Math solution needed for this problem

Question

Hi,

Here is my problem,

There are 54 beers for three people and they need to divide the beers equally. At first Laura gives 1/4 of her beers to Anton. After that Anton gives 1/2 of his beers to Tommy and Tommy gives 1/3 of his beers back to Laura. After this they have equally 18 beers per person, but how many beers did they have before that?

Answer 1

Ebola,

L = number of beers Laura started with

A = number of beers Anton started with

T = number of beers Tommy started with

first Laura gives 1/4 of her beers to Anton, that means Laura is left with 3/4 of what she started with

Laura = 3/4 L

Anton = 1/4 L + A

Tommy = T

After that Anton gives 1/2 of his beers to Tommy which leaves Anton with only 1/2 left

Laura = 3/4 L

Anton = 1/2(1/4 L + A)

Tommy = T + 1/2(1/4 L + A)

and Tommy gives 1/3 of his beers back to Laura which leaves Tommy with only 2/3 left

Laura = 3/4 L + 1/3(T + 1/2(1/4 L + A))

Anton = 1/2(1/4 L + A)

Tommy = 2/3(T + 1/2(1/4 L + A))

After this they have equally 18 beers per person

Laura = 3/4 L + 1/3(T + 1/2(1/4 L + A)) = 3/4 L + 1/3(T + 1/8 L + 1/2 A) = 3/4 L + 1/3 T + 1/24 L + 1/6 A= 18

19/24 L + 1/6 A + 1/3 T = 18

Anton = 1/2(1/4 L + A) = 18

Tommy = 2/3(T + 1/2(1/4 L + A)) = 2/3(T + 1/8 L + 1/2 A) = 2/3 T + 1/12 L + 1/3 A= 18

Now we simplify and substitute. I usually like to start with the simplest equation

1/2(1/4 L + A) = 18

1/8 L + 1/2 A = 18

1/2 A = 18 - 1/8 L

A = 36 - 1/4 L

Now substitute A in one of the other 2 equations

2/3 T + 1/12 L + 1/3 A= 18

2/3 T + 1/12 L + 1/3(36 - 1/4 L) = 18

2/3 T + 1/12 L + 12 - 1/12 L = 18

2/3 T + 12 = 18

2/3 T = 6

T = 9

Substitute T = 9 and A = 36 - 1/4 L into the first equation

19/24 L + 1/6 A + 1/3 T = 18

19/24 L + 1/6(36 - 1/4 L) + 1/3 * 9 = 18

19/24 L + 6 - 1/24 L + 3 = 18

19/24 L - 1/24 + 9 = 18

18/24 L = 9

L = 12

Lastly substitute L = 12 into the second equation.

A = 36 - 1/4 L

A = 36 - 1/4 * 12

A = 36 - 3

A = 33

Verify that the sum of L, A, T equals 54

L = 12, A = 33, T = 9

L + A + T = 54

12 + 33 + 9 = 54

54 = 54

Answer:

Laura originally had 12 beers

Anton originally had 33 beers

Tommy originally had 9 beers

I hope this helps.

Please feel free to reach out to me for any clarification.

Thanks,

Richmond