Adding and multiplying
Amy misread a question and got an incorrect answer. What should the answer have be?
Amy was asked to multiply a number by 8 and then add 5.
Instead, she added 8 to the number, then multiplied the result by 5.
This gave her 2015.
What would the answer have been if Amy had followed the instructions correctly?
This problem is adapted from the World Mathematics Championships
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 $3165$.