Why do this expressions evaluate to different results?

Question

$$\frac{12}{15} = 0.8\\ \frac{15}{12} = 1.25$$

If $15$ divided by $12 = 1.25$ shouldn't $12$ divided by $15$ be the same as $\frac{15}{12}$ and have the same result?

Can someone please explain this step by step thank you.

Answer 1

Are you using division as the operation? If so, then this example may help:

(1) I have 12 cookies, but there are 15 people in the room. I don't have enough cookies, but if I still want to share evenly, how many cookies should each person get?

(2) Now reverse, I have 15 cookies, but there are only 12 people in the room. I have more than enough cookies now. Should these people get more cookies to eat than the people in (1)?

Answer 2

If we think about the division operation it is a kind of subtraction operation.I don't understand what is the question actually.How (12/15) and (15/12) can be same.If we think from a layman way (12/15) is like you have 12 things and you want its 15 equal pieces so every piece is equal to 0.8 but (15/12) means we want to part 15 things in 12 equal pieces than every piece is equal to 1.25.I hope you understand.

Answer 3

$\frac{12}{15}$ has 15 as the divisor and thus is less than 1 and is in fact the same as $\frac{4}{5}=.8$ while the flip of this is $\frac{5}{4}=1.25$ which is the same as $1+\frac{1}{4} = 1 + 0.25$ where 12 is the divisor in this case.

Answer 4

Analogously, shouldn't $12-15=-3$ be the same as $15-12=3$ and have the same result? The answer is an emphatic "NO," of course. The operations of addition and multiplication of real numbers have some nice properties (associativity and commutativity, in particular) that simply don't hold for general operations, and in particular don't hold for subtraction and division. What you're noticing is the failure of commutativity of division of non-zero real numbers. It may be worth noting that the results you obtain are multiplicative inverses of each other (that is, $1.25\cdot0.8=1,$ the multiplicative identity) just as switching the order of a subtraction will obtain an additive inverse ($-3+3=0$, the additive identity).

Answer 5

Suppose they are the same and look for a contradiction:

$$\frac{12}{15}=\frac{15}{12}$$ would imply $$12\cdot 12=15\cdot 15$$ i.e. $$144=225$$ which is clearly false!