# Mental Math: Addition

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

- Multiplication by 11
- Multiplication by 111
- Squaring Two-Digit Numbers that End in 5
- Multiplying Complementary Pairs

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:

- The distributive law: For all real numbers $\,a\,$, $\,b\,$, and $\,c\,$, $\,a(b + c) = ab + ac\,.$
- Zero is the additive identity: For all real numbers $\,x\,,$ $\,x + 0 = x\,.$

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:

$47 + 51$ | |

$\ \ = (40 + 7) + (50 + 1)$ | Rename each number: tens plus ones. |

$\ \ = (40 + 50) + (7 + 1)$ | Re-order and re-group. When you do this mentally, you'll work with the tens first, ones last. |

$\ \ = 90 + (7 + 1)$ | ‘Hold’ the tens sum in your head, while you add the ones. |

$\ \ = 98$ | Finally, add the ones sum to the tens sum. |

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

- Pressing (1) gives you a new addition problem.
- Before pressing (2), mentally add the tens digits; hold this sum in your head.
- Before pressing (3), mentally add the ones digits.
- Before pressing (4), 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: $37 + 56$ | |

$30 + 50 = 80$ | Add tens: hold this in your head |

$7 + 6 = 13$ | Add ones: this gives you $\,1\,$ ten and $\,3\,$ ones |

$\displaystyle \begin{align} &80 + 13\cr &\ \ = 80 + 10 + 3\cr &\ \ = 93 \end{align}$ | $80\,$ 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, $\,120 + 80\,$ 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:

$119 + 79$ | |

$\ \ = (120 - 1) + (80 - 1)$ | Rename $\,119\,$ as $\,(120 - 1)\,.$ Rename $\,79\,$ as $\,(80 - 1)\,.$ |

$\ \ = (120 + 80) - 1 - 1$ | Re-order, re-group. |

$\ \ = 200 - 2$ | Compute the new (easier) problem ... |

$\ \ = 198$ | ... and adjust! |

Thought process:

- Bump $\,119\,$ up to $\,120\,$ to make it easier (you just added $\,1\,$)
- Bump $\,79\,$ up to $\,80\,$ to make it easier (you just added $\,1\,$ again)
- Add $\,120\,$ to $\,80\,$ to get $\,200\,$ (but you've added $\,2\,$)
- Adjust: You added $\,2\,$. To undo this, subtract $\,2\,$.
- $200 - 2 = 198$

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

Consider: $459 + 36\,.$ Think: $\,460 + 36\,$ is $\,496\,...$ But you added $\,1\,...$ 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:

- Pressing (1) gives you a new addition problem.
- Before pressing (2), think about what
*simpler*problem you're going to compute first. - Before pressing (3), figure out how the numbers in your simpler problem differ from the actual numbers.
- Before pressing (4), determine your
*adjustment*and your final answer.

## Practice

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