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# If 5a=9b=15c, what is the value of a+b+c?

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If 5a=9b=15c, what is the value of a+b+c? [#permalink]  13 Jul 2013, 22:06
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If 5a=9b=15c, what is the value of a+b+c?

(1) 3c-a=5c-3b
(2) 6cb=10a

*Note - my answer is different from the official answer. I'm posting it here to confirm whether I'm correct or not.
[Reveal] Spoiler: OA
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  13 Jul 2013, 22:30
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If 5a=9b=15c, what is the value of a+b+c?

Given: $$5a=9b=15c$$ --> least common multiple of 5, 9, and 15 is 45 hence we can write: $$5a=9b=15c=45x$$, for some number $$x$$ --> $$a=9x$$, $$b=5x$$ and $$c=3x$$.

(1) 3c-a=5c-3b --> $$9x-9x=15x-15x$$ --> $$0=0$$ (this means that ANY appropriate values of a, b, and c satisfy the given statement). Not sufficient.

(2) 6cb=10a --> $$6*3x*5x=10*9x$$ --> $$90x^2=90x$$ --> $$x=0$$ or $$x=1$$. Not sufficient.

(1)+(2) $$x=0$$ or $$x=1$$, thus $$a=b=c=0$$ (for $$x=0$$) --> $$a+b+c=0$$ OR $$a=9$$, $$b=5$$, $$c=3$$ (for $$x=1$$) --> $$a+b+c=17$$. Not sufficient.

Hope it's clear.
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  13 Jul 2013, 23:51
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Thanks Bunnuel.

Using the original question, I solved for a:

5a = 9b
a=9b/5

Then I substituted a into statement 2:

6cb = 10a
6cb = 10(9b/5)
6c = 90/5
30c = 90
c = 3

Once I have c , I calculated for a and b:

5a = 15(3)
a = 45/5
a = 9

9b = 15(3)
b = 5

a+b+c = 17
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  14 Jul 2013, 00:12
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jabronyo wrote:
Thanks Bunnuel.

Using the original question, I solved for a:

5a = 9b
a=9b/5

Then I substituted a into statement 2:

6cb = 10a
6cb = 10(9b/5)
6c = 90/5
30c = 90
c = 3

Once I have c , I calculated for a and b:

5a = 15(3)
a = 45/5
a = 9

9b = 15(3)
b = 5

a+b+c = 17

The problem with your solution is that you cannot reduce 6cb = 10(9b/5) by b. Doing so you exclude b=0 case:

6cb = 10(9b/5) --> cb=3b --> b(c-3)=0 --> b=0 or c=3.

Hope it's clear.
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If 5a = 9b = 15c, what is the value of a + b + c? [#permalink]  21 Aug 2013, 16:41
If 5a = 9b = 15c, what is the value of a + b + c?
(1) 3c – a = 5c – 3b
(2) 6cb = 10a
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Re: If 5a = 9b = 15c, what is the value of a + b + c? [#permalink]  21 Aug 2013, 20:12
We have from question that a=3c, 3b=5c, 5a=9b

Statement 1

3c - a = 5c - 3b is the same as 3b-a=2c, substitute a with 3c, we get 5c=3b. Hence Statement 1 is Not Sufficient.

Statement 2

6cb = 10a is the same as 6cb = 30c,
Dividing both sides with 3c gives 2b = 10 and b = 5

With the value of b we can find that c = 3 and a = 9.

Hence statement 2 is Sufficient.

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Re: If 5a = 9b = 15c, what is the value of a + b + c? [#permalink]  22 Aug 2013, 00:54
Expert's post
Asifpirlo wrote:
If 5a = 9b = 15c, what is the value of a + b + c?
(1) 3c – a = 5c – 3b
(2) 6cb = 10a

Merging similar topics. Provided OA (B) is NOT correct. Check here: if-5a-9b-15c-what-is-the-value-of-a-b-c-155926.html#p1246081

Hope it helps.
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Re: If 5a = 9b = 15c, what is the value of a + b + c? [#permalink]  22 Aug 2013, 00:55
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atewari wrote:
We have from question that a=3c, 3b=5c, 5a=9b

Statement 1

3c - a = 5c - 3b is the same as 3b-a=2c, substitute a with 3c, we get 5c=3b. Hence Statement 1 is Not Sufficient.

Statement 2

6cb = 10a is the same as 6cb = 30c,
Dividing both sides with 3c gives 2b = 10 and b = 5

With the value of b we can find that c = 3 and a = 9.

Hence statement 2 is Sufficient.

The correct answer is E, not B. Check here: if-5a-9b-15c-what-is-the-value-of-a-b-c-155926.html#p1246081 and here: if-5a-9b-15c-what-is-the-value-of-a-b-c-155926.html#p1246110

Hope it helps.
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  22 Aug 2013, 09:38
If 5a=9b=15c, what is the value of a+b+c?

(1) 3c-a=5c-3b
(2) 6cb=10a

*Note - my answer is different from the official answer. I'm posting it here to confirm whether I'm correct or not.[/quote]

Statement 1: insufficient because having infinitely many solutions
Statement 2:
6cb=10a means that
6cb=18b and 6cb=30c, so
cb=3b and cb=5c
c=3b/b=3 and b=5c/c=5 providing that c and b are not equal to 0. So I think ambiguity with 0 is resolved and a=9 so sum is 17. Sufficient
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  22 Aug 2013, 09:41
Expert's post
Temurkhon wrote:
If 5a=9b=15c, what is the value of a+b+c?

(1) 3c-a=5c-3b
(2) 6cb=10a

*Note - my answer is different from the official answer. I'm posting it here to confirm whether I'm correct or not.

Statement 1: insufficient because having infinitely many solutions
Statement 2:
6cb=10a means that
6cb=18b and 6cb=30c, so
cb=3b and cb=5c
c=3b/b=3 and b=5c/c=5 providing that c and b are not equal to 0. So I think ambiguity with 0 is resolved and a=9 so sum is 17. Sufficient

The correct answer is E, not B. Check:
if-5a-9b-15c-what-is-the-value-of-a-b-c-155926.html#p1246081
if-5a-9b-15c-what-is-the-value-of-a-b-c-155926.html#p1246110
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  24 Aug 2013, 04:57
OA is E; As correctly pointed out by bunuel

You cannot simply reduce an expression ab=bc does not imply that a=c, as it is perfectly possible for b to be zero and var a might not be equal to var c.
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  24 Aug 2013, 06:01
Bunuel wrote:

(1)+(2) $$x=0$$ or $$x=1$$, thus $$a=b=c=0$$ (for $$x=0$$) --> $$a+b+c=0$$ OR $$a=9$$, $$b=5$$, $$c=3$$ (for $$x=1$$) --> $$a+b+c=17$$. Not sufficient.

Hope it's clear.

Hi Bunuel,

When combining St 1 and St2, isn't the only acceptable solution is when x=0, since x=0 is common to both statements.
How x=1 validates both the statements?
Pls clarify?
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  25 Aug 2013, 06:29
Expert's post
imhimanshu wrote:
Bunuel wrote:

(1)+(2) $$x=0$$ or $$x=1$$, thus $$a=b=c=0$$ (for $$x=0$$) --> $$a+b+c=0$$ OR $$a=9$$, $$b=5$$, $$c=3$$ (for $$x=1$$) --> $$a+b+c=17$$. Not sufficient.

Hope it's clear.

Hi Bunuel,

When combining St 1 and St2, isn't the only acceptable solution is when x=0, since x=0 is common to both statements.
How x=1 validates both the statements?
Pls clarify?

The part you are quoting gives two examples:
$$a=b=c=0$$
$$a=9$$, $$b=5$$, $$c=3$$
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  25 Aug 2013, 19:11
Bunuel wrote:
jabronyo wrote:
Thanks Bunnuel.

Using the original question, I solved for a:

5a = 9b
a=9b/5

Then I substituted a into statement 2:

6cb = 10a
6cb = 10(9b/5)
6c = 90/5
30c = 90
c = 3

Once I have c , I calculated for a and b:

5a = 15(3)
a = 45/5
a = 9

9b = 15(3)
b = 5

a+b+c = 17

The problem with your solution is that you cannot reduce 6cb = 10(9b/5) by b. Doing so you exclude b=0 case:

6cb = 10(9b/5) --> cb=3b --> b(c-3)=0 --> b=0 or c=3.

Hope it's clear.

I made the same mistake! Thanks Brunel for pointing it out
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  30 Aug 2014, 07:44
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  31 Aug 2014, 02:27
jabronyo wrote:
If 5a=9b=15c, what is the value of a+b+c?

(1) 3c-a=5c-3b
(2) 6cb=10a

*Note - my answer is different from the official answer. I'm posting it here to confirm whether I'm correct or not.

My take is E.

5a=9b=15c
LCM(5,9,15) = 45.
=> a = 9x, b = 5y, and c = 3z

1) 3c-a=5c-3b
=> 6z = 15y-9x
or 2z = 5y - 3x

y and x must be odd for LHS to be even. so multiple solutions exist.

2) 6cb = 10a
6*5y*3z = 10*9x
x = yz

x=y=z=0
or x=y=z=1
not sufficient.

1)+2)
x=y=z=0
or x=y=z=1
not sufficient.

Hence, E.
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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  04 Sep 2014, 17:45
Hi all,

Given: 5a = 9b = 15c
to find: a+b+c

Let us suppose 5a = 9b = 15c = K (some constant)
Therefore, a= K/5, b = K/9, and c = K/15

a+b+c = K/5 + K/9 + K/15 = 17K/45
Now we have to find K

statement 1 : 3c-a = 5c-3b
3*(K/15)-(K/5) = 5*(K/15)-3*(K/9)
This reduces to 0=0
hence, statement 1 alone is not sufficient

statement 2: 6cb = 10a
I'll write this as 6cb-10a = 0

6*(K/15)*(K/9)-10(K/5) = 0
=> 2*(K^2)/45 - 2*K =
=> 2K(K/45 - 1) =
either K = 0 or K = 45
so a+b+c = 17K/45 = 0 or 17
hence, statement 2 alone is not sufficient

statement 1 and statement 2 together will not suffice because statement 1 reduces to 0.

Hence, the correct answer is E.

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Re: If 5a=9b=15c, what is the value of a+b+c? [#permalink]  16 May 2015, 05:17
Here's the simplest way that I've found to solve the problem.

(1) 3c - a = 5c -3b Simplifies to
3b = 2c + a Multiply equation by 3
9b = 6c + 3a Substitute 5a for 9b
5a = 6c + 3a Simplify
2a = 6c 5a=15c, so we 2a always equals 6c
6c = 6c This expression is always true, so (1) does not help us figure out a+b+c at all.

(2) 3cb = 5a
I converted this equation to look more similar to the question.
1/5 (15c) 1/9 (9b) = 5a
1/5 (5a) 1/9 (5a) = 5a
1/45 (25a^2) = 5a
1/9 (a^2) = a
1/9a^2 - a = 0
a(1/9a - 1) = 0
a = 0, 9. NOT SUFFICIENT

Many of the other strategies to get E are correct, but I would not think to do them on the test. (I guess I need to keep studying) This way is just basic basic algebra.
Re: If 5a=9b=15c, what is the value of a+b+c?   [#permalink] 16 May 2015, 05:17
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