AMC 8 Daily Practice Round 8

Complete problem set with solutions and individual problem pages

Problem 15 Medium

Amy, Betty, and Celia received a total of 1800 dollars in lucky money during the New Year. Amy gives \frac{1}{3} of their money to Betty, then Betty gives \frac{1}{4} of their current amount to Celia, and then Celia gives \frac{1}{5} of their current amount to Amy. At this point, they have the same amount of money. How much more money did Betty originally have than Celia?

  • A.

    25

  • B.

    35

  • C.

    45

  • D.

    100

  • E.

    125

Answer:A

Since they give money to each other, the total remains unchanged.

After all the exchanges, each person ends up with 1800 \div 3 = 600 \quad \text{(dollars)}

Celia ends up with 600, which is after giving away \frac{1}{5} of their money.

So before giving, Celia had: 600 \div \frac{4}{5} = 750 \quad \text{(dollars)}

Celia gave 150 dollars to Amy, so Amy ends up with 600 \quad \text{(dollars)}

This means Amy had 450 after giving away \frac{1}{3} of their money: \text{Amy's original amount} = 450 \div \frac{2}{3} = 675 \quad \text{(dollars)}

Amy gave 675 - 450 = 225 dollars to Betty.

Betty ends up with 600 \div \frac{3}{4} = 800 \quad \text{(dollars)} \quad \text{(before giving away } \frac{1}{4} \text{ to Celia)}

So Betty originally had: 800 - 225 = 575 \quad \text{(dollars)}

Then Celia’s original amount is: 1800 - 675 - 575 = 550 \quad \text{(dollars)}.

Therefore, Betty originally had 575 - 550 = 25 dollars more than Celia.

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