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Chunky Says Hi

Chunky is a floopy**
 * chunky also says Mr. H give your first class for math treasure loads of homework this weekend!!!!

Equivalent fractions are different fractions which name the same amount. The fractions 1/2, 2/4, 3/6, 100/200, and 521/1042 are all equivalent fractions. The fractions 3/7, 6/14, and 24/56 are all equivalent fractions. We can test if two fractions are equivalent by cross-multiplying their numerators and denominators. This is also called taking the cross-product. Example: Test if 3/7 and 18/42 are equivalent fractions. The first cross-product is the product of the first numerator and the second denominator: 3 × 42 = 126. The second cross-product is the product of the second numerator and the first denominator: 18 × 7 = 126. Since the cross-products are the same, the fractions are equivalent. Example: Test if 2/4 and 13/20 are equivalent fractions. The first cross-product is the product of the first numerator and the second denominator: 2 × 20 = 40. The second cross-product is the product of the second numerator and the first denominator: 4 × 13 = 52. Since the cross-products are different, the fractions are not equivalent. Since the second cross-product is larger than the first, the second fraction is larger than the first.

A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1. There are two methods of reducing a fraction to lowest terms. __Method 1:__ Divide the numerator and denominator by their greatest common factor. 12/30 = (12 ÷ 6)/(30 ÷ 6) = 2/5 __Method 2:__ Divide the numerator and denominator by any common factor. Keep dividing until there are no more common factors. 12/30 = (12 ÷ 2)/(30 ÷ 2) = 6/15 = (6 ÷ 3)/(15 ÷ 3) = 2/5

To change a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator of the fractional part. Examples: 2 3/4 = ((2 × 4) + 3)/4 =11/4 6 1/2 = ((6 × 2) + 1)/2 = 13/2

To change an improper fraction into a mixed number, divide the numerator by the denominator. The remainder is the numerator of the fractional part. Examples: 11/4 = 11 ÷ 4 = 2 //r//3 = 2 3/4 13/2 = 13 ÷ 2 = 6 //r//1 = 6 1/2

Method 1 - Convert to an equivalent fraction whose denominator is a power of 10, such as 10, 100, 1000, 10000, and so on, then write in decimal form. Examples:

Floopy = Robert Method 2 - Divide the numerator by the denominator. Round to the decimal place asked for, if necessary. Example: 13/4 = 13 ÷ 4 = 3.25 Example: Convert 3/7 to a decimal. Round to the nearest thousandth. We divide one decimal place past the place we need to round to, then round the result. 3/7 = 3 ÷ 7 = 0.4285… which equals 0.429 when rounded to the nearest thousandth. Example: Convert 4/9 to a decimal. Round to the nearest hundredth. We divide one decimal place past the place we need to round to, then round the result. 4/9 = 4 ÷ 9 = 0.4444… which equals 0.44 when rounded to the nearest hundredth. Which is a terminating decimal, not a repeating decimal. = =

=Equivalent Fractions =

Objective
After reviewing this unit, you will be able to:


 * Write equivalent fractions.
 * Find the prime factors of a number.
 * Write a fraction in its simplest form.

Equivalent Fractions
Equivalent fractions are fractions that //may// look different, but are equal to each other. Two equivalent fractions may have a different numerator and a different denominator. (A fraction is also equivalent to itself. In this case, the numerator and denominator would be the same.) Let's take a moment to demonstrate the concept of equivalent fractions. Follow the steps below. > ** This shaded portion represents 2/3. ** || || > ** The shaded portion is now 4/6. ** || || The shaded portion of the paper does not change, so the fraction of the paper shaded does not change. ** The fractions 2/3 and 4/6 are equivalent. **
 * # Take a sheet of paper and fold it twice, creating three equal sections. || [[image:http://cstl.syr.edu/FIPSE/fracunit/equifrac/equigif/Fig2.gif width="92" height="48" align="center"]] ||
 * # Now shade two of them.
 * # Fold the paper again, in the other direction, but down the center of the paper.
 * Equivalent fractions can be created by multiplying or dividing both the numerator and denominator by the same number. This number is referred to as a multiplier **. We can do this because, if you multiply both the numerator and denominator of a fraction by the same non-zero number, the fraction remains unchanged in value. In the demonstration above, we could get the fraction 4/6 by multiplying both the top and bottom of 2/3 by 2.

Example
Find two fractions that are equivalent to the fraction 1/2. Two fractions equivalent to 1/2 are 3/6 and 9/18. and

Example
Show that the fraction 8/12 is equivalent to the fraction 2/3. If you multiply both the numerator and denominator of 2/3 by 4, you get the fraction 8/12. Therefore, the two are equivalent.

Prime Factorization
We typically want to put fractions in their simplest form. Fractions can be simplified when the numerator and denominator have a common factor in them. Every composite number can be expressed as a product of prime numbers. These are referred to as prime factors. || Finding the prime factors of a composite number is done by dividing out the prime factors. For example, if we wish to find the prime factors of 24, we can start by dividing 24 by 2: > > If we want to find all the prime factors of 24, we must continue by finding the factors of 12. || || Now we can look at our sequence of division and list all the prime factors of 24. To review our sequence of division we have: From this we can see that 2, 2, 2, and 3 are all the prime factors of 24. A number can be written as the product of its prime factors, so 2 x 2 x 2 x 3= 24. When we report the prime factors of 24, we must list each occurrence of a number. How do we know where to start when looking for prime factors? To some extent this is trial and error, but there are some rules to help you find prime factors. .
 * ** Fundamental Theorem of Arithmetic **
 * ** A factor  of a number is a number that can be divided into the original number evenly (meaning there is no remainder). ** For example, 4 is a factor of 8. That means 8 can be divided by 4 and there is no remainder (8 ÷ 4 = 2). This means that 2 is also a factor of 8.
 * ** A prime number is a number that has only two factors, 1 and itself. ** For example, the number 2 can be divided evenly only by itself and 1, therefore, it is a prime number. The five smallest prime numbers are 2, 3, 5, 7, 11, and 13.
 * ** Numbers that are not prime numbers are referred to as composite numbers. ** The number 8 is a composite number since it has factors of 2 and 4.
 * * When we divide 24 by 2 we get a result of 12. Since 2 goes into 24 evenly, it is a factor of 24. It is also a prime factor since it can only be divided by itself and 1. So, 24 has 2 as a prime factor and 12 as a composite factor.
 * * We can also divide 12 by 2. When we do this we find that 2 and 6 are factors of 12, with 6 being another composite factor. || [[image:http://cstl.syr.edu/FIPSE/fracunit/equifrac/equigif/fig51.gif width="50" height="12" align="top"]] ||
 * * When we further divide 6 by 2 we get a result of 3. Finally, we have a result that is a prime number. So both 2 and 3 are prime factors of 6. || [[image:http://cstl.syr.edu/FIPSE/fracunit/equifrac/equigif/fig52.gif width="44" height="12" align="top"]] ||


 * ~ **Divisible By ** ||~ **Test ** ||~ **Example ** ||
 * 2 || The number is an even number. || 2248 is an even number. It will have 2 as a prime factor. ||
 * 3 || The sum of its digits is divisible by 3. || 951 has 3 as a prime factor since 9 + 5 + 1= 15, which is divisible by 3. ||
 * 5 || The number ends in 5 or 0. || 33505 ends in a 5, so it has 5 as a prime factor. ||

The first fifteen primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. One way to help you keep track of all occurrences of prime factors is creating a factor tree. Compare this with the sequence of division shown earlier. The factor tree makes it easy to identify the prime factors of a number. || || Now let's look at an example of finding the prime factors of a number.
 * The factor tree for 24 is shown at the right. The number circled at the end of each branch is a prime factor of our original number.

Example
What are the prime factors of 112? The prime factors of 112 are 2, 2, 2, 2, and 7, so 2 x 2 x 2 x 2 x 7= 112. If we wrote this as a sequence of division, it would look like: Using either of these techniques, we can find the prime factors. From the factor tree, just write down all the prime factors that are at the ends of branches. From the division sequence, we write down all the divisors and then the end result. Thus, we find the prime factors of 112 are: 2, 2, 2, 2, and 7.
 * * The first thing we notice is that 112 is an even number. This means it is divisible by 2. || [[image:http://cstl.syr.edu/FIPSE/fracunit/equifrac/equigif/fig53.gif width="62" height="12" align="bottom"]] ||
 * * If we set this up as the beginning of our factor tree, we have the diagram shown at the right. || [[image:http://cstl.syr.edu/FIPSE/fracunit/equifrac/equigif/fig11.gif width="93" height="47" align="bottom"]] ||
 * * From here, we have one branch with a prime factor, 2, and the other still has a composite number, 56. This composite number is even and therefore divisible by 2. So, we add this on to our tree. || [[image:http://cstl.syr.edu/FIPSE/fracunit/equifrac/equigif/fig12.gif width="121" height="74" align="bottom"]] ||
 * * We keep going down the factor tree dividing out the branches that have a composite number until we have all prime factors at the ends of our branches. || [[image:http://cstl.syr.edu/FIPSE/fracunit/equifrac/equigif/fig13.gif width="178" height="128" align="bottom"]] ||