A bakery owner turns on his doughnut machine at8:30am.At 11:10amthe machine has completedone third of the day’s job.At what time will thedoughnut machine complete the job?(A)1:50pm(B)3:00pm(C)3:30pm(D)4:30pm(E)5:50pm2008 AMC 10 A, Problem #1—2008 AMC 12 A, Problem #1—“Howmanyminutesdidittakethemachinetocomplete13of the job?”Difficulty:EasyNCTM Standard:Measurement Standard: understand measurable attributes of objects and theunits, systems, and processes of measurement.Mathworld.com Classification:Number Theory>Arithmetic>General Arithmetic
A square is drawn inside a rectangle.The ratio ofthe width of the rectangle to a side of the square is2:1. The ratio of the rectangle’s length to its widthis 2:1. What percent of the rectangle’s area is insidethe square?52008 AMC 10 A, Problem #2—“The ratio of length of the rectangle to a side of thesquare is 4:1 .”Difficulty:MediumNCTMStandard:AlgebraStandard:representandanalyzemathematicalsituationsandstructures using algebraic symbols.Mathworld.com Classification:Geometry>Plane Geometry>Miscellaneous Plane Geometry>Area
For the positive integern, let<n>denote the sumof all the positive divisors ofnwith the exceptionofnitself.For example,<4>= 1 + 2 = 3and<12>= 1+2+3+4+6 = 16. What is<<<6>>>?(A)6(B)12(C)24(D)32(E)362008 AMC 10 A, Problem #3—“<6>= 1 + 2 + 3 = 6.”SolutionAnswer (A):The positive divisors of6, other than 6, are 1, 2, and 3, so<6>= 1 + 2 + 3 = 6. As a consequence, we also have<<<6>>>= 6.Note: A positive integer whose divisors other than itself add up to thatpositive integer is called aperfect number.The two smallest perfectnumbers are 6 and 28.Difficulty:Medium-easyNCTM Standard:Number and Operations Standard:understand meanings of operations andhow they relate to one another.Mathworld.com Classification:Calculus and Analysis>Functions>Unary Operation
Suppose that23of 10 bananas are worth as much as8 oranges. How many oranges are worth as much as12of 5 bananas?(A)2(B)52(C)3(D)72(E)4