5.0 variations
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CHAPTER : Variations
: Variations5
5
AState the changes in a quantity with respect to the changesin another quantity in a direct variation
B Determine whether a quantity varies directly as
another quantity
C Express a direct variation as an equation in two variables
D Find the value of a variable in a direct variation given
Sufficient information
5.1 Direct Variations
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CHAPTER : Variations
: Variations
5
5
5.2 Inverse Variations
AState the changes in a quantity with respect to thechanges in another quantity in an inverse variation
B Determine whether a quantity varies inversely as anotherquantity
CExpress an inverse variation as an equation in twovariables
D Find the value of a variable in an inverse variation givenSufficient information
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CHAPTER : Variations
: Variations
5
5
5.3 Joint Variations
A Represent a joint variation by using the symbol
B Express a joint variation in the form of an equation
C Find the value of a variable in a joint variation givensufficient information
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CHAPTER
A State the changes in a quantity with respect to the changesin another quantity in a direct variation
In everyday life, there are situations where an increase in the value of aquantity will cause a corresponding increase in the value of another quantity
and vice-versa.
EX
AMPLE
The table below shows the amount Ali paid for different number ofexercise books he bought
Number of exercise books bought 2 4 8 12
Amount paid (RM) 4 8 16 24
State the change in the amount Ali paid when the number of exerciseBooks he bought is(a) doubled (b) halved (c) trebled
5.1 Direct Variations
: Variations
: Variations5
5
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CHAPTER 5
: Variations5
: Variations
AState the changes in a quantity with respect to the changes
in another quantity in a direct variation
SO
LUT
IO
N
Number of exercise books bought 2 4 8 12
Amount paid (RM) 4 8 16 24
From the table, it can be seen that the more exercise books Ali buys, thegreater the amount he has to pay.
qThe amount Ali paid is doubledwhen the number of exercise books hebought is doubled
qThe amount Ali paid is halvedwhen the number of exercise books hebought is halved
qThe amount Ali paid is trebledwhen the number of exercise books hebought is trebled
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CHAPTER 5 : Variations5 : Variations
B Determine whether a quantity varies directly as another quantity
To determine whether y varies directly as x, just check whether y is ax
constant for all given pairs of (x, y).
EXAMPLE
Based on the values of x and y given in the tables below, determinewhether y varies directly as x in each case.
x 2 3 5 6 8
y 10 15 25 30 40
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CHAPTER 5 : Variations5 : Variations
B Determine whether a quantity varies directly as another quantity
To determine whether y varies directly as x, just check whether y is ax
constant for all given pairs of (x, y).
SOLUT
ION x 2 3 5 6 8
y 10 15 25 30 40
yx 55
5 5 5 A constant
Hence, y varies directly as x.
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CHAPTER 5 : Variations5 : Variations
B Determine whether a quantity varies directly as another quantity
To determine whether y varies directly as x, just check whether y is ax
constant for all given pairs of (x, y).
EXAMPLE
Based on the values of x and y given in the tables below, determinewhether y varies directly as x in each case.
x 2 4 6 8 12
y 5 10 15 24 30
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CHAPTER 5 : Variations5 : Variations
B Determine whether a quantity varies directly as another quantity
To determine whether y varies directly as x, just check whether y is ax
constant for all given pairs of (x, y).
SOLUT
ION x 2 4 6 8 12
y 5 10 15 24 30
yx 2.52.5
2.5 3 2.5Not
a constant
Hence, y does not vary directly as x.
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CHAPTER 5 : Variations5 : Variations
C Express a direct variation as an equation in two variables
If y xn, then y = kor y = kxn, where kis the constant of variation.xn
EXAMPLE
(a)Given that y varies directly as x and y = 8 when x = 12, expressy in terms of x.
(b) If p varies directly as r3 and p = 2 when r = , find the equationwhich relates p and r.
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CHAPTER 5 : Variations5 : Variations
C Express a direct variation as an equation in two variables
SOLUT
ION
(a)Given y x.Then, y = kx, where k is the constant of variation
Given y = 8 when x = 12
= k( )
12
8 = k
12k = 2
3
Thus, y = 2x3
(a)Given that y varies directly as x and y = 8 when x = 12, expressy in terms of x.
8
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CHAPTER 5 : Variations5 : Variations
C Express a direct variation as an equation in two variables
SOLUT
ION
(a)Given p r
3
.Then, p = kr3, where k is the constant of variation
Given p = 2 when r =
= k( )3
2 = 1k
8k = 16
Thus, p = 16r3
2
(b) If p varies directly as r3 and p = 2 when r = , find the equationwhich relates p and r.
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CHAPTER 5 : Variations5 : Variations
D Find the value of a variable in a direct variation given sufficient
information
If y varies directly as xn, then y = k or y = kxn, where k is thexn
constant of variation. Hence, given sufficient information, thevalue of x or y can be determined by using y = kxn.
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CHAPTER 5 : Variations5 : Variations
D Find the value of a variable in a direct variation given sufficient
information
EXAMPLE
Given that y varies directly as x and y = 18 when x = 3, find(a) the value of y when x = 5,(b) the value of x when y = 4.
SOLUTION
Given y x y = kx
Given y = 18 when x = 3
18 = k(3)k = 6Hence, y = 6x
(a) When x = 5, y = 6(5) = 30
(b) When y = 4, 4 = 6x
4 = x6
x = 23
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CHAPTER 5 : Variations5 : Variations
D Find the value of a variable in a direct variation given sufficient
information
ANOTHER WAY
y1
x1
y2
x2
= (a) Let x1 = 3, y1 = 18 and x2 = 5
18 = y23 5y2 = 30
(b) Let x1 = 3, y1 = 18 and y2 = 4
18 = 4
3 x2x2 = 2
3
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CHAPTER 5 : Variations5 : Variations
A State the changes in a quantity with respect to the changesin another quantity in an inverse variation
In everyday life, there are situations where an increase in the value of aquantity will cause a corresponding decrease in the value of another quantity
and vice-versa.
E
XAMPLE The volume, V cm
3, of a gas is measured at several pressures, P N/cm2.The table below shows some typical values of V and P.
P (N/cm2) 10 20 30 40
V (cm2) 30 15 10 7.5
State the variation in the value of V if the value of P is(a) doubled (b) halved (c) trebled
5.2 Inverse Variations
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CHAPTER 5 : Variations5 : Variations
A State the changes in a quantity with respect to the changesin another quantity in an inverse variation
SOLU
TIO
N
P (N/cm2) 10 20 30 40
V (cm2) 30 15 10 7.5
5.2 Inverse Variations
qWhen the value of P is doubled, the value of V is halved.
qWhen the value of P is halved, the value of V is doubled.
qWhen the value of P is trebled, the value of V is multiplied by 1/3 or the value of V is three times less.
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CHAPTER 5 : Variations5 : Variations
B Determine whether a quantity varies inversely as another quantity
5.2 Inverse Variations
To determine whether y varies inversely as x, just check whether xy is aConstant for all given pairs of (x,y)
Example Determine whether y varies inversely as x, in each of the followingcases.
x 2 4 5 8
y 4 2 1.6 1
x 2 4 8 10
y 16 8 4 3.5
(a)
(b)
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CHAPTER 5 : Variations5 : Variations
B Determine whether a quantity varies inversely as another quantity
5.2 Inverse Variations
Solution
x 2 4 5 8
y 4 2 1.6 1
(a)
xy 8 8 8 8 A constant
Hence, y varies inversely as x.
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CHAPTER 5 : Variations5 : Variations
B Determine whether a quantity varies inversely as another quantity
5.2 Inverse Variations
Solution
x 2 4 8 10
y 16 8 4 3.5
(b)
xy 32 32 32 35 Not a constant
Hence, y does not vary inversely as x.
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CHAPTER 5 : Variations5 : Variations
C Express an inverse variation as an equation in two variables
5.2 Inverse Variations
If y 1 , then xny = k or y = k , where k is the constant of variation.xn xn
EXA
MPLE
(a)Given that y varies inversely as x and y = 24 when x = 2/3, expressy in terms of x.
(b) If P varies inversely as the square root of V and P =
when V = 9, find the equation which relates P and V.
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CHAPTER 5 : Variations5 : Variations
C Express an inverse variation as an equation in two variables
5.2 Inverse Variations
SO
LUT
ION
(a)Given that y varies inversely as x and y = 24 when x = 2/3, expressy in terms of x.
(a) Given y 1/x.
Then, y = k/x, where k is the constant of variation
Given y = 24 when x = 2/3
= k/( )
2/3
24 x 2/3 = k
k = 16Hence, y = 16
x
24
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CHAPTER 5 : Variations
C Express an inverse variation as an equation in two variables
5.2 Inverse Variations
S
OLUT
ION
(b) If P varies inversely as the square root of V and P = when V = 9, find the equation which relates P and V.
(b) Given P 1V
Then, P = k where k is the constant of variation.V
Given P = 1 when V = 9.
21 = k2 9
1 x 3 = k2
k = 32
Hence, P = 32V
1 = k2 3
5 : Variations
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CHAPTER 5 : Variations
D Find the value of a variable in an inverse variation given sufficientinformation
5.2 Inverse Variations
5 : Variations
If y 1 , then xny = k or y = k , where k is the constant of variation.xn xn
Hence, given sufficient information, the value of x or y can be determiedby using y = k .xn
EXAMPLE
If y varies inversely as x and y = 8 when x = 2, find(a) the value of y when x = 4,(b) the value of x when y = 10.
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CHAPTER 5 : Variations
D Find the value of a variable in an inverse variation given sufficientinformation
5.2 Inverse Variations
5 : Variations
EXAMPLE
If y varies inversely as x and y = 8 when x = 2, find(a) the value of y when x = 4,(b) the value of x when y = 10.
SOLUTION
Given y 1/x y = k/x
Given y = 8 when x = 2
18 = k/2
k = 16Hence, y = 16/x
(a) When x = 4, y = 16/4 = 4
(b) When y = 10, 10 = 16/xx = 16/10x = 1.6
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CHAPTER 5 : Variations
D Find the value of a variable in an inverse variation given sufficientinformation
5.2 Inverse Variations
5 : Variations
ANOTHER WAY
x1y1 = x2y2 (a) Let x1 = 2, y1 = 8 and x2 = 4.2(8) = 4(y2)
y2 = 4
(b) Let x1 = 2, y1 = 8 and y2 = 10.2(8) = x2(10)
x2 = 1.6
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CHAPTER 5 : Variations
A Represent a joint variation by using the symbol
5.3 Joint Variations
5 : Variations
A joint variation combines two or more variations involving three or more variables.
(a) y varies directly as p and z means y varies directly as p x z and this iswritten as y pz.
(b) y varies inversely as p and z is y 1 x 1 and this is written as y 1p z pz
(c) y varies directly as p and and inversely as z is y p x 1 and this isz
written as y pz
(d) y varies directly as p2 and and inversely as z3 is y p x 1 and this isz3
written as y p
z3
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CHAPTER 5 : Variations
B Express a joint variation in the form of an equation
5.3 Joint Variations
5 : Variations
Example Given y varies directly as x and z2, find the equation relatingx, y and z if y = 6 when x = 2 and z = 1
olution Given y xz2.Then, y = kxz2, where k is the constant of variation
Given x = 2, y = 6 and z =
1.6 = k(2)(1)26 = 2k
k = 3
Hence, the equation relating x, y and z is y = 3xz2.
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CHAPTER 5 : Variations
C Find the value of a variable in a joint variation givensufficient information
5.3 Joint Variations
5 : Variations
Example Given A varies inversely as B and the square root of C. If A = 3when B = 2 and C = 4, find the value of C when A = B = 2.
Given A 1.BC
Then, y = k , where k is the constant of variationBC
Given A = 3, B = 2 and C = 4.3 = k 24
k = 12Thus, A = 12
BC
When A = B = 2,2 = 122C
4C= 12
C = 12 = 3
4
C = 32
C = 9
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