Question

Gopi, Murthy and Hari had some amount of money.
Gopi gives half his amount to Murthy, who then
gives half of what he now has to Hari. Hari gives half
of what he now has to Gopl, who, now has exactly
what he started with. If the sum of Murthy's initial
amount and twice Hari's initial amount is 45, what
was the amount (in rupees) Gopi started with?​

Answer 1

Answer:

Rs. 30

Step-by-step explanation:

Let x, y , z be the initial amounts of Gopi, murthy and Hari.

    G                                        M               H  

=====================|===========|======

 x                                     |  y                 |        z

 x/2                                 |  y +x/2          |       z

                                        | 1/2(y+x/2)     | z + 1/2(y+x/2)

1/2(z + 1/2(y+x/2)) + x/2    | 1/2(y+x/2      |  1/2(z + 1/2(y+x/2))

//It is given gopi has the same amount he has started with

=> 1/2(z + 1/2(y+x/2)) + x/2  = x

=> z/2 + y/4 + x/8 + x/2  = x

=> z/2 + y/4 + 5x/8   = x

=> 4z+ 2y + 5x = 8x

=> 4z+ 2y = 3x === equation 1

//Also given

y + 2z = 45  equation 2

=> 2y+4z = 90

//substitute equation 1

=> 3x = 90

x = Rs. 30

Answer 2

Given : Gopi gives half his amount to Murthy, who then gives half of what he now has to Hari.  Hari gives half of what he now has to Gopi, who, now has exactly what he started with.

sum of Murthy's initial amount and twice Hari's initial amount is rupee45,

To Find : what was the amount (in rupees) Gopi started with ?​

Solution:

Let say

Gopi started with  8G

Murthy started with 4M

Hari started  with    2H

Gopi gives half his amount to Murthy,

=> Gopi now has = 8G - 8G/2  = 4G

=> Murthy has = 4M + 4G  =  4(M + G)

Murthy, who then gives half of what he now has to Hari.

=> Murthy has = 4(M + G) -  4(M + G)/2  = 2(M + G)

Hari. has = 2H + 2(M + G) = 2(H + M + G)

Hari gives half of what he now has to Gopi,

Hari has now = 2(H + M + G) - 2(H + M + G)/2  = H + M + G

Gopi Has = 4G +  H + M + G  = 5G + H  + M

8G = 5G + H  + M

=> 3G = H + M

sum of Murthy's initial amount and twice Hari's initial amount is rupee45

4M + 2(2H) = 45

=> 4(M + H) = 45

=> M + H  = 45/4

3G = H + M

=> 3G =  45/4

=> G = 15/4

=> 8G = 8 * 15/4

=> 8G = 30

Gopi started with  8G = Rs 30

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