Question : The ratio of a two-digit number and the sum of the digits of that number is 4 : 1. If the digit at the unit's place is 3 more than the digit at the ten's place, then the number is:
Option 1: $47$
Option 2: $69$
Option 3: $36$
Option 4: $25$
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Correct Answer: $36$
Solution : Let us assume that the digits of the number are $x$ and $y$ with $x$ in the ten's place. The number is $10x+y$. The digit in the unit's place is 3 more than the digit in the ten's place, i.e. $y=x+3$ ...(1) The ratio between the two-digit number and the sum of the digits of that number is $4 : 1$. $(10x+y):(x+y) = 4:1$ or, $10x+y = 4x+4y$ or, $y = 2x$ Substituting in (1), we get $2x = x+3$ $x = 3, y = 6$ The number is $36$. Hence, the correct answer is $36$.
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Question : The sum of a two-digit number and the number obtained by reversing its digits is a square number. How many such numbers are there?
Option 1: 5
Option 2: 6
Option 3: 7
Option 4: 8
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Option 1: 8 : 27
Option 2: 1 : 25
Option 3: 1 : 125
Option 4: 1 : 64
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Option 1: 6
Option 2: 5
Option 3: 4
Option 4: 0
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Option 1: 80
Option 2: 72
Option 3: 85
Option 4: 78
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Option 1: 22 : 15 : 39
Option 2: 25 : 48 : 59
Option 3: 24 : 47 : 58
Option 4: 11 : 25 : 36
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