Rounding

Rounding is a process by which we make long, detailed numbers easier to understand (or estimate with).

The most general definition of rounding is to simplify a number, by cutting off the details, to a particular place value.  When dealing with numbers in the millions, do you really need to know the details in the hundreds and thousands?  If not, you can round to the nearest million.

We often round decimals to the nearest whole number.  When we see something that is priced at \$19.99, we typically say that it costs \$20.  We don't even think of that as rounding -- it's just simplifying.  But, officially, it's rounding to the nearest whole.

So, officially, how do we round and what are the rules?

When we round, we round to a particular place value (see lesson on Place Value for naming conventions of place values).  Once we round, every digit to the right of that place value becomes a zero.  The digit we are rounding to either goes up by 1 (rounds up) or stays the same (rounds down).

This process is easiest to see in examples.

Example: 

Round $19$ to the nearest ten.

First, we like to underline the digit we are rounding to: $\underline{1}9$

Then look at the digit behind the underlined digit: $\underline{1}\overset{\downarrow}{9}$

If the digit you're looking at is 5 or greater, raise the underlined digit by 1 and make every digit to the right $0$:

$\underline{1}\overset{\downarrow}{9}\rightarrow 20$

Note: if the digit you are looking at is less than 5, keep the underlined digit the same and make every digit to the right $0$.

Example: 

Round $623$ to the nearest hundred.

First, we like to underline the digit we are rounding to: $\underline{6}23$

Then look at the digit behind the underlined digit: $\underline{6}\overset{\downarrow}{2}3$

If the digit you're looking at is 5 or greater, raise the underlined digit by 1 and make every digit to the right $0$: Not true here.

If the digit you are looking at is less than 5, you keep the underlined digit the same, and then turn every digit to the right into a $0$.

$\underline{6}\overset{\downarrow}{2}3\rightarrow 600$

It's important to remember that, when rounding, you never touch or change the digits in front of (to the left of) the digit you are rounding to!

Example: 

Round $7887$ to the nearest ten.

First, we like to underline the digit we are rounding to: $78\underline{8}7$

Then look at the digit behind the underlined digit: $78\underline{8}\overset{\downarrow}{7}$

If the digit you're looking at is 5 or greater, raise the underlined digit by 1 and make every digit to the right $0$: 

$78\underline{8}\overset{\downarrow}{7}\rightarrow 7890$

This process works in the exact same way when when working with decimals (or numbers with decimals)

Example: 

Round $76.45$ to the nearest whole.

First, we like to underline the digit we are rounding to: $7\underline{6}.45$

Then look at the digit behind the underlined digit: $7\underline{6}.\overset{\downarrow}{4}5$

If the digit you're looking at is 5 or greater, raise the underlined digit by 1 and make every digit to the right $0$: Not true here.

If the digit you are looking at is less than 5, you keep the underlined digit the same, and then turn every digit to the right into a $0$:

$7\underline{6}.\overset{\downarrow}{4}5\rightarrow 76.00$

Example: 

Round $987.346782$ to the nearest hundredth.

First, we like to underline the digit we are rounding to: $987.3\underline{4}6782$

Then look at the digit behind the underlined digit: $987.3\underline{4}\overset{\downarrow}{6}782$

If the digit you're looking at is 5 or greater, raise the underlined digit by 1 and make every digit to the right 0: 

$987.3\underline{4}\overset{\downarrow}{6}782\rightarrow 987.35$.  

Remember, with decimals you can drop trailing zeros.

Although, out of context, rounding often seems like a silly skill, students really should work to understand and master it.  As they get into higher math, students will often be asked to round their answers to specific place values (and may well get questions wrong if they can't round or round incorrectly).  So, rounding is a basic math skill that every student should have in his or her toolkit.