# The Problem

### Jessica has an odd number of stamps in her collection.
The sum of the digits in the number of stamps she has is 12. The
hundreds digit is three times the ones digit. If she has between
1000 and 2000 stamps in her collection, how many stamps does Jessica
have?

(Source: Adapted from Mathematics Teaching in the Middle School,
Nov-Dec1996)

# The Solution

### We can write a few equations that will help us to solve this
problem, from the information given to us. First, we know that
Jessica has an odd number of stamps in her collection, this is
information that we need to keep in mind as we continue to evaluate
the information that has been given to us.

### Second, we are told that the sum of the digits in her stamp
collection is 12, so we can write the following equation:

### a + b + c + d = 12

### where a, b, c, and d are the digits of the total number of
stamps she has.

### Third, we know that in Jessica's collection the hundreds digit
is three times the ones digit. That narrows the possible digits
for both of those place values. The available digits for the ones
place are 0, 1, 2, and 3 as any larger numbers would produce values
greater than 9 for the hundreds digit. Since we know that the
total number of stamps is odd, we can rule out 0 and 2 as choices
for the ones digit.

### Finally, we are told that Jessica has between 1000 and 2000
stamps in her collection. Since 2000 is an even number, we know
that Jessica really must have between 1001 and 1999 stamps in
her collection. Thus the value of a above is 1.

###

### Now we can plug in our two possible values for d, and we get
the following results:

### If d=1 then:

### 1 + 3 + c + 1 = 12

### 5 + c = 12

### c = 7

### Thus Jessica has 1371 stamps in her collection. This is one
possible solution, but we must rule out the other possibility
of d=3.

### When d=3 then:

### 1 + 9 + c + 3 = 12

### 13 + c =12

### c = -1

### This is not a possible solution, as we cannot have a negative
digit in a positive number. Thus, we now know for certain that
Jessica has 1371 stamps in her collection.