ROUNDING DECIMALS TO A SPECIFIED NUMBER OF PLACES

When you're solving a problem where an approximation suffices for the answer,
the most commonly-requested approximation looks something like this:
“please round to [beautiful math coming... please be patient] $\,2\,$ decimal places”
or equivalently,
“please round to the hundredths place”.
The purpose of this section is to discuss the concept and the technique of rounding.

Let's start with an example.
Suppose you're asked to round [beautiful math coming... please be patient] $\,x = 2.73845\,$ to two decimal places.
The number $\,x = 2.73845\,$ lies between $\,2.73\,$ and $\,2.74\,$ on the number line.
Indeed, it lies just about here:

the number 2.73845 between 2.73 and 2.74 on a number line

If we're rounding to two decimal places,
then the two “candidates” for the desired approximation are [beautiful math coming... please be patient] $\,2.73\,$ and $\,2.74\,$.
Which candidate is $\,x\,$ closest to?
Answer: [beautiful math coming... please be patient] $\,2.74\,$
So we say:
“$\,2.73845\,$, rounded to $\,2\,$ decimal places, is $\,2.74\,$”
or equivalently,
“$\,2.73845\,$, rounded to the hundredths place, is $\,2.74\,$”.

You could now be asked the follow-up question:
“Did you round up, or round down?”
When the candidate to the right of the original number is chosen,
we say that we are rounding up,
because we're moving “up” (to the right) on the number line to get the approximation.

When the candidate to the left of the original number is chosen,
we say that we are rounding down,
because we're moving “down” (to the left) on the number line to get the approximation.

To “round to [beautiful math coming... please be patient] $\,n\,$ decimal places
means to choose the closest neighbor that uses exactly $\,n\,$ decimal places.

When the two candidates are equidistant, the normal convention is to round UP;
that is, choose the candidate on the right.

EFFICIENT TECHNIQUE FOR ROUNDING

The ideas illustrated in the previous example can be efficiently implemented as follows.

ROUNDING TO $\,\,n\,\,$ DECIMAL PLACES
GENERAL PROCEDURE EXAMPLE:
Round [beautiful math coming... please be patient] $\,2.73845\,$ to $\,2\,$ decimal places.
STEP 1:   (rounding digit)

Go to the digit in the [beautiful math coming... please be patient] $\,n^{\text{th}}\,$ decimal place.
This is called the rounding digit.
Because we're rounding to [beautiful math coming... please be patient] $\,n = 2\,$ decimal places,
we want the digit in the second decimal place ($\,2.7\underline{3}845\,$).
The rounding digit is $\,3\,$.
STEP 2:   (decider digit)

Look at the next digit to the right.
This is called the decider digit.
It is used to decide whether to round up or down (next step).
What digit is to the right of the [beautiful math coming... please be patient] $\,3\,$?   ($\,2.7\underline{3}{\bf8}45\,$)
Answer: $\,8\,$
The decider digit is $\,8\,$.
STEP 3:   (rounding UP or DOWN)

If the decider digit is less than [beautiful math coming... please be patient] $\,5\,$, then round DOWN:
the rounding digit remains the same,
and all the digits to the right are dropped.

If the decider digit is [beautiful math coming... please be patient] $\,5\,$ or more, then round UP:
the rounding digit increases by $\,1\,$,
and all the digits to the right are dropped.

If the rounding digit is [beautiful math coming... please be patient] $\,9\,$ and you're rounding up,
this is a special case. See the next step!
Our decider digit is [beautiful math coming... please be patient] $\,8\,$, which is more than $\,5\,$.
We need to round UP.
The rounding digit increases by $\,1\,$ (from $\,3\,$ to $\,4\,$);
all digits to the right are dropped.
So, $\,2.73845\,$ rounds to $\,2.74\,$.
SPECIAL CASE:
ROUNDING DIGIT IS [beautiful math coming... please be patient] $\,9\,$, ROUNDING UP


If the rounding digit is [beautiful math coming... please be patient] $\,9\,$ and you're rounding up,
then the rounding digit becomes a $\,0\,$ (which you keep),
and the next digit to the left increases by $\,1\,$.
You may need to apply this rule more than once.

Whenever a [beautiful math coming... please be patient] $\,9\,$ must be increased by $\,1\,$,
then the $\,9\,$ becomes a $\,0\,$ (which you keep),
and the next digit to the left increases by $\,1\,$
.
NEW EXAMPLE:
Round [beautiful math coming... please be patient] $\,2.79845\,$ to $\,2\,$ decimal places.

The rounding digit is $\,9\,$.
The decider digit is $\,8\,$, so we round UP.
The $\,9\,$ becomes a $\,0\,$ (which we keep);
the digit to the left increases by $\,1\,$ (from $\,7\,$ to $\,8\,$).
So, [beautiful math coming... please be patient] $\,2.79845\,$ rounds to $\,2.80\,$.

Good way to think of this:
the   ‘$\,79\,$’   in   $\,2.\underline{79}845\,$   becomes an   ‘$\,80\,$’.
EXAMPLES:
[beautiful math coming... please be patient] $293.4991927\,$, rounded to $\,3\,$ decimal places, is:   $\,293.499$
(The decider digit is $\,1\,$; round down.)
[beautiful math coming... please be patient] $293.4699927\,$, rounded to $\,3\,$ decimal places, is:   $\,293.470\,$
(The decider digit is $\,9\,$; round up.)
Notice that you must include the $\,0\,$ in the thousandths place.
[beautiful math coming... please be patient] $293.4999927\,$, rounded to $\,3\,$ decimal places, is:   $\,293.500\,$
(The decider digit is $\,9\,$; round up.)
Notice that you must include the two trailing zeros.
[beautiful math coming... please be patient] $999.9999992\,$, rounded to $\,6\,$ decimal places, is:   $\,999.999999\,$
(The decider digit is $\,2\,$; round down.)
[beautiful math coming... please be patient] $999.9999998\,$, rounded to $\,6\,$ decimal places, is:   $\,1000.000000\,$
(The decider digit is $\,8\,$; round up.)
Notice that you must include all the trailing zeros.

CAREFUL!
When you are asked to round a number to [beautiful math coming... please be patient] $\,n\,$ decimal places,
then your answer must have $\,n\,$ decimal places,
even if there are zeros in those positions.

MORE EXAMPLES:
Round [beautiful math coming... please be patient] $\,0.92746\,$ to 3 decimal places.
Answer: $\,0.927$
If the decimal is less than $\,1\,$, then it is good style to put a zero in the ones place.
In the web exercise below, you must put the zero in the ones place to be recognized as correct.
Round [beautiful math coming... please be patient] $\,2.999997\,$ to $\,4\,$ decimal places.
Answer: $\,3.0000$
If you round to $\,4\,$ decimal places,
then your answer must have $\,4\,$ decimal places,
even if there are zeros in those positions.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Tables of Unit Conversion Information

 
 
Round  
  to  
  decimal place(s):
    
(an even number, please)