Transcript Document

Decimals and Fractions
Day 3
Place Value
Let’s look at position
after the decimal to help
us do some rounding!
Rounding and Estimating

When rounding a decimal you must look at
the number to the RIGHT of the place
value to which you are going to round.

If that number if 5 or greater, then you
must raise the number by one in the
position to which you are trying to round.
Example

Round 73.410 to the
nearest whole number.

Round 2145.721 to the
nearest whole number.

73.410

2145.721
“4” is NOT greater
Than 5 so no
Change is necessary
To the “3.”
A: 73
“7” IS greater than
5 so you must
change the “5” to
a “6.”
A: 2146
Example

Round 36.480 to the
nearest tenth.

Round 9641.702 to the
nearest hundredth.

36.480

9641.702
Greater than 5!
A: 36.5
Not greater than 5!
A: 9641.70
Example

Round 10.4803 to the
nearest thousandth.

Round $55.768 to the
nearest cent.

10.4803

$55.768
Not greater than 5!
A: 10.480
Greater than 5!
A: $55.77
You Try: Round 58.97360 to the
nearest
Whole Number

59
Tenth

59.0
Hundredth

58.97
Thousandth

58.974
Ten Thousandth

58.9736
Comparing Decimals
Using Models – A Graphical Approach

If you are comparing tenths to hundredths, you
can use a tenths grid and a hundredths grid.
Here, you can see that 0.4 is greater than 0.36.
Another Way…..

Line up the numbers vertically by the decimal
point.

Add “0” to fill in any missing spaces.

Compare from left to right.
Let’s put these numbers in order:
12.5, 12.24, 11.96, 12.36
12 . 50
12 . 24
11 . 96
12 . 36
After 0’s have been added to
give
Fill in the missing space
the same number of decimal
places
with
a zero.
after the decimal, you can compare
easier by “dropping” the decimal.
BUT, remember to add the decimal
back after you decide the correct
order.
11.96 < 12.24 < 12.36 < 12.5
You Try: Arrange the following
numbers from least to greatest.

0.4, 0.38, 0.49, 0.472, 0.425

0.400
0.380
0.490
0.472
0.425





400
380
490
472
425
A: 0.38 < 0.4 < 0.425 < 0.472 < 0.49
Add and Subtract Decimals
The Basic Steps to Adding or
Subtracting Decimals:

Line up the numbers by the decimal point.

Fill in missing places with zeroes.

Add or subtract.

Be sure to put the larger number on top when
subtracting.
Example: 28.9 + 13.31
28.9 0
+28.9
13.31
+ 13.31
42.21
42.21
You Try
3.04 + 0.6
3.04
 0.60
______
3.64

8 + 4.7
8 .0
 4 .7
_____
12.7
Ex: Subtract the following: 4 – 1.5

4 – 1.5
4 .0
 1 .5
____
2 .5

25.1 – 0.83
25.10
 0.83
________
24.27
Subtracting Across Zeroes

If you have several zeroes in a row, and you
need to borrow, go to the first digit that is not
zero, and borrow.

All middle zeroes become 9’s.

The final zero becomes 10.
Example: 15 – 9.372
14
9
9
10
15.000
- 9.372
________
5.628
Multiply and Divide Decimals
To Multiply Decimals:






You do not line up the factors by the decimal.
Instead, place the number with more digits on
top.
Line up the other number underneath, at the
right.
Multiply
Count the number of decimal places (from the
right) in each factor.
Use the total number of decimal places in your
two factors to place the decimal in your product.
Example: 5.63 x 3.7
1
4 2
5.63
x 37
39 4 1
+ 168 9 0
20.8 31
.
two
one
1 1
1
three
Example: 0.53 x 2.618
2.618 has more digits (4) than 0.53 (3), so it goes on top.
3
1
4
2
2 618
x 0 53
7 854
130 900
+ 000 000
1.38 754
.
.
Decimal Places
three
two
1
five
Try This: 6.5 x 15.3
3 1
2 1
15.3
x 6.5
76 5
+ 918 0
9 9.4 5
one
one
1
two
Example: 0.00325  2.5
25 0.0325
0.0013
2.5 0.00325
Example:
55.0124  0.2
2 550.124
275.062
0.2 55.0124
You Try: 0.015  0.3
3 0.15
0.05
3 0.15
Fractions
Prime Numbers

A prime number is a natural number greater
than 1 that has exactly two factors (or
divisors), itself and 1.

The number 3 is prime because it is divisible
only by the factors 1 and 3.
List of Prime Numbers in the
Natural Numbers….
st
1
50
Composite Numbers

A composite number is a natural number
that is divisible by a number other than one
and itself.

The number 9 is composite because it is
divisible by 1,3, and 9 » more than 2 factors.
Prime Factorization

Every composite number can be expressed
as the product of prime numbers.

The process of breaking a given composite
number down into a product of prime
numbers is called prime factorization.
Example: Write 2100 as a product of
primes.



Select any two numbers whose product is
2100.
Among the many choices, two possibilities
are:
21 x 100 and 30 x 70.
Let’s look at branching for both of these
possibilities using a factor tree.

Both factor trees result in the same prime
factorization:
2 35  7
2
2
Division




Divide the given number by the smallest prime
number by which it is divisible.
Divide the previous quotient by the smallest prime
number by which it is divisible.
Repeat this process until the quotient is a prime
number.
Let’s look at division for the number 2100.
It has the same answer as the
branching method…..
2 2100
2 1050
3 525
2 35  7
2
2
5 175
5 35
7
Greatest Common Divisor - GCD

The GCD is used to reduce fractions.

One technique of finding the GCD is to use prime
factorization.

The GCD of a set of natural numbers is the
largest natural number that divides (without
remainder) every number in that set.
Example: What is the GCD of 12 and
18?


A longer way to determine the GCD is to list
the divisors of each.
Divisors of 12 {1,2,3,4,6,12}
Divisors of 18 {1,2,3,6,9,18}
The common divisors are 1,2,3, and 6.
Therefore, the greatest common divisor is
6.
Prime Factorization

If the numbers are large, this method is not
practical.

The GCD can be found more efficiently by
using prime factorization.
Steps to Finding the GCD Using
Prime Factorization
1.
2.
3.
Determine the prime factorization of each
number.
List each prime factor with the smallest
exponent that appears in each of the prime
factorizations.
Determine the product of the factors found
in step 2.
Example 1: Find the GCD of 54 and
90.
3
 The prime factorization for 54 is
23
23 5

The prime factorization for 90 is

The prime factors with the smallest
exponents are
2 and 3
2
2

The product of the factors found in the last step is
2  3  18.
2

The GCD of 54 and 90 is 18.

This means that 18 is the largest natural number
that divides both 54 and 90.
You Try. Find the GCD of 315 and
450.
315 : 3  5  7
2
450 : 2  3  5
2
2
Pr ime Factors with smallest exp onents :
3  5  45
45 is the GCD.
2
Least Common Multiple - LCM

To perform addition and subtraction of
fractions, we use the LCM.

The LCM of a set of natural numbers is
the smallest natural number that is
divisible (without remainder) by each
element of the set.
Example: Find the LCM of 12 and
18?


We could start by listing all of the multiples of
each number and stop when we get to the
smallest matching multiple.
Multiples of 12: {12,24,36,48,…}
Multiples of 18: {18,36,54,….}
The LCM is 36. However, there is an easier
way using prime factorization.
Steps to Finding the LCM Using
Prime Factorization
1.
2.
3.
Determine the prime factorization of each
number.
List each prime factor with the greatest
exponent that appears in any of the prime
factorizations.
Determine the product of the factors in step
2.
Example: Find the LCM of 54 and
90.

From a previous example we found
54  2  3 and 90  2  3  5
3


2
List each prime factor with the greatest
exponent that appears in either of the prime
factorizations: 2,33 ,5
The product will give the smallest natural
number that is divisible by both 54 and 90
(The LCM):
2  3  5  2  27  5  270
3
You Try: Find the LCM of 315 and
450.
315  3  5  7 and 450  2  3  5
2
2
2
PRIME Factors with Greatest Exponents :
2, 32 , 52 , 7
The product : 2  32  52  7  3150
The LCM is 3150.