In sum-mary
What does the sum of these three numbers tell us about their product?
Age
11 to 14
Challenge level
Problem
The sum of three numbers is 2009. The sum of the first two numbers is 1004, and the sum of the last two is 1005.
What is the product of all three numbers?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 0
$$\underbrace{\_\_\_\_\quad\_\_\_\_}_{1004}\quad \_\_\_\_\qquad \therefore \quad \underbrace{\_\_\_\_\quad\_\_\_\_}_{1004}\quad \underline{1005}\\
\quad\\
\_\_\_\_\quad \underbrace{\_\_\_\_\quad \underline{1005}}_{1005}\qquad \therefore \quad \underline{1004}\quad\underbrace{\underline{\ \ \ \ 0\ \ \ \ }\quad \underline{1005}}_{1005}\\
\quad\\
\qquad\qquad\qquad\qquad\qquad\qquad 1004\times0\times1005=0$$
Alternatively:
The sum of the first two numbers and the last two numbers is 1004 + 1005 = 2009.
This counts the middle number twice.
But the sum of all three numbers is 2009, so the middle number is 0.
Hence the product of all three numbers is 0.
Using algebra
Let the three numbers be a, b and c.
We have
$a+b=1004$
$b+c=1005$
and $a+b+c=2009$
Adding the first two equations gives
$a+2b+c=2009$
and subtracting the third equationg from this gives
$b=0$
Thus the product $abc=0$.