Rounding and Estimation
 The Number Line
On of the most useful ways of displaying numbers is the number line which is
defined as follows. First draw a line. then label a number on
the line (usually 0). To the left of that number represents values
below and to the right of that number represents values above that
number. We usually make sure that the line has uniform scale.
For example if 0 is the central point, and 1 is two cm from 0, than 2 should
be 4 cm from zero.
Example
 Rounding Whole Numbers
According to the census bureau, the population of the United States is 285,868,158.
Although this number is precise, it is cumbersome and difficult to
remember. Instead of using this number, we would like to be able to
write an estimate that contains only the leftmost digit. We could use
200,000,000, but there is a better estimate. If we place this number
on the number line, we see that 285,868,158 is between 200,000,000 and
300,000,000. However the population is closer to 300,000,000.
We say that to the nearest hundred thousand, the population of the US is
300,000,000.
Without graphing how can we determine how to round? If the number is
halfway between or greater, we round up to the larger number.
Otherwise we round down to the smaller number. We can look to the second
digit to the left as the determining factor. Since 5 is the half way
point from 1 to 10, we follow the rules below:
Rounding a Whole Number
 If the second digit to the left is less than 5 we
make not change to the round off digit.
 If the second digit to the left is 5 or more we
add one to the round off digit.
Then change all the digits except the round off
digit to zeros. 
Examples
 Round 246 to the nearest hundred.
Solution
The hundreds digit is the 2, the next digit
is 4 which is less than 5.
By rule 1. we do not change the 2 and we
replace the 4 and 6
with zeros.
200
 Round 76,779 to the nearest ten thousand.
Solution
The ten thousands digit is 7 and the next
digit is 6 which is greater than 5.
By rule 2. we change the 7 to an 8
and the rest of the numbers become zeros.
80,000
 Round 43,981
to the nearest thousand.
Solution
The thousands digit is 3,
and the number to its right is 9
which is greater than 5. We change the 3
to a 4 and
everything to its right to a zero
44,000
Exercises
 Round 5,342,167 to the nearest
million
 Round 28,194 to the nearest hundred
 Rounding and Arithmetic
Example
Suppose you are planning to have a party for 293 guests at a reception hall
that charges $42 per person. If you want to figure the total bill, you
could multiply the two numbers. However, if you just want a ball park
figure, there is an easy way to quickly find the solution. We round
each number so that only the left digit is nonzero. We can say that
there are about 300 guests at $40 per person. Now the multiplication
becomes easy
300
x 40
12,000
We can conclude that the total cost will be
about $12,000.
The actual amount is $12,306 which is pretty close to the estimate.
We can use the same method to approximate any calculation.
Example
Use Rounding to estimate
592 + 421 + 389 + 830
Solution
We round each of the four numbers first and then add
600 592
rounds to 600
400 421
rounds to 400
400 389
rounds to 400
+ 800
830 rounds to 800
2200
 Application
A space ship must travel 67,231,428 km to get to mars. The ship can
travel at 62,326 km/hr. Estimate the total number of hour it will take
the ship to travel to mars.
Solution
To find the number of hours, we can divide the total km by the speed.
To make this easier, we first round the two numbers
60000 70000000
We can simplify this calculation by getting rid of four zeros from both
numbers:
1166 R 4
6 7000
6
10
 6
40
36
40
36
4
Notice that the remainder is more that half of
6. We therefore round up to the nearest whole number.
We conclude that it takes about 1167 hours to get
to mars.
Exercises
Use rounding to estimate
 345 + 278 + 523 +
289
 74,237 
28,153
 3,512 x
2,119
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