NoHalfMeasures wrote:
If a, b, c, and d are positive numbers and \(\frac{a}{b} < \frac{c}{d}\) , which of the following must be true?
I. \(\frac{(a+c)}{(b+d) }< \frac{c}{ d}\)
II. \(\frac{(a+c)}{(b+d)} < \frac{a}{b}\)
III. \(\frac{(a+c)}{(b+d)} = \frac{a}{b} + \frac{c}{d}\)
A. none
B. I only
C. II only
D. I and II
E. I and III
As each letter represents a positive number, we can cross multiply to get \(ad < cb\).
I. Cross multiply: \(ad + cd < cb + cd\)
\(ad < cb\)
Statement I must be true.
II. Cross multiply: \(ab + bc < ab + ad\)
\(bc < ad\)
We can't say for certain that this is true.
III. \(\frac{(a+c)}{(b+d)} = \frac{a}{b} + \frac{c}{d}\)
let \(a = 1, b = 2, c = 2, d = 3\)
\(\frac{3}{5} = \frac{1}{2 }+ \frac{2}{3}
\)
Not true.
Only statement that must be true is statement I.
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