MENTAL MATH: ADDITION

There are loads of ‘specialty’ math tricks for mental computation, like:

Although tricks like these are fun and impressive, they have limited usefulness in ‘real life’ because—well—
these situations just don't naturally occur that often.

On the other hand, the mental arithmetic techniques covered in this exercise are all widely applicable.
Master these techniques, and you'll go a long ways towards getting rid of any fears you might have about computing in public.
Like everything else (in math and in life)—the more you practice, the easier it will be!
There is unlimited randomly-generated practice available in these web exercises.

If you're working your way through this online algebra course, then you know by now that one theme is:

Numbers have lots of different names;
the name you use depends on what you're doing with the number.

All the mental math techniques we're going to discuss make use of this idea!
The primary skills needed are re-ordering, re-grouping, and these two familiar ‘renaming’ tools:

Most mental math techniques require mental addition, so let's practice that first.
You need to know your addition tables, through ten. (Practice here, if needed.)

The ideas for mental addition are illustrated next:

[beautiful math coming... please be patient] $47 + 51$  
   [beautiful math coming... please be patient] $= (40 + 7) + (50 + 1)$ Rename each number: tens plus ones.
   [beautiful math coming... please be patient] $= (40 + 50) + (7 + 1)$ Re-order and re-group.
When you do this mentally, you'll work with the tens first, ones last.
   [beautiful math coming... please be patient] $= 90 + (7 + 1)$ ‘Hold’ the tens sum in your head, while you add the ones.
   [beautiful math coming... please be patient] $= 98$ Finally, add the ones sum to the tens sum.

To practice, press the buttons below in the order they are numbered:

 
  Hold this sum in your head:  
  With the tens sum still in your head,
add the ones digits:
 
  Add the ones sum to the tens sum
for your final answer:
 

The prior exercise was carefully constructed so that no carrying was required.
As an example of carrying, consider this problem:

Add:   [beautiful math coming... please be patient] $37 + 56$  
[beautiful math coming... please be patient] $30 + 50 = 80$ add tens:  hold this in your head
[beautiful math coming... please be patient] $7 + 6 = 13$ add ones:  this gives you  1  ten and  3  ones
[beautiful math coming... please be patient] $80 + 13 = 80 + 10 + 3 = 93$ [beautiful math coming... please be patient] $80\,$ plus $\,10\,$ is $\,90\,$...   plus $\,3\,$...   is $\,93$

The next exercise extends the previous one; you may need to do some carrying.
No ‘hints’ this time.
Click the ‘NEW PROBLEM’ button, THINK, and then CHECK your answer:

   

Remember this key idea:
Always move from left (greatest place value) to right.
Compare this with traditional hand-computation methods, where you move from right to left.

One last idea.
Multiples of ten are easier to deal with than other numbers.
For example, [beautiful math coming... please be patient] $\,120 + 80\,$ is an easier problem than $\,119 + 79\,$.
For this reason, you'll sometimes want to use the
“TURN IT INTO A SIMPLER PROBLEM AND THEN ADJUST” rule.

Here's the math behind the method:

[beautiful math coming... please be patient] $119 + 79$  
   [beautiful math coming... please be patient] $= (120 - 1) + (80 - 1)$ rename $\,119\,$ as $\,(120 - 1)\,$;  rename $\,79\,$ as $\,(80 - 1)\,$
   [beautiful math coming... please be patient] $= (120 + 80) - 1 - 1$ re-order, re-group
   [beautiful math coming... please be patient] $= 200 - 2$ compute the new (easier) problem ...
   [beautiful math coming... please be patient] $= 198$ ... and adjust!

Thought process:
Bump [beautiful math coming... please be patient] $\,119\,$ up to $\,120\,$ to make it easier. (You just added [beautiful math coming... please be patient] $\,1$ ...)
Bump [beautiful math coming... please be patient] $\,79\,$ up to $\,80\,$ to make it easier. (You just added [beautiful math coming... please be patient] $\,1\,$ again ...)
Add [beautiful math coming... please be patient] $\,120\,$ to $\,80\,$ to get $\,200\,$. (But you've added [beautiful math coming... please be patient] $\,2$ ...)
Adjust: You added [beautiful math coming... please be patient] $\,2\,$. To undo this, subtract [beautiful math coming... please be patient] $\,2\,$.
[beautiful math coming... please be patient] $200 - 2 = 198$

The technique can be applied to just ONE of the numbers being added:

Consider: [beautiful math coming... please be patient] $459 + 36$
Think: [beautiful math coming... please be patient] $\,460 + 36\,$ is $\,496$ ...   but you added [beautiful math coming... please be patient] $\,1$ ...   subtract $\,1\,$ to adjust ...   so the answer is $\,495$

To practice the “Turn it into a simpler problem and then adjust” method,
press the buttons below in the order they are numbered:

 
  gives:  
  combining, you went:  
  and the answer is:  
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Don't Mix Up $\,3x\,$ versus $\,x^3\,$!

 
 

On this final exercise, the problems are all mixed up.
Remember: NO CALCULATOR! This is mental math!
Have fun!

Add:
    
(an even number, please)