# Upper and lower boundary mixed

If you have a calculation such as `(atimesb)-c` and you need to know the possible limits of the upper boundary and lower boundary

Apply the separate rules for each operation

and apply BIDMAS

(see our section on BIDMAS)

**Example**

The following formula has been corrected to 1 D.P.

`(23.7times31.3)-16.8`

What are the upper and lower boundaries of this calculation?

First find the upper and lower boundary of `23.7` to 1 D.P.

`23.7` to one decimal place

`23.ul7` underline the digit (`1^(st)` decimal place).

`23.ul7\0` look next door

`0` `5` or more raises the score. So `23.65` would raise to `23.7`.

`0` four or less just ignore `23.7499` it would ignore and stay at `23.7`.

This is simplified to `23.75`

So the upper and lower boundary of `23.7` is `23.75` and `23.65`.

Now find the upper and lower boundary of `31.3` to 1 D.P.

`31.3` to 1 D.P.

`31.ul3` underline the digit (`1^(st)`decimal point).

`31.ul3\0` look next door

`0` `5` or more raises the score. So `31.25` would raise to `31.3`.

`0` four or less just ignore `31.3499` it would ignore and stay at `31.3`.

This is simplified to `31.35`

So the upper and lower boundary of `31.3` is `31.35` and `31.25`.

Now find the upper and lower boundary of `16.8` to 1 D.P.

`16.8` to 1 D.P.

`16.ul8` underline the digit (`1^(st)`decimal point).

`16.ul8\0` look next door

`0` `5` or more raises the score. So `16.75` would raise to `16.8`.

`0` four or less just ignore `16.8499` it would ignore and stay at `16.8`.

This is simplified to `16.85`

So the upper and lower boundary of `16.8` is `16.85` and `16.75`.

**Summary **

Upper and lower boundary of `23.7` is `23.75` and `23.65`

Upper and lower boundary of `31.3` is `31.35` and `31.25`

Upper and lower boundary of `16.8` is `16.85` and `16.75`

**i** - So the upper boundary is`(23.7times31.3)-16.8` to 1 D.P.

**NOTE:**

Multiplication use the maximum boundary of each

Subtraction use the biggest difference between the two

This would be:

`(23.75` | `xx` | `31.35)` | `-` | `16.75` | |

Big | `xx` | Big | `-` | Small |

Applying BIDMAS.

`744.56-16.75=727.81` to 2 D.P.

**ii** - The lower boundary is `(23.7times31.3)-16.8` to 1 D.P.

**NOTE:**

Multiplication use the minimum boundary of each

Subtraction use the smallest difference between the two

This would be:

`(23.65` | `xx` | `31.25)` | `-` | `16.85` | |

Small | `xx` | Small | `-` | Big |

Applying BIDMAS.

`739.06-16.85=722.21` to 2 D.P.

**Answer: **The upper and lower boundary of this calculation is `727.81` and `722.21`.