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# If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c

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Joined: 02 Sep 2009
Posts: 59039
If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c  [#permalink]

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08 Oct 2018, 21:29
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Question Stats:

63% (03:03) correct 37% (03:12) wrong based on 163 sessions

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If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c = ?

(A) 1:2:3
(B) 3:4:5
(C) 7:8:9
(D) 7:5:9
(E) 7:6:8

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Posts: 8164
Re: If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c  [#permalink]

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09 Oct 2018, 00:46
4
2
Bunuel wrote:
If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c = ?

(A) 1:2:3
(B) 3:4:5
(C) 7:8:9
(D) 7:5:9
(E) 7:6:8

a quick and logical approach:-

choices A, B and C have b>a, so let us see
$$(b + c):(c + a) = 7:8$$......8>7 so $$c+a>b+c........a>b$$
eliminate all these choices - A,B,C

let us check D and E.. let the common ratio be y
(D) 7:5:9 .... 7y:5y:9y
(E) 7:6:8 ..... 7y:6y:8y
now a+b from (a + b):(b + c):(c + a) = 6:7:8 is 6x, if common ratio is x.
so D gives us a+b=7y+5y=12y... a multiple of 6
and E gives us a+b=7y+6y=13y... NOT a multiple of 6
D
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Re: If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c  [#permalink]

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08 Oct 2018, 22:35
3
1
Bunuel wrote:
If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c = ?

(A) 1:2:3
(B) 3:4:5
(C) 7:8:9
(D) 7:5:9
(E) 7:6:8

Let (a+b) = 6x
(b+c) = 7x
(c+a) = 8x

2a + 2b + 2c = 21x
2 (a + b + c) = 21x
2 (14) = 21x

28 = 21x
x = 4/3
a+b = 6 * 4/3 = 8

a + b + c = 14
8 + c = 14
c = 6

b + 6 = 7 (4/3)
b = 28/3 - 18/3 = 10/3
b:c = (10/3)/(6/1) = 10/18 = 5/9

a + c = 8x
a + 6 = 32/3
a = 32/3 - 18/3 = 14/3
a:c = (14/3)/(6/1) = 14/18 = 7/9

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Re: If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c  [#permalink]

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09 Nov 2018, 08:56
1
2
If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c = ?

(A) 1:2:3
(B) 3:4:5
(C) 7:8:9
(D) 7:5:9
(E) 7:6:8

since a + b + c = 14, the initial ratio can be re-written as (14 - c) : (14-a) : (14-b) = 6:7:8

rearrange the ratio as (14-a) : (14-b) : (14-c) = 7:8:6

"c" must be the smallest in order 14-c to be the biggest.

hence, c > a > b.

D
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If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c  [#permalink]

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11 Nov 2018, 10:25
1. Total of all ratios
6x + 7x + 8x = 21x

2. Recognize we have 3 equations and can solve for a,b, or c
a+b = 6x
b+c = 7x
c+a =8x

3. (equation 1) + (equation 2) - (equation 3) = 2b
6x + 7x - 8x = 2b
b= 2.5

4. use be = 2.5 to solve for a and c
a = 6x - 2.5x = 3.5x
c = 8x - 3.5x = 4.5x

5. Use ratio to recognize answer
a:b:c
3.5:2.5:4.5 = 7:5:9 => Correct answer is D
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Re: If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c  [#permalink]

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12 Nov 2018, 08:21
2
Bunuel wrote:
If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c = ?

(A) 1:2:3
(B) 3:4:5
(C) 7:8:9
(D) 7:5:9
(E) 7:6:8

Using the first two parts of the three-part ratio, we have:

(a + b)/(b + c) = 6/7

7a + 7b = 6b + 6c

b = 6c - 7a → (Eq. 1)

Using the last two parts of the three-part ratio, we have:

(b + c)/(c + a) = 7/8

8b + 8c = 7c + 7a

c = 7a - 8b → (Eq. 2)

Now if we add Eq. 1 and Eq. 2, we have:

b + c = 6c - 8b

9b = 5c

b/c = 5/9

Since in the multiple choices, only choice D has the ratio of b to c as 5 to 9, the correct answer must be D.

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Re: If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c  [#permalink]

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09 Oct 2019, 17:25
2
Bunuel wrote:
If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c = ?

(A) 1:2:3
(B) 3:4:5
(C) 7:8:9
(D) 7:5:9
(E) 7:6:8

It's a 30 seconds question if you recognise what is happening.
The sum has to be multiple of 14. Or at least 7.

(A) 1:2:3
(B) 3:4:5
(C) 7:8:9
(D) 7:5:9
(E) 7:6:8

Sum of a and b has to be a multiple of 6.

(A) 1:2:3
(D) 7:5:9
(E) 7:6:8

Sum of b and c has to be multiple of 7.
(A) 1:2:3

(D) 7:5:9

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Re: If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c   [#permalink] 09 Oct 2019, 17:25
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