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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?

May 22, 2022

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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

May 25, 2022