Source: Official Guide for the GMAT 13th Ed. Data Sufficiency; #133 Official Guide for the GMAT 2015 14th Ed. Data Sufficiency; #133

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# If x, y, and z are three-digit

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?

### 2 Explanations

1

Varun S Nair

Statement 2 is understandable; but I'm sorry, but I don't see how statement 1 is sufficient. The prompt just states that the tens digit of X is the sum of the tens digits of the two numbers Y & Z.

This does not necessarily have to mean that there is no carrying digit = just that the sum of digits is correct (as I understand, the prompt just specifies about the DIGIT, nothing about the FINAL SUM VALUE).

Does the prompt mean that the sum itself is equal and that there is NO carry at all? Please correct me if I'm wrong.

Sep 16, 2019 • Comment

Okay, so let's pretend that x = 300, y = 200 and z = 100. In this case, clearly, the hundreds digit of x is equal to the sum of the hundreds digits of y and z. So when is the initial prompt false?

Well, if the sum of the tens digits of y and z is 10 or more, then the 1 will carry to the hundreds place. In other words, if x = 300, y = 160 and z = 140, then the hundreds digit of x is not equal to the sum of the hundreds digit of y and z.

But note that, in this example, the sum of the tens digits of y and z is also not equal to the tens digit of x. Indeed, in order to carry over a 1 from the tens place to the hundreds place, the sum of the tens place of y and z must be 10 or more. But if this is true, then the sum of the tens digits of x and y cannot be equal to the sum of the tens digit of x, because the tens digit of x must be less than 10 (since it is just one digit).

So, in order for the sum of the hundreds digits of y and z to be different from the hundreds digit of z, the sum of the tens digits of y and z must be greater than 10. Statement 1 eliminates this possibility, and is thus sufficient. Mike McGarry, Magoosh Tutor