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18 Dec 2015, 07:33
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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
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
18 Dec 2015, 07:36
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a/b<c/d implies
ad<bc
Consider 1) a+c/b+d < c/d is true
then, cross multiplying
ad+dc<bc+cd
thus, ad<bc which is true and hence 1) is true.
Consider 2) a+c/b+d < a/b
implies
ab+bc<ba+ad
bc<ad
which we know to be false
3) We can't prove equality from inequality. Hence can't be 'must be true'. Bunuel, is my logic correct in here?
Hence correct answer is B
ad<bc
Consider 1) a+c/b+d < c/d is true
then, cross multiplying
ad+dc<bc+cd
thus, ad<bc which is true and hence 1) is true.
Consider 2) a+c/b+d < a/b
implies
ab+bc<ba+ad
bc<ad
which we know to be false
3) We can't prove equality from inequality. Hence can't be 'must be true'. Bunuel, is my logic correct in here?
Hence correct answer is B
General Discussion
ENGRTOMBA2018
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18 Dec 2015, 07:54
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NoHalfMeasures
If a, b, c, and d are positive numbers and a/b < c/d , which of the following must be true?
I. (a+c) / (b+d) < c/ d
II. (a+c) / (b+d) < a/b
III. (a+c) / (b+d) = a/b + c/d
a) none
b) I only
c) II only
d) I and II
e) I and III
Pls press kudos if you like the question and would like me to discuss more such questions
I. (a+c) / (b+d) < c/ d
II. (a+c) / (b+d) < a/b
III. (a+c) / (b+d) = a/b + c/d
a) none
b) I only
c) II only
d) I and II
e) I and III
Pls press kudos if you like the question and would like me to discuss more such questions
Your method is correct as well as all the 4 numbers are positive.
This question can be tackled easily by the method below:
You are given a/b < c/d ---> as all the numbers are positive you can cross multiply to get , a/c < b/d ---> (a/c) + 1 < (b/d) + 1 ---> (a+c)/c < (b+d)/d ---> (a+c) / (b+d) < c/ d which is statement I and is hence true.
Now, again consider a/b < c/d ---> d/b < c/a ---> (d/b) + 1 < (c/a) + 1 ---> (d+b)/b < (c+a)/a ---> a/b < (c+a)/(b+d) but this is not what statement II mentions, making statement II false.
Finally, for (a+c) / (b+d) = a/b + c/d , you can clearly prove this wrong by taking a,b,c,d as 1,2,3,4. So this is also not true.
For a MUST BE TRUE question, you need to get a yes ALWAYS.
Thus, only statement I is true , making B the correct answer.
Hope this helps.
