Data Sufficiency

magpie16 wrote:For nonnegative integers x and y, what is the remainder when x is divided by y?

(1) x/y = 13.8
(2) The numbers x and y have a combined total of less than 5 digits.

Target question: What is the remainder when x is divided by y?

Statement 1: x/y = 13.8
In other words, x/y = 13 4/5
Or . . . x/y = 69/5

At this point, we can see that there are several pairs of values that meet this condition. Here are three:
Case a: x = 69 and y = 5, in which case the remainder is 4 when x is divided by y
Case b: x = 138 and y = 10, in which case the remainder is 8 when x is divided by y
Case c: x = 1380 and y = 100, in which case the remainder is 80 when x is divided by y
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The numbers x and y have a combined total of less than 5 digits.
There are several pairs of values that meet this condition. Here are two:
Case a: x = 2 and y = 2, in which case the remainder is 0 when x is divided by y
Case b: x = 3 and y = 2, in which case the remainder is 1 when x is divided by y
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
Statement 1 tells us that x and y must be such that x/y = 69/5, so x and y could equal 69 and 5, or 138 and 10, or 207 and 15, or 276 and 20, etc . . .
Statement 2 tells us that the number pair must have a total of 2, 3 or 4 digits.
Only the first pair of values meets this second criterion, so it must be the case that x = 69 and y = 5
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent

satinder kaur wrote:Hi Bret,

I couldn't follow the step three where we consider both of the options. Can you please make it more clear/illustrate more.

Statement 2 tells us that the number pair must have a total of 2, 3 or 4 digits.
Only the first pair of values meets this second criterion, so it must be the case that x = 69 and y = 5

Thanks

Statement 1 tells us that x and y must be such that x/y = 69/5, so x and y could equal 69 and 5, or 138 and 10, or 207 and 15, or 276 and 20, etc . . .

Let's consider each possibility:

Case a: x = 69 and y = 5. We have a total of 3 digits (2 digits in 69, and 1 digit in 5)
Case b: x = 138 and y = 10. We have a total of 5 digits
Case c: x = 1380 and y = 100. We have a total of 7 digits

Statement 2 tells us that the number pair must have a total of 2, 3 or 4 digits.
Case b gives us a total of 5 digits and case c gives us s total of 7 digits. Since these cases break the condition in statement 2, case a is the only possible case.

I hope that helps.

Cheers,
Brent

magpie16 wrote:For nonnegative integers x and y, what is the remainder when x is divided by y?

(1) x/y = 13.8
(2) The numbers x and y have a combined total of less than 5 digits.]

We need to determine the remainder when x is divided by y.

Statement One Alone:

x/y = 13.8

We can rewrite statement one as:

x/y = 13 + 8/10

There are infinite possible values for the remainder when x is divided by y. That is because the remainder is the numerator of any equivalent fraction of 8/10. For example, the remainder can be 8 if x = 138 and y = 10, or it can be 16 if x = 276 and y = 20 (note: 276/20 = 13 + 16/20 = 13.8). Thus, statement one is not sufficient to answer the question.

Statement Two Alone:

The numbers x and y have a combined total of less than 5 digits.

Statement two does not provide enough information to answer the question. For example, x can be 3 digits and y can be 1 digit, or x can be 2 digits and y can be 2 digits. Without knowing the exact values of x and y, statement two is not sufficient to answer the question.

Statements One and Two Together:

From statement one, we know that x/y = 13.8 or x = 13.8y. Since x is at least 10 times as much as y, x has either one or two more digits than y. For example, if y = 10, then x = 138 (so x has one more digit than y); if y = 80, then x = 1,104 (so x has two more digits than y). Combining statement one with statement two, we see that either x has 3 digits and y has 1 digit or x has 2 digits and y has 1 digit.

Recall that x = 13.8y, and if y has 1 digit and x has 3 digits, a 1-digit number times 13.8 cannot yield a 3-digit whole number. Therefore, x must have 2 digits and y must be 1 digit. In that case, the only way to satisfy x = 13.8y is if y = 5 and x = 13.8(5) = 69. Hence, the remainder when x is divided y is 4 (69/5 = 13 + 4/5).

Answer: C