If x/y = c/d and d/c = b/a…
We are given two equations:
1) x/y = c/d
2) d/c = b/a
Let’s now test the expressions in the Roman numerals to determine which of them are true.
I. y/x = b/a
If we reciprocate the fractions on both sides of the first equation we have: y/x = d/c.
Since the first equation can be expressed as y/x = d/c and the second equation is d/c = b/a, then we see that y/x = b/a.
Thus, the equation in I must be true.
II. x/a = y/b
From Roman numeral I, we know that y/x = b/a.
If we multiply both sides by x, we have: y = (bx)/a. Now divide both sides by b, yielding: y/b = x/a.
Thus, the equation in II must be true.
III. y/a = x/b
From Roman numeral II, we know that x/a = y/b or y/b = x/a. However, this doesn’t mean y/a = x/b. For example, let y = 6, b = 3, x = 4 and a = 2. We see that y/b = x/a (6/3 = 4/2), but y/a ≠ x/b (6/2 ≠ 4/3).
Thus, the equation in III is not true.