Direct Variation
If a person buys some packets of sugar, the total cost is proportional to the number of packets bought.
The cost of 2 packets at Nx per packet isN2x.
The cost of 3 packets at Nx per packet isN3x.
The cost of n packets at Nx per packet isNnx.
The ratio of total cost to number of packets is the same for any number of packets bought.
This is an example of direct variation, or direct proportion. The cost, C, varies directly with the number of packets, n. In the second example, the mass, M, varies directly with the length, L.
The symbol ∝ means ’varies with’ or ‘is proportional to’. The statements in the prevous paragraph are written:
C ∝ n
M ∝ L
M ∝ L really means that the ratio M/L is constant (i.e. stays the same).
Example
1 packet of sugar costs x naira. What will be the cost of 20 packets of sugar?
Cost varies directly with the number of packets bought.
Cost of 1 packet = x naira
Cost of 20 packets = 20 X x naira
= 20x naira
Example
C ∝ n and C = 5 when n = 20. Find the formula connecting C and n.
C ∝ n means C/n = k.
0r C = kn
C = 5 when n = 20
hence 5 = k V 20
k = ¼
Thus, C = ¼n is the formula which connects C and n.
A formula such as C = ¼n is often known as a relationship between the variables C and n.
Exercise
1. 1m of wire has a mass of x g. What is the mass of 25 m of the same wire?
2. A man cycles 15 Km in 1 hour. How many will he cycle in t hours in he keeps up the same rate?
3. Eggs cost N25 each, how many will n eggs cost?
Answers
1. 25x g 2. 15t km 3. N25n
Inverse Variation
Inverse variation is the relationship of two variables such that a variable increases in its value as the other variable decreases and vice-versa i.e the two variables are inversely proportional to each other. In other words, it is defined as the mathematical expression that shows the relationship between two variables whose product is a constant.
The Inverse Variation Formula is,
Y = k/x
Some solved problems on inverse variation are given below:
Examples
Question 1: If y varies inversely with x and when y = 100, x = 25. What is the value of y when x = 10 ?
Solution
Given,
y = 100
x = 25
The inverse variation formula is,
y = kx
100 = k25
k = 100 X 25
k = 2500
Now,
x = 10
k = 2500
y = kx
y = 250010
y = 250
Question
The time taken to reach the church is inversely proportional to the driving speed. If traveled at the speed of 30 miles per hour, it takes you 2 hours to reach the church. How long will it take to reach the church at the speed of 60 miles per hour?
Solution
Given,
y = 2
x = 30
The direct variation formula is,
y = kx
2 = k30
k = 2 X 30
k = 60
Now,
x = 60
k = 60
y = kx
y = 6060
y = 1
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