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If a, b, c, and d are positive numbers and a/b < c/d , which of the fo

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If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 18 Dec 2015, 07:33
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A
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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

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If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 18 Dec 2015, 07:36
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
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If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 18 Dec 2015, 07:54
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NoHalfMeasures wrote:
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


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.
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If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 18 Dec 2015, 08:04
Thanks for your reply.

But can you also comment on the following-
St3) We can't prove equality from inequality. Hence st3 can't be 'must be true'. Is my logic correct specifically for st3?
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Re: If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 18 Dec 2015, 08:16
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NoHalfMeasures wrote:
Thanks for your reply.

But can you also comment on the following-
St3) We can't prove equality from inequality. Hence can't be 'must be true'. Is my logic correct specifically for st3?


No this is a very broad statement that you are mentioning. Take a simpler example: If I say x<3 and then ask whether x =2 is true ? You will say "yes" but here you used the inequality to arrive at your answer for an equality.

I have shown 1 way of using particular values to see that III is NOT must be true.

The other way is to assume that (a+c) / (b+d) = a/b + c/d is indeed true.

Then you end up getting \(b^2*c+a*d^2=0\) ---> \(-(a/c)= b^2/d^2\) but this can not possible as a and c are given to be of the same sign (both are positive) and as such this will make -(a/c) a negative quantity.

Additionally, you also know that \(b^2/d^2\) can never be <0 . Thus you have this contradiction, making this statement not a must be true.

Hope this helps.
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Re: If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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Bunuel I am very confused why 3rd statement is not correct. Here is my take on it -

(a+c) / (b+d) = a/b + c/d

Taking LCM -
so, (a+c)/(b+d)= (ad+cb)/(bd)

cross multiplying -
abd + bcd =abd +bcd, which is true. So the answer should be E. Can you please let me know why this approach is wrong. Thanks for all your help !
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Re: If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 27 Mar 2017, 22:14
I used real values and came to the answer. Used 1/2 and 3/4.
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If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 06 Nov 2017, 08:17
vtomar20 wrote:
bumpbot wrote:
Hello from the GMAT Club BumpBot!



I am very confused why 3rd statement is not correct. Here is my take on it -

(a+c) / (b+d) = a/b + c/d

Taking LCM -
so, (a+c)/(b+d)= (ad+cb)/(bd)

cross multiplying -
abd + bcd =abd +bcd, which is true. So the answer should be E. Can you please let me know why this approach is wrong. Thanks for all your help !


Hi vtomar20

I see a problem in your algebra.

Even i did the same mistake and eventually found what i was doing wrong.

When you simplify the expression:

(a+c) / (b+d) = a/b + c/d

you will get

abd + bcd =abd +bcd + ad^2 + cb^2

and this however is not true, since in the above expression we have an extra ad^2 + cb^2 on the right side which will make the right side bigger (and we know this because all the a, b, c, and d, are positive numbers ).

so E is out and B is the answer.

I hope its clear :-)

Thankyou
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Re: If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 11 Jan 2018, 20:55
a/b<c/d
we multiple two side by bd. remember all number are positive.
ac<cb.

this simple multiply make the fraction from into factors form. this is key to do inequality problem.
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Re: If a, b, c, and d are positive numbers and a/b < c/d , which of the fo  [#permalink]

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New post 23 Jul 2018, 00:57
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A good idea to eliminate III is if you look at option 1 it tells (a+c)/(b+d)<c/d, which we have already proved to be true.

Now go to option III (a+c) / (b+d) = a/b + c/d , which denotes (a+c) / (b+d) > c/d hence it is false as we have already proved I to be true !!
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Re: If a, b, c, and d are positive numbers and a/b < c/d , which of the fo &nbs [#permalink] 23 Jul 2018, 00:57
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