decimals and fractions

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.

7 3 . 4 1 0

Round 2145.721 to the nearest whole number.

2 1 4 5 . 7 2 1“4” is NOT greaterThan 5 so no Change is necessaryTo the “3.”

A: 73

“7” IS greater than5 so you must change the “5” to a “6.”

A: 2146

Example

Round 36.480 to the nearest tenth.

3 6 . 4 8 0

Round 9641.702 to the nearest hundredth.

9 6 4 1 . 7 0 2

A: 36.5 A: 9641.70

Greater than 5! Not greater than 5!

Example

Round 10.4803 to the nearest thousandth.

1 0 . 4 8 0 3

Round $55.768 to the nearest cent.

$ 5 5 . 7 6 8

A: 10.480 A: $55.77

Not greater than 5! Greater than 5!

You Try: Round 58.97360 to the nearestWhole Number

Tenth

Hundredth

Thousandth

Ten Thousandth

59

59.0

58.97

58.974

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, 12.3612 . 512 . 2411 . 9612 . 36

Fill in the missing space with a zero.

11.96 < 12.24 < 12.36 < 12.5

0 After 0’s have been added to give the same number of decimal placesafter the decimal, you can compare easier by “dropping” the decimal.

BUT, remember to add the decimal back after you decide the correct order.

You Try: Arrange the following numbers from least to greatest. 0.4, 0.38, 0.49, 0.472, 0.425

0.400 400 0.380 380 0.490 490 0.472 472 0.425 425

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

28.913.31+

28.9

42.21

0+ 13.31

42.21

You Try

3.04 + 0.6 8 + 4.7

64.3______

60.004.3

7.12_____

7.40.8

Ex: Subtract the following: 4 – 1.5 4 – 1.5 25.1 – 0.83

5.2____

5.10.4

27.24________

83.010.25

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

15.000- 9.372________

109914

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

5.633.7x

1

2

4

4

39098

1

16+13

1

8

1

0

1

2

two

one

three.

Example: 0.53 x 2.6182.618 has more digits (4) than 0.53 (3), so it goes on top.

2.6180.53x

4

2

58

1

700

4

90

3

13000000+457

1

831

Decimal Places

three

two

five.

Try This: 6.5 x 15.3

15.36.5x

5

1

6

2

708

1

1

3

9+54

1

99

one

one

two.

Example: 0.00325 2.5

0013.000325.05.20325.025

Example:

124.5502

2.00124.55

062.2750124.552.0

You Try: 3.0015.0

15.03 05.015.03

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 1st 50 Natural Numbers….

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:

7532 22

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

5 175

5 35

7

7532 22

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 Factorization1. Determine the prime factorization of each

number.2. List each prime factor with the smallest

exponent that appears in each of the prime factorizations.

3. Determine the product of the factors found in step 2.

Example 1: Find the GCD of 54 and 90. The prime factorization for 54 is

The prime factorization for 90 is

The prime factors with the smallest exponents are

332

532 2

232and

The product of the factors found in the last step is

The GCD of 54 and 90 is 18.

This means that 18 is the largest natural number that divides both 54 and 90.

.1832 2

You Try. Find the GCD of 315 and 450.

.454553

:expPr532:450

753:315

2

22

2

GCDtheis

onentssmallestwithFactorsime

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 Factorization1. Determine the prime factorization of each

number.2. List each prime factor with the greatest

exponent that appears in any of the prime factorizations.

3. Determine the product of the factors in step 2.

Example: Find the LCM of 54 and 90. From a previous example we found

List each prime factor with the greatest exponent that appears in either of the prime factorizations:

The product will give the smallest natural number that is divisible by both 54 and 90 (The LCM):

532903254 23 and

5,3,2 3

2705272532 3

You Try: Find the LCM of 315 and 450.

.315031507532:

7,5,3,2

:532450753315

22

22

222

isLCMTheproductThe

ExponentsGreatestwithFactorsPRIMEand