Variation is subject in mathematics that helps to evaluate or establish the relationship between two related quantities. Through variation, you are able to find out how the change in value of one quantity affects the change in value of the other. Through variation, we can establish the extent of deviation or change of the values of two related quantities with respect to each other.
There are many forms of variation. For the scope of this presentation, we will discuss the following important 4 types of variation: Direct Variation, Indirect Variation, Joint Variation, and Partial Variation.
I will focus fully on the first type of variation listed above - Direct variation. We will also see how to solve some examples of direct variation.
Direct variation describes the relationship between two quntities such that the increase in the values of one quantity results in increase of the values of the other quantity. In direct variation, the rate of increase between the values of quantity 1 and quantity 2 is constant. As such, the second quantity increases or decreases in value at exactly the same rate of increase or decrease of the first quantity.
Direct variation has 3 elements. We will represent all with letters as shown below:
First Quantity = a
Second Quantity = b
Constant = q
Symbol for direct variation = ∝
Hence, representing the direct variation of the above quantities mathematically, we have:
a ∝ b (a varies directly as b).
Now we need to introduce the constant q and turn completely make it a mathematical equation. Remember, the constant represented by q describes the rate of proportionality of a to b.
Therefore introducing the constant, we have:
a ∝ b
a = qb
This is the final equation that will be used to solve for the real values of a and b in the equation.
To solve direct variation is not difficult. You just need to understand how to approach the problem. The first thing to find is the value of the constant. To do this, you will substitute initial values of the two quantities given in the equation, thereby making the constant subject of the formula.
After finding the constant, then use the equation to solve for the values of the two quantities whose other values are also given. We will demonstrate this with real examples now.
Example 1: If X varies directly as Y and X=8 when Y = 2. Find Y when X = 40.
Solution:
We will use the initial values of the two quantities X and Y to find the constant. Then solve for Y with the final value of X.
Initial values:
X = 8,
Y = 40
Let Z = constant.
Representing it mathematically and solving for the constant Z, we have:
X ∝ Y.
X = ZY (introducing the constant Z)
8 = Zx2
Making Z the subject, we have:
Z= 8/2
Z =4
We were asked to find Y when X=40
X=ZY
40=4xY
Making Y the subject, we have:
Y = 40/4
Y = 10
Therefore, When X=40, Y=10
Example 2: a varies directly as b3 and a = 54 when b = 3. Find a when b = 5, and b when a = 128.
Solution:
Initial values:
a = 54,
b = 3
Let c = constant.
Solving for the constant, we have
a ∝ b
a= cb3 (introducing the constant c)
54 = c x 33
54 = 27c
Making c the subject, we have
c = 54/27
c = 2
Next, we need to find a when b = 5
Remember, a= cb3
a =2 x 53
a = 2 x 125
a = 250
Finding b when a = 128
Remember, a= cb3
128 = 2 x b3
128 =2b3
Making b the subject, we have:
b3 = 128/2
b3 = 64
b = (∛64) (We introduced the cube root sign to cancel the cube sign in b)
b = 4
Therefore, when a = 128, b=4
We have seen that direction variation deal with two quantities whose values increase of decrease simultaneously. As the first quantity increases in value, the second one increases too. Decrease in the first quantity also leads to decrease in the second.
In the next presentation, we will see other forms of variation and how to solve them.
