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Geometry/Direct and Inverse Variation, Cross Multiplication

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QUESTION: hello sir


i from your answer i not cleared and iam sorry to irritate behind that concept .It is too confusing to me.

Can you tell how that formulas are related to direct variation and inverse variation.Means the formula of direct x1xy2 =x2 x y1.
How it is related to if value of x increase , value of y increases in such a manner that the ratio x / y not change

How formula of inverse x1 y1 = x2 y2 is related to the defination of inverse variation ie if an increase in x causes  a decrerase in y in such a manner that the product of their corresponding values remains contant.


In this 2nd confusion related to formula is --> as we know when there is unknow in two fractions we cross multiply them but here  in terms of direct variation and inverse variation  both variations have one unknown in two fractions but here only in direct variation we are using cross multiplication method we are not using this method in inverse varaition. why

Sir youre answer is if we should not use cross multiplication in inverse variation can you tell me why and can you explain then how we should know where to cross multiply.


Sir i know that i week is over but my concept is not cleared. iam not able to understand so that s why i had explained all my doubts in this please if you not understand please read it trice or twice. please.

thanks

ANSWER: Hi Aishwarya,

Let us be clear on definitions first.
CROSS MULTIPLICATION: The process of going from a/b=c/d to ad=bc.

That's it.  That is all cross multiplication is.  The only connection between a & d, and b & c is where they are in the fractions.  In general, they have no other relation to one another.

In DIRECT VARIATION, the values of x and y increase such that the ratio x/y does not change.  There is a common ratio, let's say k.
x1/y1=k
x2/y2=k

This means
x1/y1=x2/y2.

Cross multiplying,
x1y2=x2y1.

In INVERSE VARIATION, the product of x and y does not change.  There is a common product, let's say p.
x1y1=p
x2y2=p

This means
x1y1=x2y2.

Note there is NO CROSS MULTIPLICATION in this case.  There are not even any fractions to consider.  (The positions of the 1s and 2s have NOTHING to do with cross multiplication.)

I strongly encourage the use of common sense, and not the blind reliance on formulas.  If a formula does not make sense to you, it is perfectly okay to ask, but you should not let the formula be the basis of your understanding of the concept.

As a side note, from now on do not use (x) as the multiplication sign.  Use (*) instead or use parentheses.  This is especially true when x is a variable in your equation.

Thanks,
Azeem

---------- FOLLOW-UP ----------

QUESTION: Hello sir

Sir on elast confusion behind that concept In DIRECT VARIATION, the values of x and y increase such that the ratio x/y does not change.  There is a common ratio, let's say k.
x1/y1=k
x2/y2=k

This means
x1/y1=x2/y2.

Cross multiplying,
x1y2=x2y1.

In this why we should cross multiply if their ratios are constant  and in inverse why we do common product.

Sir in this formula AND THEIR RELATION TO CONSTANT OF RATIO AND COMMON PRODUCT. iN INVERSE IF COMMON PRODUCCT IS CONSTANT THAN IN INVERSE ONE VALUE INCREASES AND OTHER ALSO DECREASES. iF IT IS INCREASES OR DECREASES THEN HOW COMMON PRODUCT IS CONSTANT. AND IN DIRECT VARIATION IF VALUE OF X INCREASE , VALUE OF Y ALSO INCREASE. iF IT INCREASE AND ICREEASES THEN HOW RATIO IS CONSTANT.

sIR I KNOW IT TO IRRITATIVE BUT PLEASE ANSWER IT .iF THIS CONCEPT IS CLEARED I CAN SOLVE QUESTIONS IN EXAM CONFIDENTLY IF NOT HOW MUCH I PRACTICE THE SUMS ALL GET CONFUSED.

tHANKYOU

Answer
Hi,

In inverse variation, the common product is constant.  The values of x and y are not.  As one increases, they other decreases so they "balance each other out."  xy=p, for any pair of x and y that fit the scenario.

Similarly, in direct variation, the common ratio is constant.  The values of x and y are not.  As one increases, the other increases as well.  x/y=p, for any pair of x and y that fit the scenario.

I do not find your questions irritating, except for your misuse of caps lock and lack of clarity (e.g. long sentences that are difficult to follow).  Going forward, please be careful with these.

Thanks,
Azeem

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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