Question 1096585: A two digit number is 3 more than 4 times the sum of its digits. if 18 is added tothis number, the sum is equal to the number obtained by interchanging the digits. find the number.
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
There are two distinct possible approaches to solving this problem:
(1) Use algebra to find what 2-digit numbers have the property that the number is 3 more than 4 times the sum of its digits, and find which one of those numbers satisfies the property that adding 18 to the number produces a number which has the same digits in reverse order; or
(2) Finding the 2-digit numbers which have the property that adding 18 to the number produces a number which has the same digits in reverse order, and finding the one for which the number is 3 more than 4 times the sum of its digits.
If you know the right number trick, the second option will get you to the answer much faster than the first. But both methods are valid; and taking a look at both of them can be useful exercises. So...
(1) using algebra....
If the 2-digit number has tens digit x and units digit y, then the value of the 2-digit number is 10x+y. We want that to be equal to 3 more than 4 times the sum of x and y:
We can easily list the combinations of x and y values that satisfy this equation and in which x and y are both single digits:
x=1 --> y=1 --> the 2-digit number is 11
x=2 --> y=3 --> the 2-digit number is 23
x=3 --> y=5 --> the 2-digit number is 35
x=4 --> y=7 --> the 2-digit number is 47
x=5 --> y=9 --> the 2-digit number is 59
Of those 2-digit numbers, there is only one which, when 18 is added to it, produces a 2-digit number with the two digits reversed: 35+18 = 53.
So the 2-digit number we are looking for is 35.
(2) the alternative method...
If you have a 2-digit number, and adding 18 to the number produces a 2-digit number with the digits reversed, then the number has a tens digit which is 2 less than the units digit. The 2-digit numbers with this property are
13 24 35 46 57 68 79
Checking these to find which of them has the property that the number is 3 more than 4 times the sum of its digits, we see that the only one satisfying that property is 35.
So again by this method the number we are looking for is 35.
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