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Re: If x and y are positive integers, what is the ratio of x to y?
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03 Mar 2020, 23:48
In DS questions on linear equations, we need to be careful about not making an assumption that we need as many equations as the number of variables. We need to look for any additional constraints given in the question which may help us solve a linear equation in two variables.
For example, in this question, there is an added constraint on x and y that they have to be positive integers. When used with the data given in the statements, this might just turn out to be sufficient.
Let’s analyse the statements now.
From statement I alone, 4x + y = 8. Whenever you have a linear equation in two variables with one of the terms on the LHS being definitely even and the RHS is also even, we can use the odds and evens concept to solve that linear equation.
In the equation given in statement I, we know that 4x will always be even regardless of the value of x; we also know that 8 is even. Therefore, y HAS TO BE even since Even + Even = Even.
If y = 0, 4x = 8 which violates the condition that x and y are positive integers (remember that 0 is not a positive integer)
If y = 2, 4x = 6 which violates the condition that x and y are positive integers.
If y = 4, 4x = 4 and hence x = 1. This is a possible combination, so let’s hold on to this solution.
If y = 6, 4x = 2 which violates the condition that x and y are positive integers.
If y = 8, 4x = 0 which violates the condition that x and y are positive integers.
Any value of y greater than 10 will make x negative and hence we don’t have to consider those values. From our analysis, we see that we have a unique value of x and y. Hence, statement I alone is sufficient to find the ratio of x and y.
Answer options B, C and E can be eliminated, possible answer options are A or D.
From statement II alone, 4x – y = 0. This means that 4x = y and hence x/y = ¼. Statement II alone is sufficient.
Note that you can also solve this equation by plugging in values based on the same principle that we used with Statement I.
If x = 1 and 4x – y = 0, then y = 4. Ratio of x and y is 1:4.
If x = 2 and 4x – y = 0, then y = 8. Ratio of x and y is 1:4.
If x = 3 and 4x – y = 0, then y = 12. Ratio of x and y is 1:4.
In all cases, we see that the ratio of x and y is 1:4. That’s essentially because of the fact the equation given has been framed in such a way as to give the ratio to be 1:4 for any value of x and y.
Statement II alone is sufficient. Answer option A can be eliminated.
The correct answer option is D.
In a DS question on equations, just because you see a beautiful pair of simultaneous equations does not mean that you have to combine the statements. Do not forget that the rules of DS say that you need to test out the individual statements alone before combining them. When you try to test out the individual statements alone is when you will start looking for additional constraints. Do this on as many equations questions as you can and you will gradually stop falling for traps like these.
Hope that helps!