2000 AMC 12 Problem 13

Question

One morning each member of Angela’s family drank an

8

-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?

3

4

5

6

7

Answer: C

Concepts:fraction linear equation bounding to limit cases

Difficulty rating: 1710

Small hint:

Suppose Angela’s own cup contains

c

units of coffee and

m

units of milk with

c + m = 1

Big hint:

Then the family total is

6c

coffee and

4m

milk, so the number of people is

6c + 4m

Solution:

Measure amounts in

8

-ounce cups, so Angela’s cup holds

c

coffee and

m

milk with

c + m = 1.

Since Angela drank a sixth of the coffee, the total coffee is

6c

; since she drank a quarter of the milk, the total milk is

4m.

The number of people equals the total number of cups,

\begin{aligned} 6c + 4m &= 6c + 4(1 - c) \\ &= 4 + 2c. \end{aligned}

This is an integer only when

2c

is an integer, and since

0 \lt c \lt 1

this forces

c = \tfrac12,

giving

4 + 1 = 5

people.

Thus, the correct answer is C.