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Pam owns an inventory of unopened packages of corn and rice

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KarishmaB
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Re: Pam owns an inventory of unopened packages of corn and rice [#permalink] Pam owns an inventory of unopened packages of corn and rice   09 Aug 2023, 01:34
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Aishyk97
KarishmaB
Konstantin1983
I hate such questions because, as in statement 1, you must be sure that such equation doesn't have unique solution. And with such numbers..How come should i guess about 12 and 6? Picking numbers is time-consuming

This question is based on the concept of "integral solutions". Since in such real world examples, x and y cannot be negative or fractional, usually these equations have a finite number of solutions.
After you get one solution, you will quickly know how many solutions the equation has. But getting the first set of values which satisfy the equation requires a little bit of brute force.
But the good thing here is that this is a DS question. You don't need to find the actual solution. The only thing you "need" is to establish that there is a single solution only. (Obviously, there has to be a solution since she does own $282 worth of packages.)

That can be done relatively easily.
First, check out this post for a conceptual discussion on this question type (case 2):
https://www.gmatclub.com/forum/veritas- ... -of-thumb/

Once you understand this, the following will make sense to you.

17x + 13y = 282
If x = 0, y is 21.something (not an integer)
If x = 1, y = 20.something
If x = 2, y = 19.something
If x = 3, y = 17.something

Now you know that there will be only one set of values satisfying this equation. Why? Because y will be less than 17 in the first set of values which satisfy this equation. So if you want to get the next set that satisfies, you will need to subtract it by 17 which will make y negative. So in any case, there will be a unique solution to this equation.
Hence statement 1 is sufficient.

Answer (A)

Hi KarishmaB Bunuel

Can you please explain the highlighted portion to me?
I know many people have provided explanations but I am finding it difficult to reach certainty that there will be only one set of solutions available. How can I be sure without doing tedious calculations?

Aishyk97 - First checkout this post to understand how to find the various possible positive integer solutions to a linear equation in 2 variables.

https://anaprep.com/algebra-integer-sol ... variables/


Now you know how to find the solution to this equation
17x + 13y = 282

The first one will be by hit and trial. So I start by putting x = 0
If x = 0, y is 21.something (not an integer)
If x = 1, y = 20.something (not an integer)
If x = 2, y = 19.something (not an integer)
If x = 3, y = 17.something (not an integer)

Next I will try x = 4, I will get y = 16.something (still not an integer)
and so on I will keep trying till I get a value that works x = 12 and y = 6

How will I get the next value? I will need to add the co-efficient of y to x and subtract the co-efficient of x from y.
So next solution will be 25, -11. Here y is negative because when I subtract 17 from 6, I get a negative value. But then this became true the moment I got y less than 17. Then did I really have to go from y = 17.something to y = 6? Whatever may have been the value of y, since it was going to be smaller than 17, another value of y would not exist if we were to reduce y.
You could also check for x = 14, 15 and 16 to ensure that we cannot go the other way i.e. reduce x and increase y.
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Re: Pam owns an inventory of unopened packages of corn and rice [#permalink] Pam owns an inventory of unopened packages of corn and rice   08 Aug 2024, 07:13
KarishmaB
Konstantin1983
I hate such questions because, as in statement 1, you must be sure that such equation doesn't have unique solution. And with such numbers..How come should i guess about 12 and 6? Picking numbers is time-consuming
This question is based on the concept of "integral solutions". Since in such real world examples, x and y cannot be negative or fractional, usually these equations have a finite number of solutions.
After you get one solution, you will quickly know how many solutions the equation has. But getting the first set of values which satisfy the equation requires a little bit of brute force.
But the good thing here is that this is a DS question. You don't need to find the actual solution. The only thing you "need" is to establish that there is a single solution only. (Obviously, there has to be a solution since she does own $282 worth of packages.)

That can be done relatively easily.
First, check out this post for a conceptual discussion on this question type (case 2):
https://www.gmatclub.com/forum/veritas- ... -of-thumb/

Once you understand this, the following will make sense to you.

17x + 13y = 282
If x = 0, y is 21.something (not an integer)
If x = 1, y = 20.something
If x = 2, y = 19.something
If x = 3, y = 17.something

Now you know that there will be only one set of values satisfying this equation. Why? Because y will be less than 17 in the first set of values which satisfy this equation. So if you want to get the next set that satisfies, you will need to subtract it by 17 which will make y negative. So in any case, there will be a unique solution to this equation.
Hence statement 1 is sufficient.

Answer (A)
­Hi Karishma,
would you please explain it in a little details that how we know for sure that there is only one set of values for a particular eqn. in eqn like these (like in this question) after getting one set of values i'm always confused if I should check for another set or not.
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Re: Pam owns an inventory of unopened packages of corn and rice [#permalink]
08 Aug 2024, 07:13
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