrac{c}{d}}{\dfrac{a}{b}} = \dfrac{bc}{ad}.[/tex]
Last edited by JeffM; 12-22-2017 at 02:28 PM.
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.
[tex]\dfrac{\dfrac{Pete}{Steve}}{\dfrac{Fred}{Bob}}[/tex] Reciprocal ==> [tex]\dfrac{\dfrac{Fred}{Bob}}{\dfrac{Pete}{Steve}}[/tex]
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.
Last edited by tkhunny; 12-22-2017 at 11:40 PM.
"Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.
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).
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