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ROUNDING DECIMALS TO A SPECIFIED NUMBER OF PLACES
Jump right to the exercises!
The concepts for this exercise are summarized below.
For a complete discussion, read the text.
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 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.
The number x = 2.73845 lies between 2.73 and 2.74 on the number line.
Indeed, it lies just about here:
If we're rounding to two decimal places,
then the two "candidates" for the desired approximation are 2.73 and 2.74 .
Which candidate is x closest to?
Answer: 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 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.
The procedure is illustrated at each step with the problem:
"Round 2.73845 to 2 decimal places."
To round to n decimal places, do the following:
- (rounding digit)
Go to the digit in the nth decimal place.
This is called the rounding digit.
(Example: Because we're rounding to 2 decimal places, the rounding digit is 3 .)
- (decider digit)
Look at the next digit to the right.
This is called the decider digit.
(Example: The decider digit is 8 .)
- If the decider digit is less than 5 , then round DOWN.
That is, the rounding digit remains the same, and all the digits to the right are dropped.
- If the decider digit is 5 or more, round UP.
That is, the rounding digit increases by 1 , and all the digits to the right are dropped.
(Example: 2.73845 rounds to 2.74 )
WHAT IF THE ROUNDING DIGIT IS 9 , AND YOU'RE ROUNDING UP?
If the rounding digit is 9 and you're rounding up,
then the rounding digit becomes 0 , and the next digit to the left increases by 1 .
You may need to apply this rule more than once.
Whenever a 9 must be increased by 1 ,
then the 9 turns into a 0 , and the next digit to the left increases by 1 .
Here are some examples:
293.4991927 , rounded to 3 decimal places, is 293.499 .
(The decider digit is 1 ; round down.)
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.
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.
999.9999992 , rounded to 6 decimal places, is 999.999999 .
(The decider digit is 2 ; round down.)
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 n decimal places,
then your answer MUST have n decimal places,
even if there are zeros in those positions.
EXAMPLES:
Round 0.92746 to 3 decimal places.
Answer: 0.927
If the decimal is less than 1, then put a zero in the ones place.
Round 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.