BTGmoderatorDC wrote:If two 2-digit positive integers have their respective tens digits exchanged, the difference between the pair of integers changes by 4. What is the greatest possible difference between the original pair of integers?
A) 76
B) 80
C) 82
D) 90
E) 94
OA C
Source: GMAT Prep
Say the first 2-digit integer is [xy] = 10x + y and the second 2-digit integer is [ab] = 10a + b such that x > a; thus, [xy] > [ab].
Difference between the original pairs = [xy] - [ab] = 10x + y - 10a - b = 10(x - a) + y - b;
After exchanging the tens digits of the integers, we have new set of integers as 10a + y and 10x + b, where 10x + b > 10a + y
Difference between the changed pairs = [xb] - [ay] = 10x + b - 10a - y = 10(x - a) - y + b;
Difference between the pair of integers = [10(x - a) + y - b] - [10(x - a) - y + b] = 2(y - b) = 4 (given)
=> y - b = 2
Note that y and b are units digits and they can have any values such that the difference between them is 2. Example: y = 2 and 0; y = 3 and 1; ... y = 9 and 7. Also, note that tens digits of the integers can be any numbers (except 0). Since we want the greatest possible difference between the original pair of integers, let's take x = 9 and a = 1.
"¢ Taking x = 9, a = 1, y = 9 and b = 7, we get [xy] = 99 and [ab] = 17. Note that you may choose any values of y and b as they can have any values such that the difference between them is 2, as discussed above.
Difference of 99 and 17 = 99 - 17 = 82 (Correct answer: Option C)
Let's cross-check this.
The new pairs of two 2-digit integers after exchanging the tens digits are 97 and 19. The difference between them = 97 - 19 = 78
We see that the difference between the pair of integers 82 - 78 changes by 4 (given).
The correct answer:
C
Hope this helps!
-Jay
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