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Kaushik786
Joined: 20 May 2018
Last visit: 13 Dec 2023
Posts: 71
Given Kudos: 71
Location: India
Schools: Booth '22 Anderson '22 CBS '22 Johnson '22
GMAT 1: 740 Q50 V41
Updated on: 12 Mar 2021, 20:54
Originally posted by Kaushik786 on 11 Mar 2021, 23:24.
Last edited by chetan2u on 12 Mar 2021, 20:54, edited 1 time in total.
Last edited by chetan2u on 12 Mar 2021, 20:54, edited 1 time in total.
Corrected the question
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P, Q and R are positive single integers. Does the three digit number PQR lie between 100 and 300 ?
1. Q=P+S, R=Q+0.75S, where S is a positive single digit integer.
2. Q=P+T+2, R=Q+3T/2, where T is a positive single digit integer.
1. Q=P+S, R=Q+0.75S, where S is a positive single digit integer.
2. Q=P+T+2, R=Q+3T/2, where T is a positive single digit integer.
12 Mar 2021, 20:53
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Kaushik786
P, Q and R are positive single integers. Does the three digit number PQR lie between 100 and 300 ?
1. Q=P+S, R=Q+0.75. S is a positive single digit integer.
2. Q=P+T+2, R=Q+3/2T. T is a positive single digit integer.
1. Q=P+S, R=Q+0.75. S is a positive single digit integer.
2. Q=P+T+2, R=Q+3/2T. T is a positive single digit integer.
Of course, the statement I is wrong the way it is written.
If R and Q are integers, R=Q+0.75 is not possible. R would not be an integer in this case.
Looking at statement II, I believe the statement I should be
1. Q=P+S, R=Q+0.75S, where S is a positive single digit integer.
Now we are concerned with only the digit in hundreds place, as tens and ones digit can take any values from 0 to 10, while hundreds digit, P, can take only 1 or 2 as values.
\(R=Q+\frac{3S}{4}\)
So S has to be a multiple of 4, that is 4 or 8.
(a) \(S=8.....R=Q+6=(P+S)+6=P+8+6\)
But R is a single digit. So S cannot be 8.
(b) \(S=4.....R=Q+3=(P+S)+3=P+4+3=P+7\)
So, R=P+7. But R is a single digit, so R can be 8 or 9. Thus P will be 1 or 2.
So PQR=1QR or 2QR
In both cases, the integer PQR will lie between 100 and 300
2. Q=P+T+2, R=Q+3/2T. T is a positive single digit integer.
\(R=Q+\frac{3T}{2}\)
So T has to be a multiple of 2, that is 2, 4, 6 or 8.
a) T=2.....R=Q+3=(P+T+2)+3=P+2+2+3=P+7
So, R=P+7. But R is a single digit, so R can be 8 or 9. Thus P will be 1 or 2.
So PQR=1QR or 2QR
In both cases, the integer PQR will lie between 100 and 300
b) T=4.....R=Q+6=(P+T+2)+6=P+4+2+6=P+12
R=P+12.
But R is a single digit. So S cannot be 8.
All other values will make R even bigger than the value in case b.
Thus only case (a) possible.
D
23 Jul 2022, 04:32
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Here is one way to do it (although it took long!)
PQR --> all positive integers, single-digit. This means P,Q,R could be 1,2,3,4,5,6,7,8,9
Is PQR between 100 and 300? This simply translates to whether P is 1 or 2.
Why? Because Q and R are never zero, so when P = 1 or 2, PQR minimum = 111, PQR maximum = 299
Let's look at the statements now.
Statement #1
Q = P+S
R = Q + 0.75S
We want to know what values P will take. So, use these equations --
P = Q - S = R - 0.75S - S = R - 1.75S
For P to be an integer, R - 1.75S must be an integer. So, 1.75S must be an integer. 1.75S = 7S/4. This takes an integer values only when S = 4 or 8
At S = 4, P = R - 7 --> P can be 1 or 2 because R can be 8 or 9 for P to be > 0 and single-digit.
At S = 8, P = R - 14 --> here, R will have to be greater than 14 for P to be a positive integer. That is not possible!
P = 1 or 2. Statement is sufficient.
Statement #2
The approach is same as that for statement 1.
Q = P+T+2 --> P = Q-T-2 = (R-3T/2)-T-2 = R - 5T/2 - 2
Again, to get integer values for P, 5T/2 must be an integer. T = 2,4,6,8.
At T = 2, P = R - 7. Same situation as statement #1. P = 1 or 2.
For T = 4,6,8, R cannot be single-digit.
Sufficient.
Answer = D
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Edit: Just saw that another similar solution is already posted. I hope this helps in any case!
PQR --> all positive integers, single-digit. This means P,Q,R could be 1,2,3,4,5,6,7,8,9
Is PQR between 100 and 300? This simply translates to whether P is 1 or 2.
Why? Because Q and R are never zero, so when P = 1 or 2, PQR minimum = 111, PQR maximum = 299
Let's look at the statements now.
Statement #1
Q = P+S
R = Q + 0.75S
We want to know what values P will take. So, use these equations --
P = Q - S = R - 0.75S - S = R - 1.75S
For P to be an integer, R - 1.75S must be an integer. So, 1.75S must be an integer. 1.75S = 7S/4. This takes an integer values only when S = 4 or 8
At S = 4, P = R - 7 --> P can be 1 or 2 because R can be 8 or 9 for P to be > 0 and single-digit.
At S = 8, P = R - 14 --> here, R will have to be greater than 14 for P to be a positive integer. That is not possible!
P = 1 or 2. Statement is sufficient.
Statement #2
The approach is same as that for statement 1.
Q = P+T+2 --> P = Q-T-2 = (R-3T/2)-T-2 = R - 5T/2 - 2
Again, to get integer values for P, 5T/2 must be an integer. T = 2,4,6,8.
At T = 2, P = R - 7. Same situation as statement #1. P = 1 or 2.
For T = 4,6,8, R cannot be single-digit.
Sufficient.
Answer = D
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Edit: Just saw that another similar solution is already posted. I hope this helps in any case!
