If x/y = 2/5, then which of the following must be true?

We are given that x/y = 2/5, and need to determine what must be true.

I. y + x = 7

This is not true. If x = 2 and y = 5, then x + y = 7; however, if x = 4, and y = 10, then x + y does not equal 7.
(Notice that x and y do not have to equal 2 and 5, respectively; if they were to equal 4 and 10, respectively, the fraction x/y would would equal 4/10, which would reduce to 2/5.)

II. (x + y)/x = 7/2

This is true since:

(x + y)/x = x/x + y/x = 1 + 5/2 = 7/2

(Notice that if x/y = 2/5, then y/x = 5/2.)

III. y^2/x^2 = 25/4

This is true. Recall that in II, we’ve mentioned y/x = 5/2. Thus, y^2/x^2 = (y/x)^2 = (5/2)^2 = 25/4,

Answer: D
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We'll simplify our given data to help us understand the logic.
This is a Precise approach.

I. Since x/y = 2/5 then x = (2/5)y so I is y + (2/5)y = (7/5)y. This doesn't have to equal 7.
Since I is false all options but (D) are eliminated and it is our answer.

Plug in values and check -

I. y + x = 7 Possible only when x = 2 and y = 5
II. (x + y)/x = 7/2 Possible for all values, plug in x = 4 & y = 10 and check
III. y^2/x^2 = 25/4 Possible for all values, plug in x = 4 & y = 10 and check

Thus, options II and III only satisfies, answer will be (D)
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Plugging option has already been explained, Algebraic method would be -

\(\frac{x}{y}=\frac{2}{5}=>\frac{y}{x}=\frac{5}{2}\) -----------------(1)

Add \(1\) to both sides of the equation (1)

\(1+\frac{y}{x}=\frac{5}{2}+1=>\frac{y+x}{x}=\frac{7}{2}\). Hence II must be true

Now, Square both sides of equation (1)

\((\frac{y}{x})^2=(\frac{5}{2})^2=>\frac{y^2}{x^2}=\frac{25}{4}\). Hence III must be true.

We are given a ratio of two variable. we cannot find the sum of the variables from ratio alone. Hence I might not be true

Option D

Given : x/y = 2/5

Check Points : Positive and negative values of x and y.
x/y = (-2).a/(-5).a = 2/5 . Assume x= -2.a and y = -5.a
x/y = (2).a/(5).a = 2/5 . Assume x=2.a and y = 5.a

Now check the answer choices.
Only options 2 and 3 seem feasible.
Hence D is the answer.

x^2/y^2 cant this be equal to -5/2 or 5/-2?
Bunuel

We are given that x/y = 2/5. If we square we'll get x^2/y^2 = 4/25. Thus, III must be true.

If we were given that x^2/y^2 = 4/25 (reverse situation), then x/y = 2/5 won't necessarily be true because x^2/y^2 = 4/25 means that |x|/|y| = 2/5. So, in this case x =+/-2 and y = +/-5 so x/y = 2/5 or x/y = -2/5.

Hence, x/y = 2/5 implies that x^2/y^2 = 4/25 but not vise-versa.

Hope it's clear.
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I solved it this way..

\(x = 2k, y = 5k\)
I.\(y+x = 7k\) (We cannot decide as it depends on k value).
II. \(\frac{(x+y)}{x} = \frac{7k}{2k} = \frac{7}{2}\)
III. \(\frac{y^2}{x^2}= \frac{49k^2}{4k^2} = \frac{49}{4}\)

Hence, only II & III.

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