Two-step problems with addition and subtraction

​​In a nutshell

Addition and subtraction problems can involve more than one calculation. Given the context of a problem, it may be necessary to add in one step, then subtract in the next for example.

Addition and subtraction methods

With two-step problems, any addition and subtraction methods you already know are applicable. It is likely that you have come across two methods already: using mental arithmetic, perhaps involving number lines, and using the column method.

Example 1

Jean is buying new furniture. She chooses a chair that costs $$£15.99$$ and a desk that costs $$£49.99$$. On her way to pay, she sees a lamp for $$£9.50$$. If she buys all three items, how much does she have to pay?

You need to add these three values. This can be done in two step: start by adding the first two together:

$$\begin {array} { c c c c c c} & \overset{1}1 & \overset{1}5 & . & \overset{1}9 & 9 \\+ & 4 & 9 & . & 9 & 9 \\ \hline & 6 & 5 & . & 9&8 \\ \hline\end {array}$$​​

Next you add the third number to the current total:

$$\begin {array} { c c c c c c} & \overset{1}6 & \overset{1}5 & . & 9 & 8 \\+ & & 9 & . & 5 & 0 \\ \hline & 7 & 5 & . & 4&8 \\ \hline\end {array}$$​​

So the final total is $$£75.48$$. ​Note: If you have three numbers to add, the order in which you add them doesn't matter. Here you would get the same answer if you added $$49.99$$ and $$9.50$$ first, then added $$15.99$$.

Take care with the order

With addition, the order doesn't matter, as seen above. More care and attention is required with when subtraction is involved. Observe:

​$$5-3+1$$​​

If you try to add the one to the three, then subtract that total from the five, get the wrong answer. Instead you must read this problem from left to right:

​$$5-3={\color{red}2}\\{\color{red}2}+1=\underline3$$

More specifically, you must be aware that the $$3$$ in this problem is not $$3$$ at all, rather it is $$-3$$. Hence you can rewrite this problem as

​$$5+(-3)+1$$​​

This is because adding a negative number is the same as subtracting the positive number. Now you have an addition problem which can be done in any order, so long as you add $$-3$$ rather than $$3$$.

Example 2

Calculate 

​$$36-21+16$$​​

You can start by subtracting $$21$$ from $$36$$. This could be done mentally with two number lines. First subtract the one unit:

​$$\begin{array}{cccccccccccccc}&&&&&-1&&&&&&& \\& &&&&\curvearrowleft &&&\\33&&34&&\underline{35}&&\underline{36}&&37&&38 \\|&&|&&|&&|&&|&&| \\ \hline\end{array}$$​​

Next subtract the two tens:

$$\begin{array}{cccccccccccccc}&&&&&-10&&-10&&&&& \\& &&&&\curvearrowleft &&\curvearrowleft&& &&&&\\-5&&5&&\underline{15}&&25&&\underline{35}&&45&&55&& \\|&&|&&|&&|&&|&&|&&| \\ \hline\end{array}$$​​

You have $$15$$ so far. Now you can add the $$16$$. This can also be done mentally. Start by adding the six units:

$$\begin{array}{cccccccccccccc}&+1&&-1&&+1&&+1&&+1&&+1&&& \\& \curvearrowright&&\curvearrowright &&\curvearrowright &&\curvearrowright&&\curvearrowright &&\curvearrowright &&\\\underline{15}&&16&&17&&18&&19&&20&&\underline{21}&& \\|&&|&&|&&|&&|&&|&&| \\ \hline\end{array}$$​​

Now add the one ten:

$$\begin{array}{cccccccccccccc}&&&&&&&+10&&&&& \\& &&&& &&\curvearrowright&& &&&&\\-9&&1&&11&&\underline{21}&&\underline{31}&&41&&51&& \\|&&|&&|&&|&&|&&|&&| \\ \hline\end{array}$$​​

So 

$$36-21+16=\underline{31}$$​​