MHB Setting up systems of equations

[TracyThomas]

I have this problem in my book:
Ann had 20(Cash) less than Betty. Ann spent 3/4 of her money, while Betty spent 4/5 of hers. Then Ann's remainder was 5/6 of Betty's remainder. If Ann had x(Cash) originally, form an equation in x and solve it.

And this solution in the answer key:

x=Ann's (Cash) x+20=Betty's (Cash)

Remainder of Ann's money = 1/4x
Remainder of Betty's money = 1/5(x+20)

1/4x = 5/6 (1/5(x + 20))
x/4 = x/6 + 50/3

12(x/4) = 12(x/6) + 12(50/3)
3x = 2x + 40
Survey says:

x = 40

The question is: How did the 50 in 50/3 get there?
(Wondering) (Worried) Can someone please explain that step so both me and my homeschool teacher can understand it?

 

[Ackbach]

Re: Ann & Betty got rich $$$--But where did the 50 come from?

I have this problem in my book:
Ann had 20(Cash) less than Betty. Ann spent 3/4 of her money, while Betty spent 4/5 of hers. Then Ann's remainder was 5/6 of Betty's remainder. If Ann had x(Cash) originally, form an equation in x and solve it.

And this solution in the answer key:

x=Ann's (Cash) x+20=Betty's (Cash)

Remainder of Ann's money = 1/4x
Remainder of Betty's money = 1/5(x+20)

1/4x = 5/6 (1/5(x + 20))
x/4 = x/6 + 50/3

I agree with the setup (usually the hard part!) You are right to question this last step, though. I would do this:

x/4 = 5/6 (x/5 + 4)
x/4 = x/6 + 20/6 = x/6 + 10/3
12 (x/4 = x/6 + 10/3)
3x = 2x + 40
x = 40.

I think it's a question of incorrect processes somehow coming to the right answer.

Does this make sense? I'm not sure I understand how the 50 got in there, but perhaps a correct (I hope) sequence of steps will clear things up.

12(x/4) = 12(x/6) + 12(50/3)
3x = 2x + 40
Survey says:

x = 40

The question is: How did the 50 in 50/3 get there?
(Wondering) (Worried) Can someone please explain that step so both me and my homeschool teacher can understand it?

 

[TracyThomas]

Re: Ann & Betty got rich $$$--But where did the 50 come from?

Thanks! I haven't found ANY errors in the Singapore Math textbook answer keys in many years of using the program, but this answer key is written by someone else, I think, so I suppose it could be in error.

I agree with the setup (usually the hard part!) You are right to question this last step, though. I would do this:

x/4 = 5/6 (x/5 + 4)
x/4 = x/6 + 20/6 = x/6 + 10/3
12 (x/4 = x/6 + 10/3)
3x = 2x + 40
x = 40.

I think it's a question of incorrect processes somehow coming to the right answer.

Does this make sense? I'm not sure I understand how the 50 got in there, but perhaps a correct (I hope) sequence of steps will clear things up.