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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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Bunuel
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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] New post   08 Oct 2018, 21:29
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A
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D
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65% (hard)

Question Stats:

65% (02:56) correct 35% (03:00) wrong based on 381 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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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] New post   08 Oct 2018, 22:35
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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

Answer choice 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] New post   09 Oct 2018, 00:46
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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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If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c [#permalink] New post   11 Nov 2018, 10:25
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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] New post   09 Nov 2018, 08:56
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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


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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Re: If (a + b):(b + c):(c + a) = 6:7:8 and a + b + c = 14, then find a:b:c [#permalink] New post   09 Oct 2019, 17:25
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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] New post   12 Nov 2018, 08:21
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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.

Answer: 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] New post   12 Jan 2021, 12:47
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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


We can PLUG IN THE ANSWERS, which represent \(a:b:c\).
Let \(x\) serve as the multiplier for each ratio.
When the correct answer is plugged in:
\((a + b):(b + c):(c + a) = 6:7:8\)

A: \(a=x, b=2x, c=3x\) --> \((a + b):(b + c):(c + a) = 3x:5x:4x = 3:5:4\) --> Eliminate A
B: \(a=3x, b=4x, c=5x\) --> \((a + b):(b + c):(c + a) = 7x:9x:8x = 7:9:8\) --> Eliminate B
C: \(a=7x, b=8x, c=9x\) --> \((a + b):(b + c):(c + a) = 15x:17x:16x = 15:17:16\) --> Eliminate C
D: \(a=7x, b=5x, c=9x\) --> \((a + b):(b + c):(c + a) = 12x:14x:16x = 6:7:8\) --> Success!

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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] New post   20 Sep 2023, 01:10
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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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