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question:Upon reaching a river, travelers asked a boat owner to take them to the opposite bank. Since the boatman was distrustful of foreign money, the travelers offered him a gold chain consisting of 6 links as payment. The boatman agreed but with the condition that he would transport all the travelers one at a time, as his boat could only carry 2 people, and payment must be made for each trip, one link at a time. Additionally, only one link could be cut. How did the travelers meet this condition?
answer:1. **Cutting the chain**: The travelers decided to cut the chain by splitting one of its links. The chain consists of 6 links. They chose to cut the 3rd link, thereby dividing the chain into three parts: - Part 1: links 1 and 2 - Part 2: the split 3rd link (1 separate link) - Part 3: covers links 4, 5, and 6 This way, they now have three separate pieces of links with the total number of links remaining 6, but one link is split. 2. **Payment for the first trip**: For the first trip, they pay the boatman the single, split link (which is effectively 1 broken link). 3. **Payment for the second trip**: For the second trip, they hand over Part 1 (consisting of 2 links, links 1 and 2). The boatman should return the change as 1 link (since Part 1 has 2 links and they had already given 1 link in the previous trip). [ text{Payment} = text{2 links} - text{1 previous link as change} ] 4. **Payment for the third trip**: For the third trip, they give the boatman Part 3 (links 4, 5, and 6). The boatman returns 2 links, since 3 links minus the 1 link already trusted with the boatman equals 2 links as the change. [ text{Payment} = text{3 links} - text{1 previous link as change} rightarrow text{2 as change} ] 5. **Payment for the fourth trip**: For the fourth trip, they again use the split link. [ text{Next link} = text{Split link} ] 6. **Payment for the fifth trip**: They pay again with Part 1, receiving 1 link back as change, as previously remembered. [ text{Payment} = text{2 links} - text{1 link as change} ] 7. **Payment for the sixth trip**: Finally, they again use the remaining existing honest single link. [ text{Payment} = text{Remaining link} ] # Conclusion: By properly calculating and meticulously arranging the sequence of payments as outlined, they completed their trips effectively and economically. [ boxed{Correct execution sequence as stated: split one link and utilizing them for sequential trips} ]
question:Evaluate (125)^{4/3}.
answer:First, recognize the structure of the expression and simplify: [ (-125)^{4/3} = ((-5)^3)^{4/3}. ] Using the properties of exponents, simplify further: [ (-5)^{3 cdot (4/3)} = (-5)^4. ] Calculate the fourth power of -5: [ (-5)^4 = 625. ] Thus, the value is boxed{625}.
question:A bin contains 7 green balls and k purple balls, where k is an unknown positive integer. A ball is drawn at random from the bin. If a green ball is drawn, the player wins 3 dollars, but if a purple ball is drawn, the player loses 1 dollar. If the expected amount won for playing the game is 1 dollar, what is k?
answer:There are 7 green balls and 7+k total balls, so the probability that a green ball is drawn is dfrac{7}{7+k}. The probability that a purple ball is drawn is dfrac{k}{7+k}. The expected value of the game is given by: E = frac{7}{7+k}(3) + frac{k}{7+k}(-1) = 1. Simplify and solve for k: frac{21}{7+k} - frac{k}{7+k} = 1. Multiply through by (7+k) to clear the denominator: 21 - k = 7 + k. Combine like terms: 21 - 7 = 2k implies 14 = 2k implies k = 7. Thus, boxed{k = 7}.
question:Alice has 10q + 2 quarters, whereas Bob has 2q + 10 quarters. If the difference in the number of their quarters is converted to nickels, how much more money does Alice have than Bob in nickels? **A)** (40(q 1) text{ nickels}) **B)** (30(q 1) text{ nickels}) **C)** (50(q 1) text{ nickels}) **D)** (35(q 1) text{ nickels}) **E)** ( none of these )
answer:1. **Find the number of quarters each person holds:** - Alice: 10q + 2 quarters - Bob: 2q + 10 quarters 2. **Calculate the difference in the number of quarters:** [ (10q + 2) - (2q + 10) = 10q + 2 - 2q - 10 = 8q - 8 ] Alice holds 8q - 8 more quarters than Bob. 3. **Convert the difference from quarters to nickels:** - 1 quarter = 25 cents, and 1 nickel = 5 cents; therefore, 1 quarter = 5 nickels. Convert the difference in quarters to nickels by multiplying by 5: [ 5 times (8q - 8) = 40q - 40 ] 4. **Conclude with the answer:** Alice has ( 40q - 40 ) nickels more than Bob, which can be expressed as: [ 40(q - 1) text{ nickels} ] The final answer is boxed{**A)** (40(q - 1) text{ nickels})}