### Brain Teasers

# How Many Dollars

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?

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?

### Hint

First get the ratio, then get their amount.### Answer

Let 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.

7X=5Y

7*(10B+A)=5*(A*10^(n-1)+B)

(5*10^(n-1)-7)A=65B

Since 65B is divisible by 5 and 5*10^(n-1)-7 is not, A must equal 5. We get:

5*10^(n-1)-7=13B

The least value of n such that 5*10^(n-1)-7 is a multiple of 13 is 6:

499993=38461*13

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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