Data Sufficiency

TheAnuja55 wrote:If x, y and z are three-digit positive integers and if x=y+z, is the hundreds digit of x equal to sum of the hundreds digit 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.

Let us assume that x = abc, y = def, and z = pqr
x = y + z implies abc = def + pqr
Hundreds digit of x = a and hundreds digit of y and z are d and p respectively
We have to find if a = d + p or not.

(1) The tens digit of x is equal to the sum of the tens digits of y and z.
Tens digit of x = b, tens digit of y and z are e and q respectively.
Then b = e + q, which implies c = f + r and a = d + p (there is no carry forward in the addition of units and tens place digit of two numbers); SUFFICIENT.

(2) The units digit of x is equal to the sum of the units digits of y and z.
Units digit of x = c, units digits of y and z are f and r respectively.
So, c = f + r but this does not imply for sure that b = e + q and hence a = d + p or not (as there can be a carry forward to the hundreds place); NOT sufficient.

The correct answer is A.

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