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by Mo2men » Tue Dec 06, 2016 1:12 pm
P and Q are both positive integers. When P is divided by Q, the remainder is some positive integer D, and when P is divided by (Q + 3), the remainder is also D. If P/Q = 1020.75 and P/(Q + 3) = 816.6, then which of the following gives the correct set of {D, Q}?

A. {6, 12}
B. {6, 15}
C. {9, 12}
D. {9, 15}
E. {15, 24}

OA : C

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by [email protected] » Tue Dec 06, 2016 9:39 pm
Hi Mo2men,

This is a layered question, but it has some built in Number Properties that you can take advantage of. Since the answer choices are numbers, we can use them along with the NPs to TEST THE ANSWERS.

While we're given a lot of information to work with, I want to start with two of the facts:
1) P and Q are POSITIVE INTEGERS
2) P/Q = 1020.75

From the answer choices, we know that Q is 12, 15 or 24. In the above fraction, we divide an integer by another integer and get a number that ends in .75 (and that can be rewritten as 3/4). Working 'backwards' from Q to P, we need the Q to be a number that eliminates the fraction so that P becomes an integer. Given the three options, Q would have to be either 12 or 24 - since (12)(3/4) = 9 and (24)(3/4) = 18.

It's interesting how that first example involves a 12 and a 9 - those are the numbers that occur in Answer C, so I'm going to TEST that Answer first...

IF....
D=9 and Q=12
how would that 'mesh' with everything we were told...

P/12 = ? r 9
P/15 = ? r 9
P/12 = 1020.75
P/15 = 816.6

With those last two fractions, we can create two equations that are set equal to P...
P = (1020.75)(12)
P = (816.6)(15)

So, are these two values equal to one another? To make the math steps 'smaller', I've factored out a 3...

(1020.75)(4)(3) = (4083)(3) = 12,249
(816.6)(5)(3) = (4083)(3) = 12,249

The values of P are the same. Given the complexity of the question, this is probably the correct answer, but we can double-check it against the other two pieces of information that we were given...

Will 12,249/12 have a remainder of 9 and will 12,249/15 also have a remainder of 9?

12,249/12 = 1020 r 9
12,249/15 = 816 r 9

This fits everything that we were told, so this MUST be the answer.

Final Answer: C

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by crackverbal » Wed Dec 07, 2016 10:21 pm
Hi Mo2men,

The best way to approach remainder questions of this type is to use the remainder formula

Dividend(D) = divisor(d) * quotient(q) + remainder(r) ------> D = dq + r

Before we solve the problem lets look at how we can represent decimal numbers in the form of a quotient and remainder. If 7 is divided by 5 then clearly the quotient is 1 and remainder is 2 and the decimal equivalent of 7/5 is 1.4.

Now the portion to the left of the decimal point always represents the quotient, so the 1 in 1.4 is the quotient. The portion to the right of the decimal point i.e. 4 does not represent the remainder but can be used to find the remainder. 0.4 when multiplied by the denominator 5 will give us the remainder 2. So when we consider a decimal, the portion to the right of the decimal point * the divisor will always give us the remainder.

Now considering the question, we are given that P/Q = 1020.75, P here is the Dividend(D) and Q is the divisor(d).

0.75 * Q = D(remainder)

We are also given that P/(Q+3) = 816.6

0.6 * (Q + 3) = D (remainder)

Now equating the left hand sides of the above equations we get

0.75Q + 0.6(Q+3)

Solving we get Q = 12. Since 0.75 * Q = D -----> D = 9.

OA : C

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by GMATGuruNY » Thu Dec 08, 2016 3:48 am
Mo2men wrote:P and Q are both positive integers. When P is divided by Q, the remainder is some positive integer D, and when P is divided by (Q + 3), the remainder is also D. If P/Q = 1020.75 and P/(Q + 3) = 816.6, then which of the following gives the correct set of {D, Q}?

A. {6, 12}
B. {6, 15}
C. {9, 12}
D. {9, 15}
E. {15, 24}
P/Q = 120.75 = 12075/100 = 4083/4.
Since P/Q = 4083/4, P must be a MULTIPLE OF 4083 and Q must be a MULTIPLE OF 4.

P/(Q+3) = 816.6 = 8166/10 = 4083/5.
Since P/(Q+3) = 4083/5, P must be a MULTIPLE OF 4083 and Q+3 must be a MULTIPLE OF 5.

Thus:
Q is a multiple of 4.
Q+3 is a multiple of 5.

Look at the answer choices.
Of the options for Q -- 12, 15 and 24 -- only 12 is a multiple of 4 such that adding 3 will yield a multiple of 5.
Thus, Q=12.
Eliminate B, D and E.

Since P/Q = 4083/4 = 12249/12, P=12249 and Q=12.
12249/12 = 1020 R9.
Thus, D=9.

The correct answer is C.
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by Matt@VeritasPrep » Thu Dec 08, 2016 7:45 pm
Another approach:

P = 1020.75 * Q

P = 816.6 * Q + 816.6 * 3

Since P = P, the two right hand sides also equal each other:

1020.75Q = 816.6Q + 2449.8

204.15Q = 2449.8

Q = 12 (use approximation: 200 * 12 = 2400, so that must be our integer)

From there, notice that P = 1020.75 * 12 = 12249 = 12000 + 240 + 9. The first two terms (12000 and 240) are divisible by 12, so the remainder is 9, and the answer is C.
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by Matt@VeritasPrep » Thu Dec 08, 2016 7:49 pm
Let me add a really spiffy quick way:

Since P = 1020Q + 0.75Q, we can surmise that the remainder, or D, = 0.75Q.

From there, look for answers in which D = 0.75Q. In A, 6 = 50% of 12, in B, 6 = 40% of 15, in D, 9 = 60% of 15, and in E, 15 = 62.5% of 24.

Those must be wrong, so only C, in which 9 = 75% of 12, can be right! :D
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