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Three-digits (OG)

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Anitochka Newbie | Next Rank: 10 Posts Default Avatar
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01 Aug 2017
Posted:
4 messages

Three-digits (OG)

Post Sat Aug 12, 2017 6:26 am
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.

Answer: A

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Post Sat Aug 12, 2017 7:28 am
Quote:
If x, y, and z are three-digit positive integers and if x = y+z, is the hundreds digits 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 z is equal to the sum f the units digits of y and z.
Let y = 2BC, z = 2EF, and x = HTU, so that the addition looks as follows:

2BC
2EF
HTU

When will it be true that H ≠ 2+2?
When we have to CARRY A 1 FROM THE TENS PLACE TO THE HUNDREDS PLACE.
To illustrate:

259
249
508

Here, because we have to carry a 1 from the tens place to the hundreds place, H = 2+2+1 = 5.

Question rephrased:

ABC
DEF
HTU

In the addition problem above, do we have to a carry a 1 from the tens place to the hundreds place?

Statement 1: The tens digit of x is equal to the sum of the tens
digits of y and z.

Since T = B+E, there is no need to carry a 1 to the hundreds place.
SUFFICIENT.

Statement 2: The units digit of x is equal to the sum of the
units digits of y and z.

Since U = C+F, we do not need to carry a 1 from the UNITS PLACE to the TENS PLACE.
But it cannot be determined whether we have to carry a 1 from the TENS PLACE to the HUNDREDS PLACE.
If T = B+E = 0+0 = 0, then there is no need to carry a 1 to the hundreds place:
If T = B+E = 9+9 = 18, then we must carry a 1 to the hundreds place.
INSUFFICIENT.

The correct answer is A.

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Post Sat Aug 12, 2017 6:30 am
Quote:
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 & 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

Answer = A

Cheers,
Brent

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