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lh.abhishek
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Solving fom single equation using interger constraint

by lh.abhishek Sun Jul 16, 2017 10:50 am



Hi, I know that in some cases of algebra that is an integer constraint and you can get the value of 2 unknown variables using just 1 equation.

I was trying to solve official DS questions and finally, this equation popped up.

15X + 18Y = 38,700

Now, on the face of I should move on, but on the back of my mind, I was unsure if this was the same case. How do i know that this cannot be solved?
Sage Pearce-Higgins
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Re: Solving fom single equation using interger constraint

by Sage Pearce-Higgins Fri Jul 28, 2017 12:05 pm

Firstly, good point about 'moving on'. In every test you need to make good decisions about where to spend your time, so the best thing for you may be just to guess and move on.

GMAT is pretty clever in that it doesn't give easy hints. You can't assume that this equation is solvable with integers just because you've seen some others that are. It might be a bluff!

In any case, 15X + 18Y = 38,700 can be solved with X = 2400, and Y = 150 (there are other solutions available, as 90 is a common multiple of 15 and 18). It took me 3 minutes to find this case!
JbhB682
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Re: Solving fom single equation using interger constraint

by JbhB682 Thu Nov 28, 2019 8:05 pm

Hi Sage -- so if i understand -- you suggest taking the risk for falling for the C-trap

If i understand you correctly, It is just not worth the extra time to calculate for EVERY equation with two variables to see if there is an integer constraint or not

------------------------------

Also -- just out of curiosity -- could you go over how you came up with

-- x = 2400 and y = 150

Thank you !
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Re: Solving fom single equation using interger constraint

by Sage Pearce-Higgins Mon Dec 02, 2019 5:27 am

I'm not suggesting that for all situations! That equation is particularly cumbersome, and guessing may be appropriate here. Perhaps it's a good example of the fact that good GMAT test-takers are rarely 100% sure of an answer, but choose an answer based on the evidence that they have.

As for finding that solution, I don't remember as it was 2 and a half years ago. Try it yourself. It's usually a good idea to list out some multiples of, say, 15 and 18 to see if you can find how they piece together.
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Re: Solving fom single equation using interger constraint

by JbhB682 Mon Dec 02, 2019 10:08 pm

15x + 18 y = 38,700

Following are my notes when I attempt to AVOID the C - trap (ensure there are no integer constraints)

Step 1) tabulate to be organized

Step 2) confirm if x and y can be integers or not (if non integers allowed for x and y -- immediately mark the equation as Insufficient)

If x and y have to be integers then

Step 3) test extremes to see if you get different integers for x and y
3a) mark x = 0 and check if 387*100 is divisible by 9 and 2
3b) mark y = 0 and check if 387*100 is divisible by 5 and 3

Step 4) look to see if L.C.M between 15 and 18 is divisible by 387*100

Step 5) IF varying integers for x and y from 3a / 3b and 4 -- mark as insufficient.

If only one integer for x and y from 3a/ 3b and 4 ..mark as sufficient and move on

Thoughts ?
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Re: Solving fom single equation using interger constraint

by JbhB682 Mon Dec 02, 2019 10:10 pm

Following is some theory i read online about this

In general, when you have two variables and one equation you will not have enough data to solve. There is at least one important exception:

✔ If your equation is of the form Ax + By = C where A, B and C are known, and
✔ if your unknowns are positive integers, and
✔ if C < the least common multiple of A & B, then the equation has only 1 solution!

Example :
Since the following equation (13c + 15b = 142) meets all three criteria I can tell at a glance that it is SUFFICIENT.
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Re: Solving fom single equation using interger constraint

by Sage Pearce-Higgins Wed Dec 04, 2019 5:25 am

That's interesting - I'd not thought about these constraints, particularly c < least common multiple of A and B. I guess that could be useful, however I think it's important to remember that we're on the fringes of GMAT here. The chance of using this rule in a GMAT test is, I would say, less than 10%.

However, simply being aware of the idea that sometimes a single equation with two variables is solvable is useful. From my experience, I would add some more categories:
-Integer constraints, as you mentioned.
-Equations which simplify. To take a crude example, x + y = 3 + y is solvable for x.
-Problems for which the question is a COMBO (there's a whole part in the Algebra section of our books about these).
-Exponent equations, especially when prime numbers are involved.
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Re: Solving fom single equation using interger constraint

by JbhB682 Wed Dec 11, 2019 12:34 pm

Hi Sage

-- could you give me an example of what you meant by exponent equation ?

Thank you
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Re: Solving fom single equation using interger constraint

by Sage Pearce-Higgins Fri Dec 13, 2019 2:27 am

Find integers x and y in the following equation: 432 = (2^x)(3^y)
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Re: Solving fom single equation using interger constraint

by JbhB682 Thu Dec 19, 2019 9:43 am

Hi Sage -- just wondering what about something like this

If Length and Breadth can be non-integers

-- S1: (Length)^2 + (Breadth)^2 = 100 &
-- S2: (Length) * (Breadth) = 48

What is the length of the rectangle

My first reaction to seeing this was there are MULTIPLE values of length / breadth when accounting for both equations (w/o putting pen to paper) - i would select E in this case

Just wondering if that is your reaction
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Re: Solving fom single equation using interger constraint

by Sage Pearce-Higgins Sat Dec 28, 2019 3:47 am

I think that you're referring to DS 48 from the Diagnostic test of the OG 2020. Here's the kind of thinking I'd encourage you to have:

-There are two equations and two unknowns here, so that it's likely you could solve these, unless there's some sort of special situation.
-There's a number squared, suggesting that there could be a positive and negative solution. However, since we're dealing with the side lengths of a rectangle, we don't need to worry about negative solutions.
-Putting the two equations together would give a quadratic that might have two positive solutions. However, that makes sense as the two possible solutions will represent two rectangles that simply have their length and width reversed. So, sure, you'll get two possible values for the length, but only one possible value for the perimeter.
-Even though there's no integer constraint in this problem, know that GMAT does favour integer solutions to problems.

Try solving these equations (a good study task, but not something to do in a test situation) and you'll hopefully get a better idea of how things work.