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Intern  Joined: 08 Jan 2013
Posts: 31
If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 63% (02:21) correct 37% (02:19) wrong based on 498 sessions

### HideShow timer Statistics If a, b, c and d are positive integers 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

Doh! I've seen my mistake now... Math Expert V
Joined: 02 Sep 2009
Posts: 56256
Re: If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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6
8
If a, b, c and d are positive integers 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

Since the numbers are positive we can safely cross-multiply. So, we are given that $$ad<bc$$.

I. $$\frac{a+c}{b+d} < \frac{c}{d}$$ --> $$ad+cd<bc+cd$$--> $$cd$$ cancels out: $$ad<bc$$. This is given to be true.

II. $$\frac{a+c}{b+d} < \frac{a}{b}$$ --> $$ab+bc<ab+ad$$ --> $$ab$$ cancels out: $$bc<ad$$. Opposite what is given, thus this option is not true.

III. $$\frac{a+c}{b+d} = \frac{a}{b} + \frac{c}{d}$$. From I we already know that $$\frac{a+c}{b+d} < \frac{c}{d}$$, thus when we add a positive value ($$\frac{a}{b}$$) to $$\frac{c}{d}$$ we make the right hand side even bigger. Thus $$\frac{a+c}{b+d} = \frac{a}{b} + \frac{c}{d}$$ cannot be true.

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Director  Joined: 07 Aug 2011
Posts: 518
GMAT 1: 630 Q49 V27 Re: If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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1
stormbind wrote:
If a, b, c and d are positive integers 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

Doh! I've seen my mistake now... a/b < c/d ; since all are positive ; a/c< b/d ; add 1 to both sides ; (a+c)/c < (b+d)/d; so we have (a+c)/(b+d) < c/d;

I. (a+c)/(b+d) < c/d -YES , we proved it above.
II. (a+c)/(b+d) < a/b - we know a/b < c/d and we also know (a+c)/(b+d) < c/d; does that mean it is always the case that (a+c)/(b+d) < a/b < c/d. cannot say with certainty .
III. (a+c)/(b+d) = a/b + c/d ; Given (a+c)/(b+d) < c/d ; adding a/b which is +ive will make the RHS more bigger . so YES.
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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1
Hi All,

This Roman Numeral question can be solved with a mix of Algebra and TESTing VALUES. Also notice that the three Roman Numerals have the SAME fraction on the "left side" - this means that you don't have to do calculations 3 times, you just have to do them ONCE and put them in all 3 Roman Numerals.

We're told that A, B, C and D are all POSITIVE and that A/B < C/D. We're asked which to the Roman Numerators MUST be true (which is the same as saying "which of these is ALWAYS TRUE no matter how many different examples you try?").

Since the variables are all POSITIVE, we can cross-multiply the inequality...

A/B < C/D

It's also worth noting that this inequality has a couple of great "weak spots": You can make the A, B and D the same value and make the C really BIG. You can also make the A, C and D the same value and make the B really BIG.

With those ideas in mind, I'm going to TEST VALUES.

IF....
A = C = D = 1
B = 100

(A+C)/(B+D) = 2/101

I. Is 2/101 < C/D = 1/1?
This is NOT enough proof to say that this is ALWAYS TRUE though....

II. Is 2/101 < A/B = 1/100?
ELIMINATE Roman Numeral II. Eliminate Answers C and D.

III. Is 2/101 = A/B + C/D = 1/100 + 1/1?
ELIMINATE Roman Numeral III. Eliminate Answer E.

We're now down to just Roman Numeral I. Notice how it has the came fraction (C/D) as the inequality from the prompt. That is INTERESTING. Let's see what happens when we cross-multiply the inequality in Roman Numeral I.....

(A+C)/(B+D) < C/D

D(A+C) < C(B+D)
AD + CD < BC + CD

The "CD"s cancel...

This is what we PROVED early on in the work. Roman Numeral I MUST be TRUE.

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Rich
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Director  Joined: 07 Aug 2011
Posts: 518
GMAT 1: 630 Q49 V27 Re: If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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We have to keep in mind the time constraint we have .

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Manager  Joined: 03 May 2013
Posts: 67
Re: If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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can please somebody explain where in m wrong here

from stem a/b <c/d = d/b < c/a
d+b /b < c+a /a
d+b/c+a < b/a

a+c/b+d > a/b form here we can deduce that 1 is true
Math Expert V
Joined: 02 Sep 2009
Posts: 56256
Re: If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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vipulgoel wrote:
can please somebody explain where in m wrong here

from stem a/b <c/d = d/b < c/a
d+b /b < c+a /a
d+b/c+a < b/a

a+c/b+d > a/b form here we can deduce that 1 is true

Please check the options. Expression in red proves that option II ((a+c)/(b+d) < a/b) is NOT true.
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Manager  Joined: 03 May 2013
Posts: 67
If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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Bunuel wrote:
vipulgoel wrote:
can please somebody explain where in m wrong here

from stem a/b <c/d = d/b < c/a
d+b /b < c+a /a
d+b/c+a < b/a

a+c/b+d > a/b form here we can deduce that 1 is true

Please check the options. Expression in red proves that option II ((a+c)/(b+d) < a/b) is NOT true.

That I understood , I proceeded as given above way, which i think mathematically correct, what i am asking , from there how we can deduce that "1" is correct OR I am wrong somewhere in that approach ( this approach is somewhat same to lucky's approach) ???
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If a, b, c and d are positive integers and a/b < c/d, which  [#permalink]

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My method was to pick easy numbers and substitute them in, since we're given all numbers are positive integers (phew):
a=1, b=2, c=3, d=4 because we need to make sure a/b < c/d (i.e. we're saying 1/2 < 3/4).

I. (a+c)/(b+d) < c/d ---> 4/6 < 3/4 (TRUE)
II. (a+c)/(b+d) < a/b ---> 4/6 < 1/2 (FALSE)
III. (a+c)/(b+d) = a/b + c/d ---> 4/6 = 1/2 + 3/4 = 5/6 (FALSE)

Answer: B (Only statement I must be true). If a, b, c and d are positive integers and a/b < c/d, which   [#permalink] 04 Jan 2019, 22:36
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