Page 13 - Mathematics_Form_Two

Rates and variations

Example 1�5 Cross multiplication gives

If y varies directly as x and x = 15 when 4 × x = 4 × 64
2
_

y = 4, find the value of y when x = 12. x = 4 × 64
4
2
Solution = 64
Given that y ∝ x . This implies that Therefore, the value of x is 64 when
FOR ONLINE READING ONLY
y = kx, where k is the constant of y is 8.
proportionality.
Making k the subject of the equation Example 1�7
gives
y
_
k = A car travels 60 kilometres using 5 litres Mathematics for Secondary Schools
x
of diesel. How many litres of diesel are
But, for any two pairs of quantities, needed to travel 150 kilometres?
y
y
2 __
1 __
k = and k = . Solution
x
x
1 y y 2
1 _
2 _
Thus, = . Let x denote the number of litres of diesel
x
x
1 2 and y denote the number of kilometres.
Given x = 15, x = 12, and y = 4, The equation for the variation becomes
2
1
1
the value of y is obtained as follows: x y
2
1 __
1 __
y x = .
____
1 2
y = x y
x
2
2
2
1 Given x = 5 litres, y = 60 km , and
Substituting the given values gives
1
1
16
______
_
y = 4 × 12 = y = 150 km, the value of x is given by:

2
2
x y
2 15 5 ____ 2
1
16
_
Therefore, the value of y is when x = y
2
5
1
x = 12 . = 5 litres × 150 km
______________
60 km
Example 1�6 = 12 . 5 litres
If x varies directly as the square of y and Therefore, 12.5 litres of diesel are
x = 4 when y = 2, find the value of x needed to travel 150 kilometres.
when y = 8.
Solution Exercise 1�3
Let x = 4, y = 2 and y = 8. The 1. If x varies directly as y and x = 16
1
1
2
variation equation is given by; when y = 10, find the value of y when
x y
2
1 __
= _____ 2 x = 20.
1
x 2
y

2 2. The surface area of a circular object
Substituting the values of x , y , and y varies directly as the square of its
1
2
1
gives radius. If its surface area is 78.5
2
2
2
4 __
__
= cm , when the radius is 5 cm, find
x
2
2 8 the surface area of the circular object
4 __
4 ___
= when the radius is 7 cm.
x
2 64
7
Student's Book Form Two
11/10/2024 20:11:08
MATHEMATIC F2 v5.indd 7
MATHEMATIC F2 v5.indd 7 11/10/2024 20:11:08