A variation is a relation between a set of values of variables, and under certain conditions, the values changes. In a variation, there are a set of values that are constant i.e. doesn’t change.
Types of Variation
There are four types of variation in mathematics and they are
- Direct Variation
- Inverse Variation
- Joint Variation
- Partial variation
In this type, the set of variables behaves the same way i.e. if one variable increases, the other will increase as well and vice-versa. Therefore, if A varies directly as B, it can be represented as A α B
Mathematically, it would then be expressed as A = KB
K is now the proportionality constant and it won’t change regardless of the condition of A and B.
In this type, the variables vary disproportionately i.e. if one increases, the other decreases and vice-versa. Therefore, if A varies inversely as B, it can be represented as A α 1/B
Mathematically, it would be expressed as A = K/B
K is the proportionality constant.
This describes a situation where one variable depends on two or more other variables and varies directly as each of them. For instance, x varies jointly as a and b. This can be written as
X α ab
X = kab
Partial variation exists when a quantity is partly constant and partly varies with another quantity. This type always leads to a simultaneous equation before you can get the two constant values.
For example, A is partly constant and partly varies as B can be written as
A = C + KB
C and K are the proportionality constant.
Examples on Variation
No 1: If x varies directly as y, and x = 6 when y = 2, what is the equation that describes this direct variation and find x when y = 5
X α y
X = ky
6 = k*2
K = 6/2 = 3
To write the equation,
X = 3y
To find x when y is 5
X = 3*5 = 15
No 2: If x varies inversely as y, and x = 9 when y = 3, write the equation for the variation and find y when x = 12
X α 1/y
X = k/y
K = xy
K = 9*3 = 27
X = 27/y
To find y when x = 12
X = 27/y
12 = 27/y
Y = 27/12 = 9/4
No 3: If a varies jointly as b and c and a = 12, when b = 6 and c = 8, find a when b = 2 and y = 6.
A α bc
A = kbc
12 = k *6*8
K = 12/48 = 1/4
A = 1/4bc
To find a when b = 2 and y = 6
A = ¼ * 2 * 6 = 3
A = 3
No 4: The cost of car service is partly constant and partly varies with the time it takes to do the work. It costs ₦3,500 for a 10 hours service and ₦2,900 for a 4 hours service.
(a)Find the formula connecting cost, ₦C with time T hours.
(b) Hence find the cost of a 14 hours service.
₦C = K1 + TK2
K1 and K2 are the proportionality constant
For the first condition
₦3,500 = K1 + 10K2 ……… (i)
₦2,900 = K1 + 4K2 ……….. (ii)
This is now simultaneous equation
Now, subtract equ (ii) from equ (i)
₦3,500 – ₦2,900 = K1 – K1 +10k2 – 4K2
₦600 = 6K2
K2 = ₦600/6 = 100
Substitute k2 = ₦100 into equ (i)
₦3500 = K1 + 10*100
₦3500 = k1 + ₦1000
K1 = ₦3500 – ₦1000 = 2500
The formula connecting the ₦C and T
₦C = 2500 + 1000T
Hence find the cost of a 14 hours service
₦C = ₦2500 + 1000*14 = ₦2500 + ₦14000 = ₦16500
The cost will be ₦16500