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Sum and Differences

Age 11 to 14 Short Challenge Level:

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:
$(33-2)+\ \quad33\ \quad+(34+2)=31+33+36=100$

$(31-1)+(33-1)+(36+2)=30+32+38=100$

$(30-2)+(32-2)+(38+4)=28+30+42=100$



You can find more short problems, arranged by curriculum topic, in our short problems collection.