# Nine in a Line

The sum of 9 consecutive positive whole numbers is 2007. What is the largest of these numbers?

## Problem

The sum of 9 consecutive positive whole numbers is 2007.

What is the largest of these numbers?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

**Answer**: $227$

**Algebra from the first number**

Let the first number be $n$, so the nine numbers are:

$n$, $n+1$, $n+2$, $n+3$, $n+4$, $n+5$, $n+6$, $n+7$, $n+8$.

We are told that they all add up to $2007$, so

$n+n$$+$$1+n$$+$$2+n$$+$$3+n$$+$$4+n$$+$$5+n$$+$$6+n$$+$$7+n$$+$$8=2007$,

$9n+36=2007$,

$9n=1971$,

$n=219$.

If the smallest number is $219$, the largest will be $219+8=227$.

**Algebra from the middle number**

Let the middle number be $n$, so the nine numbers are:

$n-4$, $n-3$, $n-2$, $n-1$, $n$, $n+1$, $n+2$, $n+3$, $n+4$.

Summing these gives $9n=2007$ so $n=223$.

Therefore the largest number, $n+4$, is $227$.

**Averages**

The average of the nine numbers is $2007\div9=223$, so they are $219$, $220$,$...$, $226$, $227$.

So the last number is $227$.

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.