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If x, y, and z are three-digit positive integers and if x =  [#permalink]

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If x, y, and z are three-digit positive integers and if x = y + z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z ?

(1) The tens digit of x is equal to the sum of the tens digits of y and z.
(2) The units digit of x is equal to the sum of the units digits of y and z.

Is it safe to conclude that the place value of an integer number (represented as a sum of different integers), depends upon only the preceding place value of integers being summed up?

for eg:
x=1000a+100b+10c+1d
y=1000e+100f+10g+1h
z=1000l+100m+10n+1p
if z=x+y then

is l only dependent upon value of b and f or some other parameters also??

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If my post was worth it, then i deserve kudos Originally posted by conty911 on 18 Sep 2012, 09:10.
Last edited by Bunuel on 18 Sep 2012, 09:16, edited 1 time in total.
Renamed the topic and edited the question.
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Joined: 02 Sep 2009
Posts: 58445
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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46
1
43
fozzzy wrote:
If x, y, and z are three-digit positive integers and if x = y + z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z ?

(1) The tens digit of x is equal to the sum of the tens digits of y and z.
(2) The units digit of x is equal to the sum of the units digits of y and z.

Is there an alternative approach for this problem?

The question basically asks whether there is a carry over 1 from the tens place to the hundreds place.

Consider the following examples:
(i)
123
234
357

Here when we add the tens digits 2 and 3 there is no carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x (3) is equal to the sum of the hundreds digits of y (1) and z (2).

(ii)
153
147
300

Here when we add the tens digits 5 and 4 and carry over 1 from the units place, we get 10, so we have carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x (3) does NOT equal to the sum of the hundreds digits of y (1) and z (1).

The first statement implies that there is no carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x is equal to the sum of the hundreds digits of y and z. Sufficient.

The second statement does not provide us with sufficient information about carry over 1 from the tens place to the hundreds place.

Hope it's clear.
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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18
1
16
x = ABC
y = DEF
z = GHI

DEF
+GHI
_____
ABC

Question: Is D + G = A? This is true if there is no carry-over from the tens digits' sum.

1. E + H = B, This means there is no carry over to hundreds position. SUFFICIENT.
2. C + F = I, This means there is no carry over to tens position BUT we do not know if there will be a carry over during the sum of tens. INSUFFICIENT.

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Originally posted by mbaiseasy on 20 Sep 2012, 00:12.
Last edited by mbaiseasy on 15 Jan 2013, 01:23, edited 1 time in total.
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Manager  Joined: 02 Jun 2011
Posts: 51
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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4
7
conty911 wrote:
If x, y, and z are three-digit positive integers and if x = y + z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z ?

(1) The tens digit of x is equal to the sum of the tens digits of y and z.
(2) The units digit of x is equal to the sum of the units digits of y and z.

Is it safe to conclude that the place value of an integer number (represented as a sum of different integers), depends upon only the preceding place value of integers being summed up?

for eg:
x=1000a+100b+10c+1d
y=1000e+100f+10g+1h
z=1000l+100m+10n+1p
if z=x+y then

is l only dependent upon value of b and f or some other parameters also??

Question is demanding 100 digit of Y + 100 digit of Z is equal to 100 digit of X, means there will not be any carryover from the sum of tens digit of Y and Z. therefore from Option 1, sum of tens digit of Y and Z equal to of X means there will not be any carryforward from here to 100 digit of Y and Z. therefore option 1 is sufficient to answer.
Option 2 unit digit sum is equal, will not give any indication whether tens digit will not carryforward any to hundered. therefore this is not sufficient.

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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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1
3
Let x= a b c
y = d e f
z= g h i

x= y+z

1 --> b= e+h which in turn implies c=f+i and so a=d+g (and this implies there is no carry forward in the addition of units and tens place digit of the two numbers)

2--> c=f+i which does not tell us if b= e+h(as there could be a carry forward bcos of this addition to the hundred place)

and so the answer is A
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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2
The concern here is the sum of the tenth digit might have a carryover, so the sum of the hundredth digit on Y & Z might not be equal to X's hundredth digit. So A is the right answer.
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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Let the 3 digit numbers be,
x=ABC
y=DEF
z=GHI

Now, its given that
DEF
+ GHI
_____
ABC
_____

Statement 1---- says that E+H=B. Substitute any digit for E and H, you will find that D+G must be equal to A. Sufficient
Statement2.......says F+I=C. E and H can be anything and in turn D and G can be anything. Not sufficient.
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Posts: 17
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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abhishekkpv wrote:
Let x= a b c
y = d e f
z= g h i

x= y+z

1 --> b= e+h which in turn implies c=f+i and so a=d+g (and this implies there is no carry forward in the addition of units and tens place digit of the two numbers)

2--> c=f+i which does not tell us if b= e+h(as there could be a carry forward bcos of this addition to the hundred place)

and so the answer is A

my confusion is the following regarding (1). maybe i am not reading the question right, but assume the following:

y: 6 4 3
z: 4 4 2

x: 1 0 8 5

so, the tens digit of x is equal to the sum of the tens digit of y + z. however, the hundreds digit, 6 + 4 = 1 0. The hundreds digit of x would be 0. can someone please explain? thank you VP  Status: Far, far away!
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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5
1
mokura wrote:

my confusion is the following regarding (1). maybe i am not reading the question right, but assume the following:

y: 6 4 3
z: 4 4 2

x: 1 0 8 5

so, the tens digit of x is equal to the sum of the tens digit of y + z. however, the hundreds digit, 6 + 4 = 1 0. The hundreds digit of x would be 0. can someone please explain? thank you Your problem is very simple : "x, y, and z are three-digit positive integers".
x cannot be 1085, it must be $$\leq{999}$$

P.S: welcome to GmatClub!
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Joined: 21 Sep 2013
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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For Statement I my problem pertains to the fact that the ten's digit of x will be expressed as ten's digit of y + ten's digit of z.
But if y=190 ,z=190 then x =380 the 1 does carry over from the ten's digit.
Although the ten's digit of x is the sum of ten's digit of y and z.

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Posts: 58445
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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Abheek wrote:
For Statement I my problem pertains to the fact that the ten's digit of x will be expressed as ten's digit of y + ten's digit of z.
But if y=190 ,z=190 then x =380 the 1 does carry over from the ten's digit.
Although the ten's digit of x is the sum of ten's digit of y and z.

The tens digit of y is 9 and the tens digit of z is 9 --> 9+9=18 not 8.

Hope it's clear.
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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Hi Guys,

I thought statement #1 can't be correct alone because (I thought) we must know that BOTH the unit & the tens digits do not carry over:

i.e.

153
147
300

vs

152
147
299

Don't we need to know that both to confirm that the hundreds digit is a 3 rather than 2?
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Joined: 02 Sep 2009
Posts: 58445
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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ColdSushi wrote:
Hi Guys,

I thought statement #1 can't be correct alone because (I thought) we must know that BOTH the unit & the tens digits do not carry over:

i.e.

153
147
300

vs

152
147
299

Don't we need to know that both to confirm that the hundreds digit is a 3 rather than 2?

But in your second example there IS a carry over 1 from the tens place to the hundreds place. No?
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If x, y, and z are three-digit positive integers and if x =  [#permalink]

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Bunuel wrote:
ColdSushi wrote:
Hi Guys,

I thought statement #1 can't be correct alone because (I thought) we must know that BOTH the unit & the tens digits do not carry over:

i.e.

153
147
300

vs

152
147
299

Don't we need to know that both to confirm that the hundreds digit is a 3 rather than 2?

But in your second example there IS a carry over 1 from the tens place to the hundreds place. No?

Ok let me clarify my question:

153
147
300

In this case the hundreds digit became 3 because the unit total 10 --> carries 1 to the tens digit --> tens digit adds to 10, so carries 1 to the hundreds digit. The hundreds digit thus = 3

vs

152
147
299

In this case the hundreds digit remains 2 because the unit total didn't exceed 9 --> nothing carries over to the tens digit (i.e. tens digit remains 9), The hundreds digit thus = remains 2

So don't we need to know the unit digit AND the tens digit to confirm whether the hundreds digit will remain 2 or pushed to 3?

(I'm not sure why I'm just not getting it!!)
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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ColdSushi wrote:
Bunuel wrote:
ColdSushi wrote:
Hi Guys,

I thought statement #1 can't be correct alone because (I thought) we must know that BOTH the unit & the tens digits do not carry over:

i.e.

153
147
300

vs

152
147
299

Don't we need to know that both to confirm that the hundreds digit is a 3 rather than 2?

But in your second example there IS a carry over 1 from the tens place to the hundreds place. No?

Ok let me clarify my question:

153
147
300

In this case the hundreds digit became 3 because the unit total 10 --> carries 1 to the tens digit --> tens digit adds to 10, so carries 1 to the hundreds digit. The hundreds digit thus = 3

vs

152
147
299

In this case the hundreds digit remains 2 because the unit total didn't exceed 9 --> nothing carries over to the tens digit (i.e. tens digit remains 9), The hundreds digit thus = remains 2

So don't we need to know the unit digit AND the tens digit to confirm whether the hundreds digit will remain 2 or pushed to 3?

(I'm not sure why I'm just not getting it!!)

Your first example does NOT satisfy the first statement: the tens digit of x is equal to the sum of the tens digits of y and z. So, it'as not valid. Sorry, cannot explanation any better.
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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1
Ok let me clarify my question:

153
147
300

In this case the hundreds digit became 3 because the unit total 10 --> carries 1 to the tens digit --> tens digit adds to 10, so carries 1 to the hundreds digit. The hundreds digit thus = 3

vs

152
147
299

In this case the hundreds digit remains 2 because the unit total didn't exceed 9 --> nothing carries over to the tens digit (i.e. tens digit remains 9), The hundreds digit thus = remains 2

So don't we need to know the unit digit AND the tens digit to confirm whether the hundreds digit will remain 2 or pushed to 3?

(I'm not sure why I'm just not getting it!!)[/quote]

Your first example does NOT satisfy the first statement: the tens digit of x is equal to the sum of the tens digits of y and z. So, it'as not valid. Sorry, cannot explanation any better.[/quote]

OMG - yes, got it!! :S
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Posts: 4
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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If the 10's digit of x is equal to the sum of the 10's digit of Y and Z, then it implies that there was no carry over from the units digits. Thus statement 2 does not provide any additional information.

In other words, if there IS a carry over from the unit's digits, the 10's digit of x will not equal to the sum of the tens digits of y and z.
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Posts: 4009
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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Top Contributor
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conty911 wrote:
If x, y, and z are three-digit positive integers and if x = y + z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z ?

(1) The tens digit of x is equal to the sum of the tens digits of y and z.
(2) The units digit of x is equal to the sum of the units digits of y and z.

Target question: Is the hundreds digit of x equal to the sum of the hundreds digits of y and z ?

Notice that there are essentially 3 ways for the hundreds digit of x to be different from the sum of the hundreds digits of y and z
Scenario #1: the hundreds digits of y and z add to more than 9. For example, 600 + 900 = 1500. HOWEVER, we can rule out this scenario because we're told that x, y, and z are three-digit integers
Scenario #2: the tens digits of y and z add to more than 9. For example, 141 + 172 = 313.
Scenario #3: the tens digits of y and z add to 9, AND the units digits of y and z add to more than 9. For example, 149 + 159 = 308

Statement 1: The tens digit of x is equal to the sum of the tens digits of y and z.
This rules out scenarios 2 and 3 (plus we already ruled out scenario 1).
So, it must be the case that the hundreds digit of x equals to the sum of the hundreds digits of y and z
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The units digit of x is equal to the sum of the units digits of y and z.
This rules out scenario 3, but not scenario 2. Consider these two conflicting cases:
Case a: y = 100, z = 100 and x = 200, in which case the hundreds digit of x equals the sum of the hundreds digits of y and z
Case b: y = 160, z = 160 and x = 320, in which case the hundreds digit of x does not equal the sum of the hundreds digits of y and z
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent
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Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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Bunuel wrote:
fozzzy wrote:
If x, y, and z are three-digit positive integers and if x = y + z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z ?

(1) The tens digit of x is equal to the sum of the tens digits of y and z.
(2) The units digit of x is equal to the sum of the units digits of y and z.

Is there an alternative approach for this problem?

The question basically asks whether there is a carry over 1 from the tens place to the hundreds place.

Consider the following examples:
(i)
123
234
357

Here when we add the tens digits 2 and 3 there is no carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x (3) is equal to the sum of the hundreds digits of y (1) and z (2).

(ii)
153
147
300

Here when we add the tens digits 5 and 4 and carry over 1 from the units place, we get 10, so we have carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x (3) does NOT equal to the sum of the hundreds digits of y (1) and z (1).

The first statement implies that there is no carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x is equal to the sum of the hundreds digits of y and z. Sufficient.

The second statement does not provide us with sufficient information about carry over 1 from the tens place to the hundreds place.

Hope it's clear.

hi
bunuel
although i have understood your method but in statement 2nd you have written
(ii)
153
147
300 so you are violating the 2nd statement,not sure , see The units digit of x is equal to the sum of the units digits of y and z., but 3+7=10 but the unit digit is 0 of X , so i think we can not use this example , the condition itself not satisfied in the example . although i can be wrong but what i understood i this .

thanks
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Joined: 02 Sep 2009
Posts: 58445
Re: If x, y, and z are three-digit positive integers and if x =  [#permalink]

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nks2611 wrote:
Bunuel wrote:
fozzzy wrote:
If x, y, and z are three-digit positive integers and if x = y + z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z ?

(1) The tens digit of x is equal to the sum of the tens digits of y and z.
(2) The units digit of x is equal to the sum of the units digits of y and z.

Is there an alternative approach for this problem?

The question basically asks whether there is a carry over 1 from the tens place to the hundreds place.

Consider the following examples:
(i)
123
234
357

Here when we add the tens digits 2 and 3 there is no carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x (3) is equal to the sum of the hundreds digits of y (1) and z (2).

(ii)
153
147
300

Here when we add the tens digits 5 and 4 and carry over 1 from the units place, we get 10, so we have carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x (3) does NOT equal to the sum of the hundreds digits of y (1) and z (1).

The first statement implies that there is no carry over 1 from the tens place to the hundreds place, thus the hundreds digit of x is equal to the sum of the hundreds digits of y and z. Sufficient.

The second statement does not provide us with sufficient information about carry over 1 from the tens place to the hundreds place.

Hope it's clear.

hi
bunuel
although i have understood your method but in statement 2nd you have written
(ii)
153
147
300 so you are violating the 2nd statement,not sure , see The units digit of x is equal to the sum of the units digits of y and z., but 3+7=10 but the unit digit is 0 of X , so i think we can not use this example , the condition itself not satisfied in the example . although i can be wrong but what i understood i this .

thanks

Dear nks2611,

Everything highlighted above is general reasoning about the stem and not specifically about the statements...
_________________ Re: If x, y, and z are three-digit positive integers and if x =   [#permalink] 25 Mar 2017, 05:57

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