2004 AMC 10A Problem 7

Question

A grocer stacks oranges in a pyramid-like stack whose rectangular base is

5

oranges by

8

oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack?

96

98

100

101

134

Answer: C

Concepts:basic counting summation

Difficulty rating: 1100

Small hint:

Each layer up has one fewer orange in each dimension than the layer below

Big hint:

Sum

5\cdot 8 + 4\cdot 7 + 3\cdot 6 + 2\cdot 5 + 1\cdot 4

Solution:

There are five layers, each one shorter and narrower than the one below. The total number of oranges is

\begin{aligned} &5\cdot 8 + 4\cdot 7 + 3\cdot 6 \\ &\quad {}+ 2\cdot 5 + 1\cdot 4 \\ &= 40 + 28 + 18 + 10 + 4 = 100. \end{aligned}

Thus, the correct answer is C.