 Decimals and Fractions

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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 to the nearest whole number. Round to the nearest whole number. “7” IS greater than 5 so you must change the “5” to a “6.” “4” is NOT greater Than 5 so no Change is necessary To the “3.” A: 73 A:

Example Round to the nearest tenth. Round to the nearest hundredth. Not greater than 5! Greater than 5! A: A:

Example Round to the nearest thousandth. Round \$ to the nearest cent. \$ Greater than 5! Not greater than 5! A: A: \$55.77

You Try: Round 58.97360 to the nearest
Whole Number Tenth Hundredth Thousandth Ten Thousandth 59 59.0 58.97 58.974

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 . 5 After 0’s have been added to give the same number of decimal places after the decimal, you can compare easier by “dropping” the decimal. BUT, remember to add the decimal back after you decide the correct order. Fill in the missing space with a zero. 11.96 < < < 12.5

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

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 + 13.31 42.21 42.21

You Try

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

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 two x 3.7 one 1 1 39 4 1 1 + 16 8 9 2 . 8 3 1 three

Example: 0.53 x 2.618 2.618 has more digits (4) than 0.53 (3), so it goes on top. 3 4 Decimal Places 1 2 2.618 three x 0.53 two 1 7 8 5 4 13 9 + 00 . 1 3 8 7 5 4 five

Try This: 6.5 x 15.3 3 1 2 1 15.3 one x 6.5 one 1 7 6 5 + 9 1 8 . 9 9 4 5 two

Example:

Example:

You Try:

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:

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 1050 3 525 5 175 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 {1,2,3,4,6,12} Divisors of {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
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.
The prime factorization for 54 is The prime factorization for 90 is The prime factors with the smallest exponents are

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.

You Try. Find the GCD of 315 and 450.

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
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 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):

You Try: Find the LCM of 315 and 450.