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Jamboree and GMAT Club Contest: The sum of the two digit numbers

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Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 09:54
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D
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Jamboree and GMAT Club Contest Starts



QUESTION #1:

X Y
+Y X
________

The sum of the two digit numbers above is a three digit number PQ5, where each letter X, Y, P, and Q represents a different non zero digit. Which of the following can be the value of X?

I) 7
II) 8
III) 9

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I , II and III


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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 10:17
4
1
Digit P has to be 1 as we are adding 2 two digit numbers
Y+X=15


XY= 78 or 87
YX= 87 or 78

So X can be either 7 or 8

If X=9 , then
69
+96
165
In this case Q will be 6 . So the digits won't be distinct.

Answer D
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 10:27
1
ok, so we have
XY+
YX
= PQ5, where all the digits are distinct.
from this, we can conclude that X+Y is equal to either a 1 digit number or a 2 digit number, the units digit of which would be 5.

from the given possibilities:
I x = 7, then we must have Y equal to 8 so that the units digit would be 5.
78+87 = 165 - since all the digits are distinctive, then yes, X can be 7.

II x = 8, this works the same way as I. if x =8 then y = 7, and we get 87+78 = 165. again, this satisfies the conditions.
If we stumble and don't know what to do next, we can just eliminate A, B, and C, and try to make a smart guess with a chance of getting the right answer 1/2.

III x = 9, let's see, if X =9, then Y must be 6.
96+69 = 165. Same thing as I and II right? no! it is not. The condition is that X, Y, P, and Q are distinct digits, which mean that no digit should be repeated. In this case we have Q has the same digit as Y. This does not satisfy the condition, and thus, the answer is D.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 10:44
I choose E as an answer.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 11:19
1
QUESTION #1:

_X Y
+Y X
________

The sum of the two digit numbers above is a three digit number PQ5, where each letter X, Y, P, and Q represents a different non zero digit. Which of the following can be the value of X?

I) 7
II) 8
III) 9

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I , II and III




_X Y
+Y X
________


X+Y = 5

Possible values of x and Y are (X,Y)

(1,4)
(2,3)
(3,2)
(4,1)


The above pair can not produce 3 digit distinct numbers when they are added, hence we neglect these options.

(6,9)
(7,8)
(8,7)
(9,6)


The sum of the two digit numbers above is a three digit number PQ5

_X Y
+Y X
________

Check for the possible options -

(6,9)

69+96 = 165

(7,8)

78+87 = 165

(8,7)

87+78 = 165

(9,6)

96 + 69 = 165

Given the conditions only the folowing set of numbers product 3 digit distinct numbers in the form of PQ5 and the sum of its units digit is 5

(6,9)
(7,8)
(8,7)
(9,6)

So, the possible values of X can be 6,7,8 and 9.

Among the given options all 7 ,8 & 9 ca be the possible values of X.

Hence IMHO answer is (E) all 1,2 and 3.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 11:43
1
Given in question:

XY
+YX
i,e sum of two digit number in this fashion is equal to three digit number PQ5, where P,Q,X,Y are non-zero digit and vakue for X is asked.

We start of checking with X=7,
for getting 5 in sum, Y need to be 8.
Sum =78 +87 =165 .( three digit number )

ii) X=8
we have Sum= 87 + 78 =165

iii) X=9
for getting 5 at ones place, Y=6.
Sum= 96 + 69 =165.

Therefore, in all cases we get the required sum i,e three digit number in form PQ5.
Correct answer is E
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 12:12
X and y should add upto 15, check for x=7 and 9. Both satisfies the given conditions, hence answer choice E
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 12:50
XY
+YX

sum of both numbers is equal to PQ5

So it only possible if sum two number numbers are greater than 50 as only then three digit number is possible

So checking with the answer choices
When X=7 then y can be 8
so 78+87= 165
When X=8 y =7
87+78=165
Also when X=9 Y =6
96+69=165

So E is correct answer
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post Updated on: 08 Nov 2015, 03:53
4
X Y
Y X
_______
PQ5

Here both the two digit numbers are made of x & y.

So, value of x+y can only be 15. [as none of x & y is 0 and unit digit of 3-digit sum is 5]

Let's try out the value of x with 3 given options.

x + y = sum of digits ......>> sum of both numbers
7 + 8 = 15 ..................>> 78 + 87 = 165
8 + 7 = 15 ..................>> 87 + 78 = 165
9 + 6 = 15 ..................>> 96 + 69 = 165 !!! x, y, p & q all should be different digit. NOT acceptable

Thus only two values [7 & 8] for x satisfy the condition.

Option (D) is correct.
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Originally posted by musunna on 07 Nov 2015, 13:39.
Last edited by musunna on 08 Nov 2015, 03:53, edited 1 time in total.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 13:54
1
10x+y+10y+x=100p+10q+5
11(x+y)=100p+10q+5...(1)

Therefore, pq should have a remainder of 5 when divided by 11.
So pq could be 16, 27, 38...

We take pq as 16 and divide 165 with 11 (equation 1). Therefore, x+y=15. So x could be 7 then y could be 8 and vice versa. Re-check that p,q,x and y have different digits. I and II could be true.

We take pq as 27 and divide 275 with 11. x+y=25. But x and y are different digits and the highest that they can be added to is 17 (9+8=17). We stop calculating here. III doesn't seem to be true.

Answer: D
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 14:20
1
D is correct

both 7 and 8 can be placed in X and Y positions respectively
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 14:54
1
X+Y=can be 5 or 15. We cannot have that X+Y=5, since we would have that the sum of XY and YX is less than 100. E.g: 14+41=55, 23+32=55. Therefore, X+Y=15. Let’s analyze possible scenarios
(X, Y)= {(6,9);(7,8);(8,7);(9,6)}. Let’s see how each of these scenario fit within our conditions:
  • (6,9)-X=6,Y=9,P=1,Q=6. Since we are told that all numbers are different, this scenario is not feasible. Similarly, the last scenario is not feasible, since we would have Q=Y.
  • (7,8)-X=7,Y=8,P=1,Q=6. All variables are different, so this solution is fine. The same can be told about the scenario in which (X=8,Y=7).
Overall, it seems that X can be 7 or 8; therefore, the answer for this question is D.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 20:43
The answer is E.
> x/y= 9/6
or
>x/y=7/8

In both the cases the sum will involve carry over and thus the digits will be different.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 22:06
1
The answer is (D) I and II only.

XY
+YX
____
PQ5
_________
Question asks, what value X can take?

1) Option 1 says : 7, so if X can be 7 then number we can write it as 78+87 as then only we can satisfy the third condition which is PQ5. Also, If we add 78+87=165 , so it follows, P=1, Q=6 , X=7, and Y=8. They are distinct and non-zero. Other combination 71, 72, 73, 74.....does not satisfy the sum condition PQ5 so they were eliminated at first. So X can take value 7.

2. Similarly, 87 is the only possible number which satisfy all the conditions. So X can take value 8.

3. The possible number will be 96+69= 165 however, this does not satisfy the third condition which says P,Q,X,Y are distinct. Q and Y in this case becomes equal. So this cannot be true.

The best answer choice is D.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 07 Nov 2015, 23:54
let x =7 y=8 then PQ5 = 165
let X=8 y =7 then PQ5 =165
Let X=9 y = 6 then PQ5 = 165

thus x can b 7, 8 , 9

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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 08 Nov 2015, 00:06
10X + Y + 10Y + X = PQ5
11(X + Y) = PQ5
PQ5 has to be a multiple of 11, should end with 5 and has to be a 3 digit integer.. i.e PQ5 = 55(2K+1) --> 165, 275, 385
But the maximum value of PQ can be 19 since XY and YX are 2 digit numbers..
So we have only 165 as the possible value of PQ5..
PQ = 16. Since 1 is carried over --> 16 - 1 = 15
15 can be obtained by the combination of (7+8); (8+7) and (9+6)..
So value of X can be 7, 8 and 9

Answer: E
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 08 Nov 2015, 02:11
78+87 = 165 x=7
87+78 = 165 x=8
69+96 = 165 x=6
96+69 = 165 x=9

ANS E
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 08 Nov 2015, 03:49
1
Since the unit digit of the sum is 5, the possible (X,Y) pairs can be (7,8); (8,7); (6,9) and (9,6)

Note that for any of the pairs listed above the sum will be 165 so P=1 and Q =6. Since we are given that P, Q, X and Y are different digits. X and Y cannot be 6,9 as Q=6. Therefore the two possible pairs are (7,8); (8,7) so X can 7 or 8.

Ans should be D
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 08 Nov 2015, 04:42
1
Hi

The answer (D) I and II only.

When we take X as 7, then for the sum of the two numbers (XY and YX) to have 5 at the units' place, Y must be 8.
In this case, the sum of the numbers is 78+87 = 165
All digits P(1), Q(6), X(7) and Y(8) are different and satisfy the criteria mentioned in the question.
Hence, X can take the value 7.

Now, when we take X as 8, then for the sum of the two numbers (XY and YX) to have 5 at the units' place, Y must be 7.
In this case, the sum of the numbers is 87+78 = 165
All digits P(1), Q(6), X(8) and Y(7) are different and satisfy the criteria mentioned in the question.
Hence, X can take the value 8.

Now, when we take X as 9, then for the sum of the two numbers (XY and YX) to have 5 at the units' place, Y must be 6.
In this case, the sum of the numbers is 96+69 = 165
But, Q and Y have the same value 6 and hence the criteria laid out in the question for all variables to have unique values is not met.

The answer is therefore (D)I and II only.
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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers  [#permalink]

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New post 08 Nov 2015, 06:44
1
Ans E

Since xy + yx=pq5

Therefore both the digits must be more than 50
e.g.

87+78=165
78+87 =165
96+69=165

We need to note that sum of unit digits should be a multiple of 5.

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Re: Jamboree and GMAT Club Contest: The sum of the two digit numbers &nbs [#permalink] 08 Nov 2015, 06:44

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