You are here:

# Geometry/Direct and Inverse Variation

Question
QUESTION: Hello sir

Sir my question was re;ated to dirct variation and inverse.

For eg Inverse variation--> 34 workers take 18 days to build a house. If the work is to be competed in 10 days how many workers are needed.

Direct variation--->In a family five members need 2 liters of milk . Then how much 10 members need.

In thses both fractions are formed and value of numerator or denominator are missing. But then also why only we cross multiply in direct variation. Why we do different in inverse variation.

Ie dIRECT = 5 MEMBERS=2LITERS
10 MWMBERS=X

aCCORDING TO DIRECT VARIATION FORMULA WE DO CROSS MULTIPLICATION 5xX =10x2.

iN INVERSE VARIATION--> 34 WORKS = 18 DAYS
x       =  10 DAYS

iN THIS WE WILL DO 34X18 = 10xX

sO IN THIS IAM ASKING THAT IN BOTH VARIATIONS WE WILL FORM FRACTIONS AND VALUE OF NUMERATOR OR DENOMIANTOR IS MISSING. tHEN WHY DO WE FOLLOW DIFFERENT RULES FOR BOTH. wHY DO WE CROSS MULTIPLICATION IN dIRECT AND AND FRONT MULTIPLICATION IN INVERSE.

tHANKYOU

Thank you for providing sample questions.
First and foremost, I encourage the use of common sense.  You should never blindly use a formula, because it might not apply in that context.

In these cases, there is a rate in question: the unit rate (litres per family member, or days per worker).  To answer the question, you must set up the rate on both sides of the equation, then solve for the unknown.  It is far simpler to do this than it is to use any "formula" you may have been shown.  (If you are expected to show your work using a particular equation, I would still recommend you do what I show below, but be sure to include a step where you have the expected equation.)

Q: A family of 5 needs 2 litres of milk.  Then how much does a family of 10 need?
Here, the equality is for litres of milk per family member.

2L / 5 members = x / 10 members
2/5 = x/10

Then solve for x.  If you have difficulty solving for x in an equation such as this, then ask me a separate question and we will address it there.  It does not matter how you solve for x (cross-multiplication, multiplication by inverse, whatever), as long as you do so correctly.

Q: 34 workers take 18 days to build a house.  How many workers would be needed to build a house in 10 days?
Here, the equality is in days per worker.  (You set up the equation incorrectly in your question.)

18 days / 34 workers = 10 days / x workers
18/34 = 10/x

Then solve for x.

Note that in both of these questions, you could have set up the inverse relations (e.g. 5 members / 2L = 10 members / x ) and it still would have worked out.  I just set up the equations in the way that was logical to me.

I hope this has been helpful.  As I mentioned above, if actually solving the equations is troubling you (or cross-multiplication in particular), then ask a separate question for that.  In as far as setting up relations for direct variation and inverse variation, I hope this method works for you.

Regards,
Azeem

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

QUESTION: hello sir

Sir from above situation one question is very confusing me that why only in direct variation we will cross multiply ie we will multiply numerator of first fraction into denominator 2nd fraction and denominator of first fraction into numerator of 2nd fraction. But why do we not follow the same rule of direct variation in inverse variation.

Honestly, I would not even think about it in terms of direct or inverse variation.  I think you are being more confused by the names of the methods than how to actually set it up and solve.

I will refer back to your original examples, as you have set them up:

Ie dIRECT = 5 MEMBERS=2LITERS
10 MWMBERS=X

CORRECT

aCCORDING TO DIRECT VARIATION FORMULA WE DO CROSS MULTIPLICATION 5xX =10x2.
CORRECT

iN INVERSE VARIATION--> 34 WORKS = 18 DAYS
x       =  10 DAYS

CORRECT

iN THIS WE WILL DO 34X18 = 10xX

INCORRECT

Why did you do suddenly change things for the last part?  You should have done the same thing has with the litres of milk, which would give you 34 x 10 = 18 x X, which will get you to the right answer.

Inverse and direct variation have to do with DIVISION, not multiplication.  If you are being taught to set it up as a multiplication, that is needlessly confusing.  It's like starting the problem from halfway through.
If you wish to solve the equations by cross multiplying, that's fine.  The ironic thing is you are mentioning numerators and denominators, but you have not shown me any fractions.

What you should do is write me out the fractions.  Show me the fractions as you think they should be.  Don't cross-multiply, just write the fractions.  If you can do that correctly, then we can take it from there.  If you cannot write the fractions properly, then that is a separate issue, and we will take it from there instead.

I am having trouble seeing where you are getting confused, and I think this is a way for me to pinpoint your difficulty and help you through it.

Please ask a follow-up where you show me the fractions.  I want to see them in the form (numerator1) / (denominator1) = (numerator2) / (denominator2)

Thanks,
Azeem

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

QUESTION: Hello  sir

Sir leave my confusion sir in brief you explain sir what is direct and inverse variation and the formulas of direct = X1 X Y2 = X2 x y1
INVERSE = X1 x y1 = X2 x y2

Sir  my confusion is simple that why we do different method for direct and different method for inverse because direct and inverse variation, is based on concept unitary method and in unitary method for eg for 1 kg 50 rupees for 5 kg how much ?

In this we do 1:50
5:?
1 by 50 = 5 by x

1 x 5
50  x

there fore we do 1 x X = 50 x5

X= 50 x 5
1

x= 250 rupees.

This method is called unitary method . Same is used in direct and inverse variation. So in unitary method there is only one method that i shown in above ex ie cross multiplication then in inverse variation we are not doing that cross mutiplication we are doing multiplication of numerator = myltiplication of denominator. Why

Hi Aishwarya,

"INVERSE = X1 x y1 = X2 x y2"

This equation is extremely confusing.  It only holds true if you define x's and y's in a really illogical way.  Simply put, don't do this.  Don't use this equation.  You'll go wrong more often than you go right.  This is precisely why you set up the example with workers and days incorrectly, in spite of having identified the ratios properly.

"multiplication of numerator = myltiplication of denominator".  This is not a thing.  This should not be done.  If you can show me an example of this being used and leading to the correct answer, then I could take a look, but honestly, this is confusing and should not be done, period.

a/b = c/d , then you can cross multiply to obtain ad = bc.  (a over b equals c over d implies a times d equals b times c).
Set up the fractions, then solve.  Don't let anyone force confusing, incorrect equations on you.

(If you have a solved example of the "inverse variation formula" being used, I can try to make some sense of it.  However, if you are capable of setting up the ratios--which you are--you are capable of getting to the correct answer.)

Regards,
Azeem

Geometry

Volunteer

#### 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.