### Pyramids

What are the missing numbers in the pyramids?

### Paving the Way

A man paved a square courtyard and then decided that it was too small. He took up the tiles, bought 100 more and used them to pave another square courtyard. How many tiles did he use altogether?

### Chess

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

# Adding and Multiplying

##### Age 11 to 14 Short Challenge Level:

Answer: 3165

Working backwards
Added 8 then multiplied by 5, got 2015
After adding 8: 2015$\div$5 = 403
Original number: 8 less than 403, which is 395

Multiply by 8: 395$\times$8 = 3160
Add 5: 3160 + 5 = 3165

Using algebra to find what number Amy started with
If Amy started with $n$,
Add $8$ then multiply by $5$ and get $2015$, so $(n+8)\times5=2015$
\begin{align}n+8&=2015\div5\\n+8&=403\\n&=403-8\\n&=395.\end{align}
Amy was supposed to multiply $n$ by $8$ and then add $5$, so she should have found $8n+5$.
$$8n+5=8\times395+5=3160+5=3165.$$

Using algebra to find the value of the correct expression
When Amy added $8$ to the number and then multiplied by $5$, she got $2015$, so $(n+8)\times5=2015$, where $n$ is the number that Amy started with.
Amy was supposed to multiply $n$ by $8$ and then add $5$, so she should have found $8n+5$.

We want to get from $(n+8)\times5$ to $8n+5$. Knowing $(n+8)\times8$ would be helpful, because \begin{align}(n+8)\times8&=8n+64\\&=8n+(5+59)\\&=(8n+5)+59\end{align}

If $(n+8)\times5=2015$, then $n+8=2015\div5=403$, so $(n+8)\times8=403\times8=3224$.

So $3224$ is $59$ more than $8n+5$, so $8n+5=3224-59=3165$. So Amy would have got $3165$.

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