# Solving Variation Problems

Updated on August 7, 2018

SOLVING VARIATION PROBLEMS

DIRECT VARIATION

It is a special function which can be expressed as the equation y = kx where k is a constant. The equation y = kx is read "y varies directly as x" or "y is proportional to x. The constant k is called the the constant of variation or constant of proportionality.

Illustration Number One :

The circumference ( C ) of a circle varies directly as the diameter ( d ). The direct variation is written as C = ∏ d . The constant of variation is ∏ .

Illustration Number Two :

A teacher makes \$10 per hour. The total wage of the teacher is directly proportional to the number of hours (h ) worked. The equation of variation is W = 10h. The constant of proportionality is 10.

A direct variation equation can also be written in the form y = kx^n, where n is a positive number. For example the equation y = kx^2 is read " y varies directly as the square of x..

Illustration Number Three :

The area ( A ) of a circle varies directly as the square of a radius ( r ) of the circle.

The direct variation equation is A = ∏ r ^2. The constant of variation is ∏.

Sample Problem Number One :

Given that V varies directly as r and that V = 50 when r = 5 , find the constant of variation and the equation of variation.

First, write the basic direct variation equation: V = kr

Then, replace V and r by the given values : 50 = k * 5

Solve for k : (1/5) 50 = 5k (1/5 ==è k = 10

Write the direct variation equation by substituting the values of k into the basic direct variation equation: V = 10r

Sample Problem Number Two :

The tension ( T ) in a spring varies directly as the distance ( x ) it is stretched.

If T = 20 lbs. when x = 5 inches. Find T when x = 10 inches.

Write the basic direct variation equation : T = kx

Replace T and x by the given value then solve for k :

20 = 5 * k =è (1/5) 20 = 5k ( 1/5) ===è k = 4

Write the direct variation equation by substituting the value of k into the basic direct

Variation equation : T = 4x

To find T when x = 10 inches. Substitute 10 for x in the equation to solve for T.

T = 4 (10) = 40 lbs.

INVERSE VARIATION

It is a function which can be expressed as the equation y = k/x where k is a constant. The equation y = k/x is read " y varies inversely as x" or "y is inversely proportional to x ." In general, an inverse variation equation can be written y = k/x^n where n is a positive number .

Illustration Number Four :

The equation y = k/x^2 is read "y varies inversely as the square of x."

Given that P varies inversely as the square of x and that P = 10 when x = 2,

Find the variation constant and the equation of variation :.

Set the inverse variation equation : P = k/x^2

Substitute the given values to corresponding variables in the equation and solve for k :

10 = k/2^2 ==è ( 4 ) 10 = k/4 (4) =è k = 40

The constant of variation is 40. The inverse variation equation is P = 40/x^2

Sample Problem Number Three :

The length ( L ) of a rectangle with fixed area is inversely proportional to the width.

If L = 10 W = 4, find the length when w = 7.

Write inverse variation equation : L = k/W

Substitute the given values to the equation and solve for k :

10 = k/4 =è k = 40

L = 40/7 or L = 5 and 5/7

JOINT VARIATION

It is a variation wherein a variable varies directly as the product of two or more

other variables. A joint variation can be expressed as the equation Z = kXY, where K is a constant . The equation is read as "Z varies jointly as X and Y.

Illustration Number Five :

The area ( A ) of a triangle varies jointly as the base and the height. The joint variation equation is written as A = ½ bh. The constant of variation is ½.

COMBINED VARIATION

It is a variation wherein two or more types of variation occurs at the same time.

For example in Physics, the volume ( V ) of a gas varies directly as the temperature ( T ) and inversely as the pressure ( P ). The combind variation equation is written as

V = k T/ P

Sample Problem Number Four :

The pressure P of a gas varies directly as the temperature T and inversely as the volume ( V ). When T = 50 degrees and V = 200 in ^3 P = 30lb/in. Find the pressure of a gas when T = 70 degrees and V = 300 in^3.

Write first the basic combined variation equation : P = kT/V

Replace the variables by the given values then solve for k :

30 = k (50)/ 200

30 * 200 = 50*k

(1/50) 6000 = 50 k (1/50)

k = 6000/50 = 120

P = 120 T/V =è P = 120 (70)/300

(1/300) 300 P = 8400 (1/300)

P = 8400/300 = 28 lb/ in.

SOURCE : INTERMEDIATE ALGEBRA

AN APPLIED APPROACH

By Aufmann/

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• ysa

5 years ago

the time of travel of a free falling body varies directly as the square root of the distance it falls

• Don

6 years ago

If you have a square puzzle with 1000 pieces of equal size, how many pieces are in the border?

• alex andrea

6 years ago

tnx.. for the article.................

• orvil bayona

6 years ago

thank you very much,.,this would be a big help for my upcoming exams

• tm...

6 years ago

nice one ngbgay kaalaman.......

• mr.gentleman

6 years ago

i confused w/ combined variation but this is nice tnks u

• lamidi yusuf olatunji

6 years ago

thankr

• kim :)

6 years ago

nice ?

• lloyd

7 years ago

help on this! given that Z varies directly as the square of y, find the increase in Z when y is increased by 40%.

• lj

7 years ago

I wanna know if there are more kinds of variations cause I"m a math lover

• miriammacmac

8 years ago

thank u soooo much......

• may

8 years ago

tnx for the article.......

• kristina_10

9 years ago

its a big help.......

• eloisabeth

9 years ago

thank; for the article that you've been posted.......

it helps the student understand more about in the world....of math

sophies i love u tedy bear

• HS Jarmmethian 08-09

9 years ago

Thanks for your great article ma'am.. Exactly what I'm looking for! I badly needed this... Buti na lang nandito lang ang sagot sa pinareresearch ni Ma'am Nica =)..

• AUTHOR

Maria Cristina Aquino Santander

10 years ago from Manila

Thans MrMarmalade for appreciating this hub.

10 years ago from Sydney

Never did like Algebra. Leave it all up to Son 3.

Anyway like or not this is a good hub.

I sent it on to my mathematical genius

• AUTHOR

Maria Cristina Aquino Santander

10 years ago from Manila

Thanks dsaaaer for taking tim to visit this hub.

• dsasser

10 years ago from US

useful algebra

working