# If (a – b + c) : (b – c + 2d) : (2a + c – d) = 2 : 3 : 5, then find (3a + 3c – 2d) : d.

given that:

(a – b + c) : (b – c + 2d) : (2a + c – d) = 2 : 3 : 5, then find (3a + 3c – 2d) : d

a-b+c = 2k .............(1)

b-c+2d = 3k ...........(2)

2a+c-d = 5k ...........(3)

a+2d = 5k

a=5k-2d .........(4)

using (3)

c = 5k+d-2a

c = 5k+d-2(5k-2d)

c = 5d -5k .......(5)

using (2)

b-c+2d = 3k

3a+3c-2d =3(5k-2d)+3(5k-5d)-2d

3a+3c-2d = 15k -6d +15k-15d-2d

3a+3c-2d = 30k-23d =d

30k =24d

5k = 4d ...... ....(6)

using ....(4), (5) and (6)

a = 2d, and c = d

therefore

3a+3c-2d =6d +3d -2d =7d

Hence ratio is 7:1

Shortcut:-

Given:   a-b+c = 2k        b-c+2d = 3k      2a+c-d = 5k

now divide all the equation on both side by  "d"  and say

a/d = x   b/d =y    c/d =z    k/d = w

Then the given equation changes to ,

x-y+z = 2w   ------------------   (1)

y-z+2 = 3w  -------------------(2)

2x+z-1 = 5w -------------------(3)

We need to find out  (3a+3c-2d):d   which is equal to   3 (a/d) + 3(c/d) - 2  =  3x+3z-2.

i.e we need to find  3(x+z)-2 .

NOW add equation 1 and 2, we get  x = 5w-2.

Put this value of x in equation 3 , we get  z= -5w+5.

Put these value of x and z in  3x+3z-2.
we get , 3( x+z) - 2 =  3( 5w - 2 - 5w + 5 ) -  2 =  3(3) -2 = 9-2 = 7 is the answer.

Now since the question is in ration form we can write it as 7: 1.

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