Two different two-digit numbers are written beside each other such tha

Question

Two different two-digit numbers are written beside each other such that the larger number is written on the left. When the absolute difference of the two numbers is subtracted from the four-digit number so formed, the number obtained is 5481. What is the sum of the two two-digit numbers?

(A) 68

(B) 70

(C) 73

(D) 118

(E) 187

(A) 68

(B) 70

(C) 73

(D) 118

(E) 187

nick1816 · Answer

let 2 numbers are 'ab' and 'cd', such that ab>cd.
abcd-(ab-cd)=5481
100*ab+cd-ab+cd=5481
99*ab+2*cd=5481

99*ab and 5481 are multiples of 9, hence cd must be a multiple of 9.

At cd=18, we get ab=55

Sum of the two 2-digit numbers= 55+18=73

MahmoudFawzy · Answer

let  ab>cd

To subtract the 4-digit minus the absolute difference of the 2 2-digits:
 100ab + cd - (ab-cd) = 5481 
 99ab + 2cd = 5481

There are two possibilities for ab: 
if  (ab-cd) > cd  ---> in this case,  ab = 55  --->  2cd = 5481 - (55*99) = 36  --->  cd = 18  ( possible )
if  (ab-cd) < cd  ---> in this case,  ab = 54  --->  2cd = 5481 - (54*99) = 135  --->  cd = 67.5  ( impossible )

then  ab + cd = 55 + 18 = 73 
  C

philipssonicare · Answer

Hovkial 
What is the question source?

nick1816 
Why stop at cd=18 and not all other multiples of 9?

Mahmoudfawzy83 
What is your reasoning for 'two possibilities for ab'?

nick1816 · Answer

99ab+2cd=5481 
 cd= \frac{-99}{2}ab+\frac{5481}{2}  is an equation of a straight line, having slope -99/2
two closest integral solutions to (55, 18) on above mentioned line are-

1. cd= 18+99=  117 , and ab= 55-2=53
2. cd= 18-99=  -81 , and ab=55+2=57

As cd is a 2-digit positive integer, hence there is no other integral solution possible in which both ab and cd are positive 2-digit number.

MahmoudFawzy · Answer

Hi  philipssonicare

(1) it is given that ab>cd, so (ab - cd) is a positive number , let's call this positive number x.
x can be as low as 1 (for example if ab = 99 and cd = 98), and can be as high as 89 (for example if ab = 99 and cd = 10)

(2) it is given that abcd - x = 5481
lets focus on the hundreds and thousands digits while subtracting:
if cd> x, then ab will stay as they are after subtraction (because there will be no need to borrow from the hundreds digit) . i.e ab = 54

if cd< x, then ab will decrease by 1 after subtraction(because we will need to borrow from the hundreds digit). i.e ab = 55 before subtraction

there is no other options

vanam52923 · Answer

nick1816 
how did u decide that 99ab is multiple of 9

nick1816 · Answer

99ab is not a 4 digit-number; it's 99*ab, that is obviously a multiple of 9. 
I'll put  *  in my solution , in order to remove ambiguity.

vanam52923 · Answer

Ty so much

andreagonzalez2k · Answer

From the data we deduce that first number must be  55 , because:
-  54<81  (and first number must be bigger than second one)
- and, to obtain  5481  you have substracted the difference of two two-digit numbers (that must be at most a two digit number).

So second number is:  10a+b

Difference it from 55:  55-10a-b

Add the difference to  5481  to obtain the 4-digit number:

5481+55-10a-b = 5500 +10a+b 
 5536-5500 = 20a+2b 
 10a+b = 18

As a and b must be a digit, the only solution is a = 1 and b = 8, so the second number is  18 .

55+18 = 73

GMATGuruNY · Answer

Alternate approach:

Let the original 4-digit integer = ABCD

When the absolute difference of the two numbers is subtracted from the four-digit number so formed, the number obtained is 5481. 
Thus, ABCD > 5481

Two different two-digit numbers are written beside each other such that the larger number is written on the left. 
Since AB > CD and ABCD > 5481, the least possible option for ABCD = 5510.

When the correct value for ABCD has been determined, the difference between ABCD and 5481 will be equal to the absolute difference between AB and CD.

If ABCD = 5510, we get:
Difference between 5510 and 5481 =  29 
Absolute difference between AB and CD = |55-10| =  45 
Doesn't work.
The two colored values are not equal but have a difference of 16.

To close the 16-value gap between the two colored values, the blue value must INCREASE by 8, while the red value DECREASES by 8.
For the difference between ABCD and 5481 to increase by 8, ABCD must increase from 5510 to 5518, yielding the following:
Difference between 5518 and 5481 =  37 
Absolute difference between AB and CD = |55-18| =  37 
Now the two colored values are equal.

Since ABCD = 5518, we get:
AB+CD = 55+18 = 73

C

Dufa · Answer

How is everyone getting least possible value of 5510 please

Posted from my mobile device

GMATGuruNY · Answer

Since 5481 is the result after subtraction, the original number must be greater than 5481.
In the original number, the first two digits must form an integer greater than the integer formed by the last two digits.
Since 54 cannot be greater than any integer between 82 and 99, inclusive, it is not possible that the original number is between 5482 and 5499, inclusive.
Thus, the original number must be greater than 5500. (5500 itself is not viable, given that the last two digits must form a two-digit integer.)
Since the smallest option for the integer formed by the last two digits is 10, we get:
Least possible option for the original number = 5510

chetan2u · Answer

I. Logical method

If the number after subtraction is 5481, the first two digits give the larger number 
The difference between any 2 two digit numbers can never be more than 99-10 or 89, so largest possible value of 4-digit number before subtraction will be <5481+89
So, the larger number can be 54 or 55.
If larger number is 54, the 4-digit number would be less than 5454, but we are given the 4 digit number is greater than 5481.
 Hence, the larger number is 55.

Now, let the smaller number be ab. 
55ab is nothing but 5500+ab => 5500+ab-(55-ab)=5481
5445+2ab=5481
2ab=36
ab=18

Sum of the two numbers= 55+18=73

II. Use the options

The difference of two 2-digit numbers can never be a 3-digit number. Eliminate D and E

A. 68
If the larger number is 54, then smaller is 68-54=14. The difference between them is 54-14=40.
Is 5414-40=5481?……NO
If the larger number is 55, then smaller is 68-55=13. The difference between them is 55-13=42.
Is 5513-42=5481?……NO

B. 70
If the larger number is 54, then smaller is 70-54=16. The difference between them is 54-16=38.
Is 5416-38=5481?……NO
If the larger number is 55, then smaller is 70-55=15. The difference between them is 55-15=40.
Is 5515-40=5481?……NO

C. 73
If the larger number is 54, then smaller is 73-54=19. The difference between them is 54-19=35.
Is 5419-35=5481?……NO
If the larger number is 55, then smaller is 73-55=18. The difference between them is 55-18=37.
Is 5518-37=5481?……YES

Answer is C

Foreheadson · Answer

I would just add the 3rd option. 
the difference is odd, so the sum must be odd too. only C and E are odd. E is more than 100. we know first number is around 55. so no solution is necessary.

Posted from my mobile device

Crytiocanalyst · Answer

2 numbers are 'yz' and 'xy',
Given that yz>cd.
yzxy-(yz-xy)=5481

100*yz+xy-yz+xy=5481

99*yz+2*xy=5481...1

Since yz has to be greater than 54

let us assume it to be 55 then the other number when put in 1 we get

At xy=18, we get yz=55

Sum of the two 2-digit numbers= 55+18=73

Hence IMO C

Diyaj2403 · Answer

Thanks for the explanation  MahmoudFawzy . Just one thing - how did we arrive at ab  = 55 or 54? Thanks in advance

Two different two-digit numbers are written beside each other such tha

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