ICAI Notes- Equations | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


EQUATIONS
2
CHAPTER
After studying this chapter, you will be able to:
? Understand the concept of equations and its various degrees – linear, simultaneous,
quadratic and cubic equations;
? Know how to solve the different equations using different methods of solution; and
CHAPTER OVERVIEW
Definition of Equation
Simple
Equation
Applications Equations
Simultaneous
equations in
two unknowns
Methods of Solving
Three Linear Equations
with three Variables
Quadratic
Equation
Roots of the
Quadratic Equation
Constructions of
Quadratic Equation
Nature of
the Roots
Methods of Solution
Cross Multiplication
Method
Elimination
Method
© The Institute of Chartered Accountants of India
Page 2


EQUATIONS
2
CHAPTER
After studying this chapter, you will be able to:
? Understand the concept of equations and its various degrees – linear, simultaneous,
quadratic and cubic equations;
? Know how to solve the different equations using different methods of solution; and
CHAPTER OVERVIEW
Definition of Equation
Simple
Equation
Applications Equations
Simultaneous
equations in
two unknowns
Methods of Solving
Three Linear Equations
with three Variables
Quadratic
Equation
Roots of the
Quadratic Equation
Constructions of
Quadratic Equation
Nature of
the Roots
Methods of Solution
Cross Multiplication
Method
Elimination
Method
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
2.2
Equation is defined to be a mathematical statement of equality. If the equality is true for
certain value of the variable involved, the equation is often called a conditional equation
and equality sign ‘=’ is used; while if the equality is true for all values of the variable
involved, the equation is called an identity.
For Example: 
+ 2 + 3
+ = 3
32
xx
holds true only for x =1.
So it is a conditional. On the other hand, 
+2 +3 5 +13
+ =
32 6
xx x
is an identity since it holds for all values of the variable x.
Determination of value of the variable which satisfy an equation is called solution of the
equation or root of the equation. An equation in which highest power of the variable is 1
is called a Linear (or a simple) equation. This is also called the equation of degree 1. Two or
more linear equations involving two or more variables are called Simultaneous Linear
Equations. An equation of degree 2 (Highest Power of the variable is 2) is called Quadratic
equation and the equation of degree 3 is called Cubic Equation.
For Example: 8x+17(x–3) = 4 (4x–9) + 12 is a Linear equation.
3x
2
 + 5x +6 = 0 is a Quadratic equation.
4x
3
 + 3 x
2
 + x–7 = 1 is a Cubic equation.
x
 
+ 2y = 1,  2x + 3y = 2 are jointly called Simultaneous equations.
A simple equation in one unknown x is in the form ax + b = 0.
Where a, b are known constants and a  0
Note: A simple equation has only one root.
Example: 
4
3
x
-1 =
14
15
x +
19
5
.
Solution: By transposing the variables in one side and the constants in other side we have
4
3
x
 – 
14
15
x
= 
5
19
+1 or 
(20-14) 19 5
15 5
?
?
x
  or 
6 24
15 5
?
x
.
 ?
24x15
 = 12
5x6
x
Choose the most appropriate option (a) (b) (c) or (d).
# 
© The Institute of Chartered Accountants of India
Page 3


EQUATIONS
2
CHAPTER
After studying this chapter, you will be able to:
? Understand the concept of equations and its various degrees – linear, simultaneous,
quadratic and cubic equations;
? Know how to solve the different equations using different methods of solution; and
CHAPTER OVERVIEW
Definition of Equation
Simple
Equation
Applications Equations
Simultaneous
equations in
two unknowns
Methods of Solving
Three Linear Equations
with three Variables
Quadratic
Equation
Roots of the
Quadratic Equation
Constructions of
Quadratic Equation
Nature of
the Roots
Methods of Solution
Cross Multiplication
Method
Elimination
Method
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
2.2
Equation is defined to be a mathematical statement of equality. If the equality is true for
certain value of the variable involved, the equation is often called a conditional equation
and equality sign ‘=’ is used; while if the equality is true for all values of the variable
involved, the equation is called an identity.
For Example: 
+ 2 + 3
+ = 3
32
xx
holds true only for x =1.
So it is a conditional. On the other hand, 
+2 +3 5 +13
+ =
32 6
xx x
is an identity since it holds for all values of the variable x.
Determination of value of the variable which satisfy an equation is called solution of the
equation or root of the equation. An equation in which highest power of the variable is 1
is called a Linear (or a simple) equation. This is also called the equation of degree 1. Two or
more linear equations involving two or more variables are called Simultaneous Linear
Equations. An equation of degree 2 (Highest Power of the variable is 2) is called Quadratic
equation and the equation of degree 3 is called Cubic Equation.
For Example: 8x+17(x–3) = 4 (4x–9) + 12 is a Linear equation.
3x
2
 + 5x +6 = 0 is a Quadratic equation.
4x
3
 + 3 x
2
 + x–7 = 1 is a Cubic equation.
x
 
+ 2y = 1,  2x + 3y = 2 are jointly called Simultaneous equations.
A simple equation in one unknown x is in the form ax + b = 0.
Where a, b are known constants and a  0
Note: A simple equation has only one root.
Example: 
4
3
x
-1 =
14
15
x +
19
5
.
Solution: By transposing the variables in one side and the constants in other side we have
4
3
x
 – 
14
15
x
= 
5
19
+1 or 
(20-14) 19 5
15 5
?
?
x
  or 
6 24
15 5
?
x
.
 ?
24x15
 = 12
5x6
x
Choose the most appropriate option (a) (b) (c) or (d).
# 
© The Institute of Chartered Accountants of India
2.3 EQUATIONS
1. The equation –7 x + 1 = 5 – 3 x will be satisfied for x equal to:
a) 2 b) –1 c) 1 d) none of these
2. The root of the equation 
 + 4 - 5
 +  = 11
4 3
x x
 is
a) 20 b) 10 c) 2 d) none of these
3. Pick up the correct value of x for 
2
 = 
30 45
x
a) x = 5 b) x = 7 c) x =1
3
1
d) none of these
4. The solution of the equation 
 + 24 
 = 4 + 
5 4
x x
a) 6 b) 10 c) 16 d) none of these
5. 8 is the solution of the equation
a) 
+ 4 - 5
+ = 11
4 3
x x
b) 
4 10
8
2 9
x x ? ?
? ?
c) 
24
4
5 4
x x ?
? ?
d) 
-15 5
4
10 5
x x ?
? ?
6. The value of y that satisfies the equation 
4
7 y
9
1 y
-
6
11 y ?
?
? ?
 is
a) –1 b) 7 c) 1 d) –
7
1
7. The solution of the equation (p+2) (p–3) + (p+3) (p–4) = p(2p–5) is
a) 6 b) 7 c) 5 d) none of these
8. The equation 
12 +1 15 1 2 5
= +
4 5 3 1
x x x
x
- -
-
 is true for
a) x=1 b) x=2 c) x=5 d) x=7
9. Pick up the correct value x for which 
? ?
1 1
+ =0
0.5 0.05 0.005 0.0005
x x
a) x=0 b) x = 1 c) x = 10 d) none of these
 ILLUSTRATIONS:
1. The denominator of a fraction exceeds the numerator by 5 and if 3 be added to both the
fraction becomes 
4
3
.  Find the fraction.
© The Institute of Chartered Accountants of India
Page 4


EQUATIONS
2
CHAPTER
After studying this chapter, you will be able to:
? Understand the concept of equations and its various degrees – linear, simultaneous,
quadratic and cubic equations;
? Know how to solve the different equations using different methods of solution; and
CHAPTER OVERVIEW
Definition of Equation
Simple
Equation
Applications Equations
Simultaneous
equations in
two unknowns
Methods of Solving
Three Linear Equations
with three Variables
Quadratic
Equation
Roots of the
Quadratic Equation
Constructions of
Quadratic Equation
Nature of
the Roots
Methods of Solution
Cross Multiplication
Method
Elimination
Method
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
2.2
Equation is defined to be a mathematical statement of equality. If the equality is true for
certain value of the variable involved, the equation is often called a conditional equation
and equality sign ‘=’ is used; while if the equality is true for all values of the variable
involved, the equation is called an identity.
For Example: 
+ 2 + 3
+ = 3
32
xx
holds true only for x =1.
So it is a conditional. On the other hand, 
+2 +3 5 +13
+ =
32 6
xx x
is an identity since it holds for all values of the variable x.
Determination of value of the variable which satisfy an equation is called solution of the
equation or root of the equation. An equation in which highest power of the variable is 1
is called a Linear (or a simple) equation. This is also called the equation of degree 1. Two or
more linear equations involving two or more variables are called Simultaneous Linear
Equations. An equation of degree 2 (Highest Power of the variable is 2) is called Quadratic
equation and the equation of degree 3 is called Cubic Equation.
For Example: 8x+17(x–3) = 4 (4x–9) + 12 is a Linear equation.
3x
2
 + 5x +6 = 0 is a Quadratic equation.
4x
3
 + 3 x
2
 + x–7 = 1 is a Cubic equation.
x
 
+ 2y = 1,  2x + 3y = 2 are jointly called Simultaneous equations.
A simple equation in one unknown x is in the form ax + b = 0.
Where a, b are known constants and a  0
Note: A simple equation has only one root.
Example: 
4
3
x
-1 =
14
15
x +
19
5
.
Solution: By transposing the variables in one side and the constants in other side we have
4
3
x
 – 
14
15
x
= 
5
19
+1 or 
(20-14) 19 5
15 5
?
?
x
  or 
6 24
15 5
?
x
.
 ?
24x15
 = 12
5x6
x
Choose the most appropriate option (a) (b) (c) or (d).
# 
© The Institute of Chartered Accountants of India
2.3 EQUATIONS
1. The equation –7 x + 1 = 5 – 3 x will be satisfied for x equal to:
a) 2 b) –1 c) 1 d) none of these
2. The root of the equation 
 + 4 - 5
 +  = 11
4 3
x x
 is
a) 20 b) 10 c) 2 d) none of these
3. Pick up the correct value of x for 
2
 = 
30 45
x
a) x = 5 b) x = 7 c) x =1
3
1
d) none of these
4. The solution of the equation 
 + 24 
 = 4 + 
5 4
x x
a) 6 b) 10 c) 16 d) none of these
5. 8 is the solution of the equation
a) 
+ 4 - 5
+ = 11
4 3
x x
b) 
4 10
8
2 9
x x ? ?
? ?
c) 
24
4
5 4
x x ?
? ?
d) 
-15 5
4
10 5
x x ?
? ?
6. The value of y that satisfies the equation 
4
7 y
9
1 y
-
6
11 y ?
?
? ?
 is
a) –1 b) 7 c) 1 d) –
7
1
7. The solution of the equation (p+2) (p–3) + (p+3) (p–4) = p(2p–5) is
a) 6 b) 7 c) 5 d) none of these
8. The equation 
12 +1 15 1 2 5
= +
4 5 3 1
x x x
x
- -
-
 is true for
a) x=1 b) x=2 c) x=5 d) x=7
9. Pick up the correct value x for which 
? ?
1 1
+ =0
0.5 0.05 0.005 0.0005
x x
a) x=0 b) x = 1 c) x = 10 d) none of these
 ILLUSTRATIONS:
1. The denominator of a fraction exceeds the numerator by 5 and if 3 be added to both the
fraction becomes 
4
3
.  Find the fraction.
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
2.4
Let x be the numerator and the fraction be 
+ 5
x
x
. By the question 
+ 3
=
+ 5 + 3
x
x 4
3
or
4x + 12 = 3x + 24 or x = 12
The required fraction is 
12
.
17
2. If thrice of A’s age 6 years ago be subtracted from twice his present age, the result would
be equal to his present age. Find A’s present age.
Let x years be A’s present age. By the question
2x–3(x–6) = x
or 2x–3x + 18 = x
or –x + 18 = x
or 2x = 18
or x=9
? A’s present age is 9 years.
3. A number consists of two digits the digit in the ten’s place is twice the digit in the unit’s
place. If 18 be subtracted from the number the digits are reversed. Find the number.
Let x be the digit in the unit’s place. So the digit in the ten’s place is 2x. Thus the number
becomes 10(2x) + x. By the question
20x + x– 18 = 10x + 2x
or 21x – 18 = 12x
or 9x = 18
or x = 2
So the required number is 10 (2 × 2) + 2 = 42.
4. For a certain commodity the demand equation giving demand ‘d’ in kg, for a price ‘p’ in
rupees per kg. is d = 100 (10 – p). The supply equation giving the supply s in kg. for a price
p in rupees per kg. is s = 75( p – 3). The market price is such at which demand equals
supply. Find the market price and quantity that will be bought and sold.
Given d = 100(10 – p) and s = 75(p – 3).
Since the market price is such that demand (d) = supply (s) we have
100 (10 – p) = 75 (p – 3) or 1000 – 100p = 75p – 225
or – 175p = – 1225.  
- 1225
p = 7
- 175
? ?
.
So market price of the commodity is ` 7 per kg.
? the required quantity bought = 100 (10 – 7) = 300 kg.
and the quantity sold = 75 (7 – 3) = 300 kg.
© The Institute of Chartered Accountants of India
Page 5


EQUATIONS
2
CHAPTER
After studying this chapter, you will be able to:
? Understand the concept of equations and its various degrees – linear, simultaneous,
quadratic and cubic equations;
? Know how to solve the different equations using different methods of solution; and
CHAPTER OVERVIEW
Definition of Equation
Simple
Equation
Applications Equations
Simultaneous
equations in
two unknowns
Methods of Solving
Three Linear Equations
with three Variables
Quadratic
Equation
Roots of the
Quadratic Equation
Constructions of
Quadratic Equation
Nature of
the Roots
Methods of Solution
Cross Multiplication
Method
Elimination
Method
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
2.2
Equation is defined to be a mathematical statement of equality. If the equality is true for
certain value of the variable involved, the equation is often called a conditional equation
and equality sign ‘=’ is used; while if the equality is true for all values of the variable
involved, the equation is called an identity.
For Example: 
+ 2 + 3
+ = 3
32
xx
holds true only for x =1.
So it is a conditional. On the other hand, 
+2 +3 5 +13
+ =
32 6
xx x
is an identity since it holds for all values of the variable x.
Determination of value of the variable which satisfy an equation is called solution of the
equation or root of the equation. An equation in which highest power of the variable is 1
is called a Linear (or a simple) equation. This is also called the equation of degree 1. Two or
more linear equations involving two or more variables are called Simultaneous Linear
Equations. An equation of degree 2 (Highest Power of the variable is 2) is called Quadratic
equation and the equation of degree 3 is called Cubic Equation.
For Example: 8x+17(x–3) = 4 (4x–9) + 12 is a Linear equation.
3x
2
 + 5x +6 = 0 is a Quadratic equation.
4x
3
 + 3 x
2
 + x–7 = 1 is a Cubic equation.
x
 
+ 2y = 1,  2x + 3y = 2 are jointly called Simultaneous equations.
A simple equation in one unknown x is in the form ax + b = 0.
Where a, b are known constants and a  0
Note: A simple equation has only one root.
Example: 
4
3
x
-1 =
14
15
x +
19
5
.
Solution: By transposing the variables in one side and the constants in other side we have
4
3
x
 – 
14
15
x
= 
5
19
+1 or 
(20-14) 19 5
15 5
?
?
x
  or 
6 24
15 5
?
x
.
 ?
24x15
 = 12
5x6
x
Choose the most appropriate option (a) (b) (c) or (d).
# 
© The Institute of Chartered Accountants of India
2.3 EQUATIONS
1. The equation –7 x + 1 = 5 – 3 x will be satisfied for x equal to:
a) 2 b) –1 c) 1 d) none of these
2. The root of the equation 
 + 4 - 5
 +  = 11
4 3
x x
 is
a) 20 b) 10 c) 2 d) none of these
3. Pick up the correct value of x for 
2
 = 
30 45
x
a) x = 5 b) x = 7 c) x =1
3
1
d) none of these
4. The solution of the equation 
 + 24 
 = 4 + 
5 4
x x
a) 6 b) 10 c) 16 d) none of these
5. 8 is the solution of the equation
a) 
+ 4 - 5
+ = 11
4 3
x x
b) 
4 10
8
2 9
x x ? ?
? ?
c) 
24
4
5 4
x x ?
? ?
d) 
-15 5
4
10 5
x x ?
? ?
6. The value of y that satisfies the equation 
4
7 y
9
1 y
-
6
11 y ?
?
? ?
 is
a) –1 b) 7 c) 1 d) –
7
1
7. The solution of the equation (p+2) (p–3) + (p+3) (p–4) = p(2p–5) is
a) 6 b) 7 c) 5 d) none of these
8. The equation 
12 +1 15 1 2 5
= +
4 5 3 1
x x x
x
- -
-
 is true for
a) x=1 b) x=2 c) x=5 d) x=7
9. Pick up the correct value x for which 
? ?
1 1
+ =0
0.5 0.05 0.005 0.0005
x x
a) x=0 b) x = 1 c) x = 10 d) none of these
 ILLUSTRATIONS:
1. The denominator of a fraction exceeds the numerator by 5 and if 3 be added to both the
fraction becomes 
4
3
.  Find the fraction.
© The Institute of Chartered Accountants of India
BUSINESS MATHEMATICS
2.4
Let x be the numerator and the fraction be 
+ 5
x
x
. By the question 
+ 3
=
+ 5 + 3
x
x 4
3
or
4x + 12 = 3x + 24 or x = 12
The required fraction is 
12
.
17
2. If thrice of A’s age 6 years ago be subtracted from twice his present age, the result would
be equal to his present age. Find A’s present age.
Let x years be A’s present age. By the question
2x–3(x–6) = x
or 2x–3x + 18 = x
or –x + 18 = x
or 2x = 18
or x=9
? A’s present age is 9 years.
3. A number consists of two digits the digit in the ten’s place is twice the digit in the unit’s
place. If 18 be subtracted from the number the digits are reversed. Find the number.
Let x be the digit in the unit’s place. So the digit in the ten’s place is 2x. Thus the number
becomes 10(2x) + x. By the question
20x + x– 18 = 10x + 2x
or 21x – 18 = 12x
or 9x = 18
or x = 2
So the required number is 10 (2 × 2) + 2 = 42.
4. For a certain commodity the demand equation giving demand ‘d’ in kg, for a price ‘p’ in
rupees per kg. is d = 100 (10 – p). The supply equation giving the supply s in kg. for a price
p in rupees per kg. is s = 75( p – 3). The market price is such at which demand equals
supply. Find the market price and quantity that will be bought and sold.
Given d = 100(10 – p) and s = 75(p – 3).
Since the market price is such that demand (d) = supply (s) we have
100 (10 – p) = 75 (p – 3) or 1000 – 100p = 75p – 225
or – 175p = – 1225.  
- 1225
p = 7
- 175
? ?
.
So market price of the commodity is ` 7 per kg.
? the required quantity bought = 100 (10 – 7) = 300 kg.
and the quantity sold = 75 (7 – 3) = 300 kg.
© The Institute of Chartered Accountants of India
2.5 EQUATIONS
Choose the most appropriate option (a) (b) (c) or (d).
1. The sum of two numbers is 52 and their difference is 2. The numbers are
a) 17 and 15 b) 12 and 10 c) 27 and 25 d) none of these
2. The diagonal of a rectangle is 5 cm and one of at sides is 4 cm. Its area is
a) 20 sq.cm. b) 12 sq.cm. c) 10 sq.cm. d) none of these
3. Divide 56 into two parts such that three times the first part exceeds one third of the second
by 48. The parts are.
a) (20, 36) b) (25, 31) c) (24, 32) d) none of these
4. The sum of the digits of a two digit number is 10. If 18 be subtracted from it the digits in
the resulting number will be equal. The number is
a) 37 b) 73 c) 75 d) none of these numbers.
5. The fourth part of a number exceeds the sixth part by 4. The number is
a) 84 b) 44 c) 48 d) none of these
6. Ten years ago the age of a father was four times of his son. Ten years hence the age of the
father will be twice that of his son. The present ages of the father and the son are.
a) (50, 20) b) (60, 20) c) (55, 25) d) none of these
7. The product of two numbers is 3200 and the quotient when the larger number is divided
by the smaller is 2.The numbers are
a) (16, 200) b) (160, 20) c) (60, 30) d) (80, 40)
8. The denominator of a fraction exceeds the numerator by 2. If 5 be added to the numerator
the fraction increases by unity. The fraction is.
a) b) 
1
3
c) 
9
7
d) 
5
3
9. Three persons Mr. Roy, Mr. Paul and Mr. Singh together have ` 51. Mr. Paul has ` 4 less
than Mr. Roy and Mr. Singh has got ` 5 less than Mr. Roy. They have the money
as.
a) ( ` 20, ` 16, ` 15) b) ( ` 15, ` 20, ` 16)
c) ( ` 25, ` 11, ` 15) d) none of these
10. A number consists of two digits. The digits in the ten’s place is 3 times the digit in the
unit’s place. If 54 is subtracted from the number the digits are reversed. The number is
a) 39 b) 92 c) 93 d) 94
11. One student is asked to divide a half of a number by 6 and other half by 4 and then to add
the two quantities. Instead of doing so the student divides the given number by 5. If the
answer is 4 short of the correct answer then the number was
a) 320 b) 400 c) 480 d) none of these.
© The Institute of Chartered Accountants of India