Question 2: Four 2-digit
numbers are made up from the
digits 1, 2, 3,..., 8 with no
repetition. The following are the
hints about these numbers: -In
the smallest number, the ones
digit is twice the tens digit. - In
the largest number, the sum of
ones and tens is 9. - In the
second largest number, the tens
digit is 5 more than the ones
digit. -There is only 1 odd
number among these 4
numbers. Find the second
smallest number.
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given:
1) four 2-digit numbers made up from digits 1 - 8 without repetition
2) there is only one odd number
ANALYZATION
1) smallest 2-digit number = 10a + b
a) b = 2a -> a = b/2
b) this must be even number, because the ones digit is twice of tens digit
c) ones digit could be 2, 4, 6 or 8
2) largest 2-digit number = 10c + d
a) c + d = 9
b) this must be even or odd number, because there sum is 9, in order to get a sum of odd number, you need one odd number and one even number
3) 2nd largest 2-digit number = 10e + f
a) e = f + 5 -> e - f = 5
b) this must be even or odd number, because there difference is 9, in order to get a difference of odd number, you need one odd number and one even number
POSSIBLE PAIRS:
1) a = b/2 (smallest)
2 = 2(4) (smallest possible pair)
3 = 2(6)
4 = 2(8)
2) c + d = 9 (largest)
1 + 8 = 9 (eliminated from possibility)
2 + 7 = 9 (eliminated from possibility)
3 + 6 = 9
4 + 5 = 9
5 + 4 = 9
6 + 3 = 9
7 + 2 = 9
8 + 1 = 9 (largest possible pair)
3) e - f = 5 (2nd largest)
9 - 4 = 5 (eliminated from possibility)
8 - 3 = 5 (eliminated from possibility)
7 - 2 = 5
6 - 1 = 5
(3) is the only pairs that least possibilities, 61 or 72..
1ST ASSUME:
lets assume that the 2nd largest number is 72
so the largest possibile number will be 81
numbers 1, 2, 7 and 8 already use
from the possible pairs in (1), 36 is the only we can pick after eliminating all possibilities
numbers 1, 2, 3, 6, 7 and 8 already use, not used 4 and 5
there are two possible numbers in 4 and 5.. 45 and 54
but we already have one odd number
therefore we must use 54
so the numbers we get are:
smallest = 36
2nd smallest number = 54
2nd largest number = 72
largest number = 81
2ND ASSUME:
lets assume that the 2nd largest number is 61
so the largest possibile number can be 72 (because there is only one odd, and 61 is already a odd number)
numbers 1, 2, 6 and 7 already use
from the possible pairs in (1), 48 is the only we can pick after eliminating all possibilities
numbers 1, 2, 4, 6, 7 and 8 already use, not used 3 and 5
there are two possible numbers in 3 and 5.. 35 and 53, which both odd numbers
THEREFORE: WE MUST CONSIDER THE FIRST ASSUME
WHERE:
smallest = 36
2nd smallest number = 54
2nd largest number = 72
largest number = 81