<< Chapter < Page Chapter >> Page >
In this section, you will:
  • Solve direct variation problems.
  • Solve inverse variation problems.
  • Solve problems involving joint variation.

A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.

Solving direct variation problems

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formula e = 0.16 s tells us her earnings, e , come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See [link] .

s , sales price e = 0.16 s Interpretation
$4,600 e = 0.16 ( 4,600 ) = 736 A sale of a $4,600 vehicle results in $736 earnings.
$9,200 e = 0.16 ( 9,200 ) = 1,472 A sale of a $9,200 vehicle results in $1472 earnings.
$18,400 e = 0.16 ( 18,400 ) = 2,944 A sale of a $18,400 vehicle results in $2944 earnings.

Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation . Each variable in this type of relationship varies directly with the other.

[link] represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formula y = k x n is used for direct variation. The value k is a nonzero constant greater than zero and is called the constant of variation . In this case, k = 0.16 and n = 1. We saw functions like this one when we discussed power functions.

Graph of y=(0.16)x where the horizontal axis is labeled, “s, Sales Price in Dollars”, and the vertical axis is labeled, “e, Earnings, $”.

Direct variation

If x and y are related by an equation of the form

y = k x n

then we say that the relationship is direct variation    and y varies directly    with, or is proportional to, the n th power of x . In direct variation relationships, there is a nonzero constant ratio k = y x n , where k is called the constant of variation    , which help defines the relationship between the variables.

Given a description of a direct variation problem, solve for an unknown.

  1. Identify the input, x , and the output, y .
  2. Determine the constant of variation. You may need to divide y by the specified power of x to determine the constant of variation.
  3. Use the constant of variation to write an equation for the relationship.
  4. Substitute known values into the equation to find the unknown.

Solving a direct variation problem

The quantity y varies directly with the cube of x . If y = 25 when x = 2 , find y when x is 6.

The general formula for direct variation with a cube is y = k x 3 . The constant can be found by dividing y by the cube of x .

k = y x 3 = 25 2 3 = 25 8

Now use the constant to write an equation that represents this relationship.

y = 25 8 x 3

Substitute x = 6 and solve for y .

y = 25 8 ( 6 ) 3 = 675
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask