Q1 : The product of two consecutive numbers is 72. Find the numbers ?
Solution:
Let two consecutive numbers be x and x + 1.
Product of two consecutive numbers = 72.
x ( x +1) = 72.
x2 + x - 72 = 0
x2 + 9x - 8x - 72 = 0
x(x + 9) - 8(x + 9) = 0
(x + 9)(x - 8) = 0
x + 9 = 0 or x - 8 = 0
x = -9 or x = 8
If x = -9 then x + 1 = -9 + 1 = -8
If x = 8 or x + 1 = 8 + 1 = 9
Therefore the two consecutive numbers are -9, -8 or 8, 9.
Q2: The product of two consecutive even numbers is 24 . What are the numbers?
Solution : Let two consecutive even numbers be x and x + 2 .
Their product is 24.
x (x + 2) = 24
x2 + 2 x - 24 = 0
x2 + 6 x - 4x - 24 = 0
x(x + 6 ) - 4( x + 6) = 0
(x - 4 )( x + 6) = 0
x - 4 = 0 or x + 6 = 0
x = 4 or x = -6
If x = -6 then x + 2 = -6 +2 = -4
If x = 4 then x + 2 = 4 + 2 = 6.
Therefore the required numbers are -6 , -4 or 4, 6.
Q3 : The product of two consecutive odd numbers is 35. Find them .
Solution : Let the two consecutive odd numbers are x , x+ 2.
Their product = 35.
x(x+2) = 35.
x2 + 2x - 35 =0.
x2 + 7x - 5x - 35 = 0
x(x + 7) - 5 ( x + 7) =0
(x + 7 )( x - 5) = 0
x + 7 = 0 or x - 5 = 0
x = - 7 or x = 5
If x = -7 then x + 2 = -7 + 2 = -5.
If x = 5 then x + 2 = 5 + 2 = 7.
Therefore required numbers are -7, -5 or 5, 7.
Q4: The difference of two numbers is 3 and their product is 18. Find them.
Solution : Let two numbers be x and y .
Given x - y = 3...........(1)
xy = 18
y = 18/x ...................(2)
Substituting (2) in (1) we have,
x - 18/x = 3
x2 - 18 = 3x
x2 - 3x - 18 = 0
x2 -6x + 3x - 18 = 0
x ( x - 6) + 3 ( x - 6 ) = 0
( x - 6 )(x + 3) = 0
x - 6 = 0 or x + 3 = 0
x = 6 or x = -3
If x = 6 then y = 18/6 = 3.
If x = -3 then y = 18 /-3 = -6
Therefore required numbers are 3, 6 or -6, -3.
Q5: The sum of the squares of two consecutive even numbers is 20 . Find them.
Solution: Let x and x + 2 are two consecutive even numbers.
From given x2 + (x + 2)2 = 20
x2 + x2 + 4x + 4 = 20
2x2 + 4x + 4 - 20 = 0
2x2 + 4x -16 =0
x2 +2x - 8 = 0
x2 +4x - 2x - 8 = 0
x(x + 4) -2( x + 4) = 0
(x + 4)(x - 2) =0
x = -4 or 2
If x = -4 then x + 2 = -4 +2 = -2
If x = 2 then x +2 = 2 + 2 = 4
Therefore required numbers are -4 , -2 or 2, 4 .
Q6: Find the number which is less than its square by 132.
Solution : Let the number be x.
From given
x = x2 - 132
x2 - x -132 =0
x2 -12x +11x -132 = 0
x( x - 12 ) + 11 ( x - 12) = 0
(x - 12)(x + 11) = 0
x - 12 =0 or x + 11 = 0
x = 12 or -11
Q8: The length of a hall exceeds its breadth by 8 m and its perimeter is 56m . Then find the dimensions of the hall.
Solution : Let the breadth of the hall be (b) = x m .
Then length(l) = ( x + 8 )m
Perimeter of a rectangular hall (P) = 2 (l + b)
P = 56 m
2 ( x + 8 + x ) = 56
2x +8 = 56/2
2(x + 4) = 28
x + 4 = 28/2
x = 14 - 4
x = 10
Length = x + 8 = 10 + 8 = 18 m, and breadth = x = 10 m.
Therefore the dimensions of rectangular hall is 18m and 10m .
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