2008 AMC 10A ws-25.pdf - A bakery 8:30 am one third...

A bakery owner turns on his doughnut machine at
8:30 am . At 11:10 am the machine has completed
one third of the day’s job. At what time will the
doughnut machine complete the job?
(A) 1:50 pm (B) 3:00 pm (C) 3:30 pm (D) 4:30 pm (E) 5:50 pm
2008 AMC 10 A, Problem #1 —
2008 AMC 12 A, Problem #1 —
“How many minutes did it take the machine to
complete
1
3
of the job?”
Difficulty: Easy
NCTM Standard: Measurement Standard: understand measurable attributes of objects and the
units, systems, and processes of measurement.
Mathworld.com Classification: Number Theory > Arithmetic > General Arithmetic

A square is drawn inside a rectangle. The ratio of
the width of the rectangle to a side of the square is
2:1. The ratio of the rectangle’s length to its width
is 2:1. What percent of the rectangle’s area is inside
the square?
5
2008 AMC 10 A, Problem #2 —
“The ratio of length of the rectangle to a side of the
square is 4:1 .”
Difficulty: Medium
NCTM Standard: Algebra Standard: represent and analyze mathematical situations and
structures using algebraic symbols.
Mathworld.com Classification: Geometry > Plane Geometry > Miscellaneous Plane Geometry
> Area

For the positive integer n , let <n> denote the sum
of all the positive divisors of n with the exception
of n itself. For example, < 4 > = 1 + 2 = 3 and
< 12 > = 1+2+3+4+6 = 16 . What is <<< 6 >>> ?
(A) 6 (B) 12 (C) 24 (D) 32 (E) 36
2008 AMC 10 A, Problem #3 —
“ < 6 > = 1 + 2 + 3 = 6 .”
Solution
Answer (A): The positive divisors of 6 , 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 that
positive integer is called a perfect number . The two smallest perfect
numbers are 6 and 28.
Difficulty: Medium-easy
NCTM Standard: Number and Operations Standard: understand meanings of operations and
how they relate to one another.
Mathworld.com Classification: Calculus and Analysis > Functions > Unary Operation

Suppose that
2
3
of 10 bananas are worth as much as
8 oranges. How many oranges are worth as much as
1
2
of 5 bananas?
(A) 2 (B)
5
2
(C) 3 (D)
7
2
(E) 4