Rounding Numbers

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#### How do you round a number to the nearest tenth (10th), hundredth (100th), thousandth (1000th), etc.?

Here are two explanations from the Elementary Fractions and Decimals area of our archives, followed by links to more information:

Rounding Decimals
```  The first thing to know about rounding decimals is that it works
exactly like rounding whole numbers. To round to the nearest tenth, you
do the same thing you do to round to the nearest ten, except that you
are working with a different digit in the number. I can use the same
picture to explain both:

|                        |              X-------->|
+----+----+----+----+----+----+----+----+----+----+
140  141  142  143  144  145  146  147  148  149  150

This shows that 148 is between 140 and 150, and is closer to 150
because it's bigger than 145.

The next picture shows that 1.48 is between 1.40 and 1.50, and is closer
to 1.50 because it is bigger than 1.45:

|                        |              X-------->|
+----+----+----+----+----+----+----+----+----+----+
1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50

To round to the nearest ten, you drop all digits to the right of the
tens place, replacing them with zero, so that you are left with a
multiple of 10. Then you check to see whether your original number is
actually nearer to the next multiple of 10, by checking whether the
next digit (the leftmost one you dropped) was 5 or more:

148 rounded to nearest 10:
|\
| \ replace 8 with 0
V  \
140   8 is greater than 5, so add 10:
|
|
V
150

What all this means is that 148 is between 140 and 150, but is closer
to 150 so we round it to that.

Now, to round to the nearest tenth, you do the same thing, but using
the tenths digit rather than the tens:

1.48 rounded to nearest 0.1:
|\
| \ replace 8 with 0
V  \
1.40   8 is greater than 5, so add 0.1:
|
|
V
1.50

Again, this means that 1.48 is between 1.4 and 1.5, but is closer to
1.5. If the number had been 1.45, exactly between the two, you would
still round up to 1.5. If it were 1.43, you would have rounded down,
leaving it at 1.40, because 3 is less than 5.

The only real difference is that when you put zeroes to the right of a
decimal, you can ignore them completely because they don't affect the
number. So you can write the answer as 1.5 rather than 1.50.

- Doctor Peterson, The Math Forum
```

Rounding to the Nearest...
```  We always want to keep two things in mind when we round any number.
The first is what we are rounding to. In your example we are rounding
to tenths, so we should end up with one digit to the right of the
decimal point, because that is the tenths place. Now immediately you
can see that 451 is not an acceptable answer, but 451.7 or 451.8
could be.

The second thing to keep in mind is the method we are using to round
by. In your problem that is the nearest method. In the nearest method
we look at the digit to the immediate right of the digit we will round
to (so for this example we will look at the hundredths digit), and if
it is a five or greater (5 through 9) we round up (add 1 to the digit
we are rounding and drop all the digits to the right). If the digit
to the right of our rounding digit is less than 5 (0 through 4) we
round down (drop all digits to the right of digit we are rounding).

So for our example:

number to be rounded to nearest tenth      451.7576
number to the right of the tenths digit    5
since the number is a 5 we round up   451.8

number to be rounded to nearest tenth      451.7476
number to the right of the tenths digit    4
since the number is a 4 we round down      451.7

How about rounding 451.7576 to the nearest hundredth?

number to be rounded to nearest hundredth      451.7576
number to the right of the hundredths digit    7
since the number is a 7 we round up       451.76

Now you try: (solutions below)

a) round 72.3478 to the nearest tenth
b) round 15.4201 to the nearest tenth
c) round 5.74999 to the nearest tenth
d) round 87.5555 to the nearest hundredth

I hope this is a bit clearer now. If anything was not clear in my
explanation, just write back noting what was confusing, and I will try
to make it clearer. To check your work, here are the answers to the
questions above:

a) 72.3
b) 15.4
c) 5.7
d) 87.56 .

I hope that you have more fun in class!  Good luck,

-Doctor Keith,  The Math Forum
```

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