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 $722.21$ .