Need your help. Fractions. x-1/y / y-1/x this is a fraction one of these answers must be correct a. x/y; b.y/x; c.1; d.-1 e.-x/y

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kjcdb8er's profile pic

kjcdb8er | Teacher | (Level 1) Associate Educator

Posted on

When you have a complicated fraction, it helps to separate it into distinct parts. And always remember that dividing is the same as multiplying by the inverse (i.e. A/B = A*(1/B)). So in your problem, notice that you can separate it by the center divide:

x - 1/y / y - 1/x = (x - 1/y) / (y - 1/x) = A/B

A = x - 1/y = (xy -1)/y          (use least common denominator to factor)

B = y - 1/x = (xy - 1)/x

1/B = x/(xy - 1)                 (now to take the inverse, just flip B upside down)

A*(1/B) = (xy - 1)/y * x/(xy - 1)       (the (xy-1)s cancel)

A/B = x/y

popocatepetl's profile pic

popocatepetl | College Teacher | eNotes Newbie

Posted on

x-1/y / y-1/x.  Rewrite so that all denominators are the same to get

xy/y-1/y / yx/x-1/x

Then, combine to get

( xy-1)/y  /  (xy-)/x    multiply top and bottom by the                                            factor xy.   To get

x(xy-1) / y(xy-1)         simplify by canceling (xy-1) to get

x/y             so the answer is   A.

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

The answer depends upon how you group it. A typical grouping or putting bracket is as below:

(x-1/y)/(y-1/x)=[(xy-1)/y]   /    {(xy-1)/x}

=x/y, as (xy-1) gets cancelled. So a is the choice. But I do not agree with this as this is a forced answer as I did my own way of grouping or braketing which is not permissible.

If I choose  to put the brackets like: {x-[1/(y/y)-1]}/x , then the answer is {x-0}/x=1, Now it seems the choice at c is O.K.

But strictly under the rules of operations of maths,  in an expression like:  x-1/y/y-1/x , there is no choice  for us of grouping or braketing arbitrarily .You may say that you have written like:

x-1/y   /   y-1/xy . But space is not relevant operation in maths and does not affect the priority of operations.Mathematics has its own strict rules and priority of operations according to which we have to simplify:

Here, x  is the first term.

- is an operation as well as a connector between the first and the second term.

1/y/y  is the 2nd term which means  (1/y) divided by y and the result is 1/y^2. There are 2 divisions of equal priority and so you should do first the division on the left and then on the result operate the second division.

- is an operation and a connector between the second and third term.

1/x is the 3rd term .

Therefore the correct answer is (x)-(1/y^2)-(1/x) = {(x^2-1)/x}-{1/y^2}. Unfortunately no answer is correct.

Test: Take x=20 and y = 5 in the above expression and feed to your computer  or a  scientic calculator which process strictly according to the priority rules of operations of maths and see the result:

So the value of the expression x-1/y/y-1/x for x=20 and y=10 is 20-1/10/10-1/20 =19.94   by Ms Exel or a scientific Calculator.

20-1/10^2-1/20 =20-0.01-0.05=19.99-0.05=19.94.

But according to the choice given

a) x/y=20/10=2

b)y/x=10/20

c)1 and

d)-1

e)-x/y= -20/10=-2

Thus none of the choices is correct.

 

 

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