The first thing we do when we add fractions is to find a common denominator. The denominator is the number on the bottom of a fraction. Fractions have common denominators when their denominators are all equal. 1/4, 3/4 and 7/4 Have the common denominator of 4. We can not just change the denominators and not change the numerators though. The Numerator is the number in the top part of a fraction. Fraction Means part of a number. When we say we have 1/2 that means 1 divided by 2. We can visualize this by drawing a box, drawing a line through that box so that it is cut in 2 equal parts, shading 1 of those parts.
How much of our box is shaded?... 1/2. Lets say we have a box and the shaded area of the box is 2/2. We can shade in the un-shaded part of our first box, so our whole box is shaded.
2 parts of 2 are shaded, which is = to 1. This is very useful to know in finding a common denominator so that we can add our fractions. 2/2 means 2 divided by 2. Whenever we divide a number by itself, the result (quotient) is one. Also whenever we multiply a number by 1, the result (product) is the value number we multiplied by 1. 3x1=3 For example. Lets go through a simple problem to see how to get our common denominator. 1/2 + 1/4 =_? If we simply changed the 2 in 1/2 to 4 making it 1/4 we don't have the same value for 1/2 anymore and our answer will come out wrong too.
In the illustration above there is a graphic representation of the fractions in the problem right next some of the fractions. You can see a red box around the graphical representations. In the first part of the illustration before the green squiggly line is the original problem, 1/2+1/4=. In the second part of the illustration, we multiply 1/2x2/2 in order to make our denominator 4. When you multiply fractions, just go across the top and multiply the numerators. Go across the bottom and multiply the denominators. 1/2x2/2=2/4. In the third part of the illustration, is a graphics showing the results of 1/2X2/2 as 2/4. Look at it closely and compare it to the graphics representing 1/2; see how 1/2 is = to 2/4. Remember that 2/2=1 and any time you multiply a number by 1 the product is the value of the number you multiplied by 1. In this case the value remains the same but the denominator and numerator are changed. Next we just go across the top and add 2+1. We just leave the denominator the same. Look at the sum of 1/2+1/4 which is 3/4 and its graphical representation. You can imagine just adding the green units of the 2/4 representation to the 1/4 representation. We can also prove to our selves that 1/2 does = 2/4 by finding the decimal value of 1/2 and then finding the decimal value of 2/4 and comparing them. We find the decimal value by dividing 1 by 2 for the value of 1/2 and dividing 2 by 4 for the value of 2/4 although we haven't shown how to do this just yet, you could use a calculator just to see for now.
The reason we need to make all the denominators = can also be seen looking at the addition illustration. We don't add one kind of unit to another. If we are adding feet, we wouldn't add 3 inches directly to 1 foot. There are 12 inches in one foot so 1 foot can be represented as 12/12 feet and 3 inches can be represented as 3/12 feet. Now we can add the numerators 3+12=15, leave the denominator the same so our sum is 15/12 feet or 15 inches. We can draw boxes with 12 equal divisions or graduations on them to represent feet and show the addition of some fractions of feet. Lets try 8 inches + 2 Inches.
If you do an addition or subtraction problem and it doesn't state in the problem what kind of units you are working with, yards, feet, pounds, miles, it is assumed that all the types of units in the problem are the same. If a problem does state what the units are and they are not the same, you will have to convert them so that they are all the same kind of units before you can add them or subtract them. This is not always true with multiplication and division. Much of the time we are converting from one kind of a unit to another using multiplication and division. For instance to change feet to inches, we multiply the number of feet by 12. 2Ft X 12In. = 24 In. To change inches to feet we divide. 24in/12=2Ft. Lets look at finding the common denominator for any 2 fractions. We have to multiply 1 or both our fractions by some value that will make both denominators equal to each other and keep the value of the original fractions the same.
1.)Look and see if the smaller of the 2 denominators will go evenly into the larger denominator. That is to say will the larger denominator divided by the smaller denominator be an even number with no remainder. If it does divide evenly like in 1/2+1/4=.
A.)4 divided by 2=2 therefore we can multiply 1/2x2/2=2/4 and our denominators will be equal.
2.)If one denominator won't divide into the other evenly.
A.)Multiply the first fraction in your problem by the value of the second fraction in your problem's denominator over itself and replace the first fraction with your new fraction.
B.)Multiply the second fraction in your problem by the value of the first fraction's denominator over itself and replace the second fraction in your problem with its new value.
This illustration shows instructions for 2.) A.) and B.) Once you have all your denominators the same just go across the top and add the numerators. The denominators do not change when we add fractions. As you do the practice problems you will sometimes get answers or sums with the numerator larger than the denominator. 7/8+5/6=? Gives such a sum. 82/42 The sum of 7/8+5/6 is called an improper fraction, because fraction means part of a number and 82/42 is greater than 1. 42/42=1. The first step in reducing 82/42 is to make it a proper fraction. Divide 82 by 42. It equals 1 40/42. 82-42=40, 40-42 won't go so 82 divided by 42 is 1 with 40/42 left over. Next we reduce the fraction portion of our sum by finding numbers that divide into the numerator and the denominator equal amounts of times with no remainders. Ask yourself will 2 go into 40 evenly? Yes; 40 divided by 2 = 20. Will 2 go into 42 evenly? Yes; 42 divided by 2 = 21. Rewrite our reduced version. 1 20/21 Repeat the process over and over until you can no longer reduce by 2. Follow the steps below to reduce any proper fraction to its simplest form. 1.) Ask yourself will 2 go into the numerator and denominator evenly? If it will divide the numerator and denominator by 2 as many times as they can be divided by 2 evenly. 2.) Ask yourself can numerator and denominator both be divided by 3 and if so, divide them both by 3 until you can no longer reduce by 3. 3.) Ask yourself if both numerator and denominator can be divided by 5 and if so, reduce by 5 until you can no longer reduce by 5. 4.) Ask yourself if both numerator and denominator can be reduced by 7 and if so, reduce by 7 until you can't reduce by 7 anymore. 5.) Try and reduce by 11 and reduce as many times as possible. Continue checking the odd numbers and increasing them until you reach the value of the numerator. 6.) Try and reduce by the value of the numerator. If you follow these 6 steps you'll always find the simplest form of the fraction. You can do these steps in any order and sometimes it will be quicker to start at step 6, but do check each possibility. In step 5 odd numbers are numbers that can't be divided by 2 without having any remainders. It is necessary to try all the odd numbers until you reach the value of the numerator because the numerator and denominator may be products of prime numbers or the numerator may be a prime number and be a factor of the denominator. A prime number is a number that is divisible by 1 and its own value without any remainder and not divisible by any other whole numbers without a remainder. 3 is prime. 3/3=1 and 3/1=3 but there are no other whole numbers we can divide it by and not have a remainder. 1, 2, 3, 5, 7, 11, 13, 17, 19 Are also prime numbers. 2 Is prime also and it seems strange that it is because all the other prime numbers are odd and not divisible by 2. 2 Meets the requirements of the definition though. 2 Can be divided by itself and 1 and no other numbers without leaving a remainder. As of the year 2007 no one has come up with a pattern for finding what the next prime number will be on a number line. Look at the spacing of the prime numbers on a number line. 1 Space between 1 and 2. 1 Space between 2 and 3. Then 2 spaces between 3 and 5. Then 2 spaces between 5 and 7. Next 4 spaces between 7 and 11. You can make a large number line with hundreds of numbers and look at the spacing and there doesn't seem to be any predictable pattern. This makes you have to try all those prime numbers until you reach the value of the numerator while reducing. A factor of a number is a whole number that when multiplied by some other whole number, equals that number. 6 Is a factor of 42 because 6x7=42. 7, 2 And 21 are also factors of 42; 21x2=42. A whole number is a number that doesn't have any fractional part. 2 1/2, 3.25 are not whole numbers because they contain fractional parts. 1,2,3 and 4 are whole numbers and do not have any fractional parts. Integer means the same thing as whole number and will be used in its place in many math problems
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Copyright 2008 Robert Lee Thomas
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