Page 16 - Mathematics_Form_Two

Rates and variations

k Thus, m and d vary inversely, that is,
The equation y = can be re-arranged to 1 __ k __
x m ∝ , which implies that m = .
d
d
get xy = . k If x and y are the first set of
1
1
inversely related variables, then the second Thus, k = md .
set of values x and y can be computed
2
2
using the relationship, 2 2 . x y For the two pairs of quantities ( m , d )
1
1
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and ( m , d ), it follows that,
x y
x y =
2
2
Mathematics for Secondary Schools Thus, x y = x y or = . Thus, d = m d . 1
11
m d = m d .
1
2
2
1
__
2 __
1
y
x
2 2
1 1
1
2
____
1
1
m
2
2
Example 1�8
Given d = 12 days, m = 10 men and
1
If x varies inversely as y and x = 2 when
m = 15 men, the value of d is:
2
2
y = 3, find the value of y when x = 18.
10 men × 12 days
d = ________________
2
Solution = 8 days 15 men
1 __
k __
The statement x ∝ implies that x = , Therefore, it takes 8 days for 15 men to
y
y
where k is the constant of proportionality. assemble the same machine.
Making k the subject of the equation
gives, Example 1�10
xy = k, which implies that x y = x y .
1 1
2 2
When x = 2, y = 3, and x = 18, the The intensity of light is inversely
1
1
2
value of y is proportional to the square of the distance
2
x y 2 × 3 D from the light source. Calculate the
1 __
____
y = = _____ = percentage change in intensity under the
1 1

x
3
18
2
2
following conditions:
1 _
Therefore, the value of y is when the (a) The distance is halved.
3
value of x is 18. (b) The distance is increased by 30%.
Solution
Example 1�9 I ∝ 1
D 2
If it takes 12 days for 10 men to assemble k
a machine, how long does it take 15 men I = D 2 (i)
with the same ability to assemble the (a) If the distance D is halved, the new
same machine? distance is D = D .
Solution 1 2 k
Let m be the number of men and d be The new intensity I =  D  2
1
the number of days. It is obvious that 15  
men will take less time to assemble the  4k 2 
machine than 10 men. = (ii)
D 2
10
Student's Book Form Two
11/10/2024 20:11:09
MATHEMATIC F2 v5.indd 10
MATHEMATIC F2 v5.indd 10 11/10/2024 20:11:09