Sum and differences
Three numbers add up to 100. The difference between the larger two is 12 and the difference between the smaller two is 2. What are the numbers?
Age
11 to 14
Challenge level
Problem
Three numbers add up to 100.
The difference between the two larger numbers is 12.
The difference between the two smaller numbers is 2.
What are the numbers?
This problem is taken from the World Mathematics Championships
Student Solutions
Answer: $28, 30, 42$
Using algebra
Middle number is $n$
$\begin{align}n-2\ \ +\ \ n\ \ +\ \ n+12 &= 100\\
\Rightarrow 3n+10&=100\\
\Rightarrow 3n&=90\\
\Rightarrow n&=30\end{align}$
So $n=30$, $n+12=42$ and $n-2=28$.
Guess and check - differences
Try some numbers which are the right distances apart
$\begin{align}30 + 32 + 44 &= 106\\
&=100 + 6\\
&= 100+3\times2\end{align}$
$\therefore (30-2)+(32-2)+(44-2)$ should work
$28+30+42=100$
Guess and check - sum
Try some numbers which add up to $100$
$33+33+34=100$
Need to make largest number larger, always balancing out changes:
$(33-2)+\ \quad33\ \quad+(34+2)=31+33+36=100$
$(31-1)+(33-1)+(36+2)=30+32+38=100$
$(30-2)+(32-2)+(38+4)=28+30+42=100$