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# What is the ratio of a two-digit number and its sum of digits?

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Math Expert
Joined: 02 Sep 2009
Posts: 44284
What is the ratio of a two-digit number and its sum of digits? [#permalink]

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06 Aug 2017, 01:05
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Question Stats:

49% (01:17) correct 51% (01:14) wrong based on 73 sessions

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What is the ratio of a two-digit number and its sum of digits?

(1) The ratio is the minimum possible.
(2) The number has 9 as its unit digit.
[Reveal] Spoiler: OA

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Manager
Joined: 02 Nov 2015
Posts: 174
GMAT 1: 640 Q49 V29
Re: What is the ratio of a two-digit number and its sum of digits? [#permalink]

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06 Aug 2017, 01:48
As per I understand the question, the answer should be A.
Let us take a number XY.

Because the ratio will be minimum only when x and y are equal.

So given any equal value of X and Y the answer will always be 5.5.
I mean any two digit number can be expressed as 10x+y. Let x and y b equal.

So (10x+y)/(x+y) = 11x/2x= 5.5, which is the least value .

If I have missed something. Kindly correct me.

* Questions of Bunuel are always great and I have turned out to be a big fan of Him.

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Intern
Joined: 05 Mar 2017
Posts: 2
Re: What is the ratio of a two-digit number and its sum of digits? [#permalink]

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06 Aug 2017, 02:25
I think the answer is A as the only smallest ratio will be 19/10.correct me if i am wrong.
Director
Joined: 18 Aug 2016
Posts: 630
GMAT 1: 630 Q47 V29
GMAT 2: 740 Q51 V38
Re: What is the ratio of a two-digit number and its sum of digits? [#permalink]

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06 Aug 2017, 03:28
Bunuel wrote:
What is the ratio of a two-digit number and its sum of digits?

(1) The ratio is the minimum possible.
(2) The number has 9 as its unit digit.

Let the ratio of 2-digit number and its sum of digit be
(10x+y)/(x+y) = 1 + (9x/(x+y)) (x cannot be 0)

(1) ratio is minimum

9x/(x+y) will be minimum if x = 1 and y = 9 i.e. 9/10 which will become 1 + 9/10 = 19/10

if x = 2 and y = 9 then ratio will be 18/11 (not minimum) and so on in all other cases it will exceed 1

Hence Sufficient

(2) y = 9
then
1+ (9x/(x+y))
but we do not know the value of x ( any digit from 1-9)

Not Sufficient

Hence A
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Joined: 02 Apr 2014
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Re: What is the ratio of a two-digit number and its sum of digits? [#permalink]

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02 Nov 2017, 20:17
Let the number be $$ab$$ => $$10a + b$$

question: $$(10a + b) / (a + b)$$ => $$(9a + a + b)/(a+b)$$ => $$(9a/(9a + b)) + 1 =$$ ?

Statement 1:
the above equation $$(9a/(9a + b)) + 1$$ is minimum when a = 1 and b = 9
so ratio = $$(9/10) + 1$$ is minumum => a = 1, b = 9 => number is 19 => sufficient

Statement 2: units digit is 9, but no idea about tens digit => InSuff

Re: What is the ratio of a two-digit number and its sum of digits?   [#permalink] 02 Nov 2017, 20:17
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