7162
/www/nrich/html/content/id/7162/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>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?<br></br>
<br></br>
<br></br>
If you liked this problem, <a href="http://nrich.maths.org/642&part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br>
<br></br>
</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
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 $0$.<br></br>
<br></br>
[<span style="font-style: italic;">Alternatively</span>: Let the three
numbers be <span style="font-style: italic;">a</span>, <span style="font-style: italic;">b</span> and <span style="font-style: italic;">c</span>.<br></br>
<br></br>
We have <br></br>
<div style="margin-left: 120px;">$a+b=1004$</div>
<div style="margin-left: 120px;">$b+c=1005$</div>
<div style="margin-left: 40px;">and
$a+b+c=2009$</div>
<div style="margin-left: 40px;"> </div>
<div>Adding the first two
equations gives</div>
<div style="margin-left: 80px;"> $a+2b+c=2009$</div>
<div style="margin-left: 80px;"> </div>
<div>and subtracting the third
equationg from this gives</div>
<div style="margin-left: 120px;"> $b=0$</div>
<div> </div>
<div>Thus the product
$abc=0$.]</div>
<br></br>
<br></br></mdoxml>
2
3
0
1
1
0
0
Weekly Problem 14 - 2011
Weekly Problem 14 - 2011
Algebra
Using symbols
Algebra
Creating expressions/formulae
Algebra
Simultaneous equations