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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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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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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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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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 !

I used real values and came to the answer. Used 1/2 and 3/4.

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
Arunabh saxena

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.

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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Pick numbers... noticing the LHS is the same for all 3 roman numerals.
a=1/b=1 < c=2/d=1

I. 1+2/1+1 < 2/1
3/2 < 2, TRUE

II. 3/2 < 1, FALSE

III. 3/2 = 3, FALSE

Pick some other numbers to retest I.
a=1/b=3 < c=2/d=3
1+2/3+3 < 2/3
3/6 < 2/3
1/2 < 2/3, TRUE
Good enough, B it is.

I solved it by taking convenient values.
Lets take a= 4 ; b=2 ; c=12 ; d = 3

c/d = 4 which is greater than a/b=2

I must be true as ,
(a+c) /( b+d) = 16 / 5 = 3.2
c/d = 4
So (a+c) /( b+d) < c/d

II may not be true ,

(a+c) /( b+d) = 16 / 5 = 3.2
a/b = 2
2 is not greater than 3.2

III may not be true ,

(a+c) /( b+d) = 16 / 5 = 3.2

a/b + c/d = 2 +4 =6

Only I must be true.
B is the answer.

Please give me Kudos if you liked my explanation.

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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