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Answers
Step-by-step explanation:
66.
The digit at hundreds place be X
The digit at tens place be Y
The digit at ones place be Z
Then the three digits number = 100X+10Y+Z
The new number obtained by reversing the digits
= 100Z+10Y+X
Given that
New number = Original number +27
=> 100Z+10Y+X = 100X+10Y+Z +27
=> 100Z+10Y+X-100X-10Y-Z = 27
=> 99Z-99X = 27
=> 99(Z-X) = 27
=> Z-X = 27/99
=> Z-X = 3/11
The difference of the last two digits of the number = 3/11
67.
Let the two numbers be X and Y
X > Y
The difference of the two numbers=5
=> X-Y = 5
=> X = 5+Y -----(1)
Difference of their squares = 65
=> X²-Y² = 65
=> (5+Y)²-Y² = 65
=> 25+10Y+Y²-Y² = 65
=> 25+10Y = 65
=> 10Y = 65-25
=> 10Y = 40
=> Y = 40/10
=> Y = 4
On Substituting the value of Y in (1)
X = 5+4 = 9
Therefore the two numbers are 9 and 4
68.
Let the three consecutive integers be X,X+1,X+2
4 times the first integer =4(X) = 4X
Half of the second = (X+1)/2
Twice the third = 2(X+2) = 2X+4
Given that
4times the first + one half the second - twice the third = 24
=> 4X+{(X+1)/2}+2X+4 = 24
=> 6X+4+{(X+1)/2} = 24
=> (12X+8+X+1)/2 = 24
=> (13X+9)/2 = 24
=> 13X+9 = 24×2
=> 13X+9 = 48
=> 13X = 48-9
=> 13X = 39
=> X = 39/13
=> X = 3
and X+1 = 3+1 = 4
X+2 = 3+2 = 5
The three consecutive integers are 3,4 and 5
69.
Given equation is 7X+3Y = 123
=> 3X = 123-7Y
=>123-7Y is a multiple of 3
Given that X,Y>0
=>123-7Y> 0
=> 123-7Y+7Y> 0+7Y
=> 123 > 7Y
=> (123/7) > (7Y/7)
=> 123/7 > Y
=> 17.57> Y
The possible integer values of Y are 3,6,9,12,15
The number of integer solutions of the given equation is 5
70.
Let the present age of Sabrina be X years
The present age of Michael = X+6 years
Five years ago their ages wI'll be
X-5 and X+6-5
=> X-5 years and X+1 years
Given that
Five years ago
Michael was thrice as old as Sabrina
=> X+1 = 3(X-5)
=> X+1 = 3X-15
=>1+15 = 3X-X
=> 16 = 2X
=> X = 16/2
=> X = 8 years
Present age of Sabrina = 8 years
Present age of Michael = 8+6 = 14 years