Dave had some 50-cent coins. Wayne had some $1 coins. Dave gave 50% of his coins to Wayne and Wayne gave 50% of his coins to Dave. Wayne used some of the 50-cent coins to buy a book that cost $5. Dave used some of his $1 coins to buy a bag that cost $16. After that the ratio of the number of 50-cent coins to $1 coin Dave had was 1:2 and the number of 50-cent coins to the number of $1 coins Wayne had was 1:5. How much did Dave have at first?

Guest May 22, 2022

#1**+1 **

Dave had some 50-cent coins. Wayne had some $1 coins. Dave gave 50% of his coins to Wayne and Wayne gave 50% of his coins to Dave. Wayne used some of the 50-cent coins to buy a book that cost $5. Dave used some of his $1 coins to buy a bag that cost $16. After that the ratio of the number of 50-cent coins to $1 coin Dave had was 1:2 and the number of 50-cent coins to the number of $1 coins Wayne had was 1:5. How much did Dave have at first?

slight simplification:

Dave had some 50-cent coins. I let that be 2x (50c) coins

Wayne had some $1 coins. I let that be 2y ($1) coins

Dave gave 50% of his coins to Wayne and Wayne gave 50% of his coins to Dave.

Now they each have x (50c) + y ($1)

Wayne used some of the 50-cent coins to buy a book that cost $5. So wayne spends 10 of his 50c peices

Dave used some of his $1 coins to buy a bag that cost $16. So Dave spends 16 of his $1 coins

So now

Wayne has (x - 10) (50c coins) + y ($1coins) which means that Wayne has $ \((\frac{x-10}{2}+y)\)

Dave has x (50c coins) + (y-16) ($1coins) which means that Dave has \($(\frac{x}{2}\;+\;\;y-16)\)

After that the ratio of the number of 50-cent coins to $1 coin Dave had was 1:2

\(\frac{x}{y-16}=\frac{1}{2}\\ 2x=y-16\\ y=2x+16\\\)

and the number of 50-cent coins to the number of $1 coins Wayne had was 1:5.

\(\frac{x-10}{y}=\frac{1}{5}\\ y=5x-50\)

Solve simultaneously

\(2x+16=5x-50\\ 66=3x\\ x=22\\ y=44+16=60\)

How much did Dave have at first?

He had 2x 50c coins so initially he had 44 fifty cent peices = $22

__Check:__

Started with Dave having 44 fifties and Wayne having 120 dollars

Dave

44 fifties

22 fifties and 60 ones

22 fifties and 44 ones Ratio 1 to 2 good

Wayne

120 dollars

22 fifties and 60 ones

12 fifties and 60 ones Ratio 1 to 5 good

Melody May 25, 2022