How Many DollarsMath brain teasers require computations to solve.
Sandy and Sue each has a whole number of dollars. I ask them how many dollars they have.
Sandy says: "If Sue gives me some dollars, we'll have the same amount of money. But if I give Sue the same number of dollars, she'll have twice as much money as I have."
Sue says: "And if you remove the first digit of my wealth and place it to the end, you'll get Sandy's wealth."
If neither of them has more than 1 million dollars, how many dollars do they each have?
HintFirst get the ratio, then get their amount.
AnswerLet X and Y be Sandy's and Sue's amounts of money, respectively. Let Z be the number of dollars in Sandy's statement.
Then X+Z=Y-Z and 2(X-Z)=Y+Z. Solving this we get X=5Z and Y=7Z.
According to Sue's statement, let A be the first digit of Sue's wealth, B be the remaining digits, and n be the number of digits, then Y=A*10^(n-1)+B and X=10B+A.
Since 65B is divisible by 5 and 5*10^(n-1)-7 is not, A must equal 5. We get:
The least value of n such that 5*10^(n-1)-7 is a multiple of 13 is 6:
The next is 12, which makes them have more than 1 million dollars. Hence n=6 and B=38461.
Therefore, Sandy has 384615 dollars and Sue has 538461 dollars.
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