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 zSince 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 zCase 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 zSince we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer =
Cheers,
Brent
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