2003 AMC 10B Problem 12

Question

Al, Betty, and Clare split

\$1000

among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of

\$1500.

Betty and Clare have both doubled their money, whereas Al has managed to lose

\$100.

What was Al’s original portion?

\$250

\$350

\$400

\$450

\$500

Let Al, Betty, and Clare start with

a,

b,

and

c,

so

a+b+c=1000

The final total is

(a-100)+2(b+c)=1500;

substitute

b+c=1000-a

Let

a,

b,

and

c

be the original portions. Then

a+b+c=1000

and

(a-100)+2(b+c)=1500.

Substituting

b+c=1000-a

into the second equation,

a-100+2(1000-a)=1500,

so

a=400.

Thus, the correct answer is C.

Problem 12 in other years