How do I devide a rational number? Such as fractions and such?

Question

God I'm sure that I learned this in High School but I just can't remeber how, and the text book is no help. Thx in advance

Answer 1

(a/b) / (c/d) =
(a/b) * (d/c)

For example:

(2/5) / (3/7) = (2/5) * (7/3) = 14/15

Answer 2

TURN IT INTO A DIGITAL NUM.(5.5=51/2) THEN DIVIDE

Answer 3

Rational Numbers
by Shelley Walsh ©2000

What are fractions?
Fractions are numbers that are used to represent parts of things. The meaning of a fraction goes like this. You divide up a whole into a number of pieces and that number is the denominator and it is put at the bottom. Then you take a number of those pieces and that number is called the numerator, and it is put at the top. For example in the fraction 3/5, you divide up the whole into 5 pieces and each piece is a fifth of the whole. Then the number 3/5 means the part of the whole that you get when you take 3 of those fifths, so it is called 3 fifths and written 3/5.

You can also think of fractions as division problems because you are dividing the number in the numerator into the denominator number of equal pieces and that is exactly what division does.
Determining Which Fraction is Biggest
It is good when working with any kind of number to have some kind of feel for how big it is. One way to get a feel for this with fractions is to compare them with each other. When one fraction is a lot bigger than the other you can often do this simply by picturing how big a piece they refer to, or you can do some estimating by asking yourself such things as whether they are bigger or smaller than certain simple fractions like 1/2 or 1/4. If the denominator is more than twice the numerator then the fraction is smaller than 1/2. But if the numbers are quite close together this might not be so easy, so another way to do it is to put them under a common denominator, something I will talk about when I tell you about adding and subtracting fraction.
Reducing to Lowest Terms and Equivalent Fractions
It is possible to write down many different fractions that represent the same sized piece, and such fractions are called equivalent. You can do this simply by cutting up all the pieces into an equal number of pieces. The effect of this on the fraction is to multiply the numerator and denominator by the same number, so multiplying numerator and denominator by the same number produces equivalent fractions. For example if you take 2/3 and cut each of the thirds into 5 equal pieces, then the new pieces are 15th, and it takes 10 of those pieces to make up the 2/3, so 2/3 is equivalent to 10/15.

It might help you better understand this if you draw some pictures of your own with different fractions or maybe for some more fun try this on a cake some time. Anyway, once you have convinced yourself that this works you see that it means that you can take any fraction and write as many equivalent fractions to it as you want by multiplying numerator and denominator by the same number.
This will be useful later when we add fractions, but going the other direction is more often what you want to do when you are writing your final answers to problems, because it is best to write fractions with as small a numbers as possible. Writing an equivalent fraction with the numerator and denominator as small as possible is what we mean by reducing a fraction to lowest terms, and this is a very important skill to have. To do this you must divide out as large a common factor as you can until there are no more common factors left. Ideally you want to divide by the greatest common factor, but if you don't all you have to do is reduce some more until there is nothing that will go into both numerator and denominator evenly anymore.

Addition and Subtraction
When you add fractions with like denominators you are adding up various numbers of pieces of like size. Remember, the denominator represents the size of the pieces and the numerator represents the number of pieces. When you are adding you want to know what the size that you get when you put the two sizes together, so you just add the numerators to find the total number of pieces. The size of the pieces hasn't changed, so you keep the same denominator. Similarly with subtraction you are taking away pieces, so you just subtract the numerators, and again the size of the pieces doesn't change, so the denominator is still the same.
The more difficult task comes when the denominators are different, because then you are adding or subtracting different sized pieces, so it is not at all clear what kind of fraction you can use to represent the total. The way to deal with this is to cut up your pieces into smaller pieces so that all the pieces are the same size. To do this you need to write the fractions as equivalent fractions with a common denominator. Remember the only way to write an equivalent fraction for a fraction which is in lowest terms is to multiply top and bottom by the same number. This means that the common denominator must be a multiple of all of the denominators in the problem. The name for such a creature is a common multiple. Any common multiple will do and one way you can always find a common multiple is to multiply the numbers, but it is best to use the smallest possible number which is the least common multiple, often abbreviated LCM. See GCFs and LCMs for help with finding the LCM.

So to add or subtract numbers with different denominators you need to find the least common multiple of the denominators and then write each fraction as an equivalent fraction that has that number as its denominator. To do that look at each denominator and ask yourself what number you need to multiply the denominator by to get the LCM and then multiply the numerator and denominator by that number. So for example if your denominators are 8 and 12, then your LCM is 24, so you need to multiply the one with the denominator of 8 top and bottom by 3 and the one with the 12 top and bottom by 2.

Mixed Numbers
A mixed number represents a number of wholes plus a part.
To convert a mixed number to an improper fraction you have to convert the whole number to a fraction with the same denominator as the fraction part. To do that you have to realize that each whole is the denominator number of pieces, so for example each whole in 4 and 2/3 is 3 thirds. This means that to figure out how many fraction pieces there are in the whole number part of a mixed number you need to multiply the whole number by the denominator. But then there is also the fraction part. Putting this all together, the denominator is the same as the denominator of the fraction part, and to get the numerator you multiply the whole number by the denominator and then add the numerator.

To convert an improper fraction to a mixed number you regard the fraction as a division and express the remainder as a fraction. To express a remainder as a fraction, the remainder is the numerator and the divisor is the denominator.

Addition and Subtraction of Mixed Numbers
There are two ways to add and subtract mixed numbers.
One way is to just convert them into improper fractions and add or subtract like usual. This sounds like the simplest way, and it is fine to do if the whole numbers are not too big like 1 or 2, but when you have big whole numbers like 23 or 149, this will involve a lot of unnecessary multiplying of big numbers and there is a better way.

The better way to do it, especially for when you have large whole number parts is to add or subtract the fractions and whole numbers separately and treat the fraction parts and the whole number parts sort of like you treat the digits of numbers. For addition this means that if you add up the fraction part and it turns out to be an improper fraction, then you convert the fraction part into a mixed number and carry the whole number part of this number to the ones place of the whole numbers. For subtraction it means that sometimes you will have to borrow. If the fraction part of the second number is larger than that of the first number then since you can't subtract a larger number from a smaller number you have to borrow from the ones place of the whole number part. So, for example, you might have to convert 4 and 5/6 into 3 and 11/6. To do this you would take one of the wholes and break it up into 6ths. 1 is 6/6, so you add the 6/6 to the 5/6 to get 11/6. A good quick way to figure out what the new fraction part is going to be when you borrow this way is to simply add the numerator and the denominator. Doing that in this example would give you 5+6=11, so the numerator is 11 just like I said it was.

Multiplication
Before we learn how to multiply fractions we ought to think a bit about what it means. Addition of fractions is just combining the pieces together, but what about multiplying? Multiplying whole numbers is repeated addition, but how do you repeat something a fraction number of times? That doesn't really make sense, but perhaps another interpretation of multiplication will help. If you multiply a fraction like say 2/3 by a whole number like 4, one way you could see it is that you are making 4 copies of the 2/3, sort of like you were representing 4 wholes, but 2/3 is the whole.

And when you do that, you can see from the picture that you get a total of 8 of those 1/3 pieces, or the fraction 8/3. Now let's suppose instead that you wanted to multiply 2/3 by 4/5. Then you could do it in the same way representing 4/5, but again thinking of 2/3 as the whole. I will make the original whole bigger so that it will be easier to see the pieces.

Now we can use this picture to see how to perform the operation of fraction multiplication. Notice that in splitting up the whole twice, the total number of pieces that the whole gets divided into is the product of the denominators, 3 times 5, or 15, and the number of pieces we are taking is the number of rows times the number of columns, which is the product of the numerators. So to multiply fractions, you multiply the numerators and multiply the denominators, straight across. And no matter how many fractions you have to multiply, you do it the same. Just multiply all the numerators and all the denominators, and you'll do just fine.
Cross Canceling
Cross Canceling is really, really important to do. With fractions multiplying is nicer than adding or subtracting in many ways, and one of them is that you can take care of all the reducing to lowest terms before you do the problem. The reason for this is that reducing to lowest terms is taking out common factors from the numerator and the denominator, and you can find this a lot easier before the multiplying is done. Because before the multiplying is done the numerator and denominator are partly factored, and you are looking for common factors. This means that many times if you multiplied out first and then tried to reduce to lowest terms, you would be multiplying and then refactoring (unmultiplying) in order to find the common factors, and that would be quite a waste. To avoid doing this you need to remember that all of the numerators will be factors of the numerator of the product and all of the denominators will be factors of the denominator of the product, so you can cancel out any factor of anything in a numerator with any factor of anything in a denominator. This is often called cross canceling, because if the original fractions were in lowest terms, and you only had two fractions, then all your canceling would be on the diagonals, but really you can cancel anything upstairs with anything downstairs. Then if you really have done all the canceling you can, after you multiply your numbers out, the final answer will automatically be in lowest terms and you won't have to do any further reducing. The only thing you might have to do with it further is convert it to a mixed number if it is an improper fraction.
Mixed Numbers
Multiplying fractions is mostly a lot easier than adding fractions, because you don't have to mess with common denominators, so I suppose it is only justice that something should be harder about it. What's harder about it is that for multiplying you do absolutely have to convert your mixed numbers to improper fractions. There is no nice way to multiply mixed numbers.
Multiplying and Adding
It is important to remember that multiplying fractions is very different from adding fractions and not get them confused. Remember for multiplying you don't need a common denominator and you multiply both the numerators and the denominators. For adding you need a common denominator and you add only the numerators.
Division
Reciprocals
the reciprocal of a number is the number you can multiply by it and get 1. So the reciprocal of 2 is 1/2 because when you multiply them everything cross cancels and you get 1. Similarly the reciprocal of 2/3 is 3/2. When you multiply any fraction by the fraction you get by turning it upside down everything will cross cancel and you will get 1. So the reciprocal of any fraction is the fraction you get by turning it upside down. The reciprocal is also sometimes called the inverse or more properly the multiplicative inverse.
How to Divide Fractions
When you divide two fractions like

you are looking for something you can multiply 6/7 by to get 3/4, just like when you divide

you are looking for something you can multiply by 8 and get 56. For

you can get this by multiplying

because look, if you multiply that by 6/7 you get

and the 7/6 and 6/7 cross cancel away, and you indeed get 3/4. So you can turn all division problems into multiplication problems by inverting the second fraction. So to divide, multiply by the reciprocal.
Rational Numbers
The rational numbers are the positive and negative fractions and whole numbers. A nice short and precise way of expressing this is that a rational number is any number that can be written as p/q where p and q are integers and q is not 0. This including the integers themselves, because q can be 1. See How to Add, Subtract, Multiply, and Divide Integers for information about the integers. Negative fractions work pretty much the same as positive fractions. You just have to follow the rules of positive and negative numbers (See How to Add, Subtract, Multiply, and Divide Integers ) as well as those for fraction arithmetic. How to Add, Subtract, Multiply, and Divide Integers
by Shelley Walsh ©2000

The Meaning of Negative Numbers
Negative numbers are numbers that are below 0. Now what kind of sense does it make to talk about numbers that are below 0? Well, you have heard of temperatures being below zero, but even that is kind of artificial because if you used the Kelvin scale, you wouldn't have to deal with them anyway. There are other things that kind of correspond to negative numbers like below sea level, but it's not like you're gong to deal with temperature or elevation all of the time. Perhaps a better reason to have negative numbers is the purely mathematical one, that it makes our number system more complete and easier to work with. Particularly when you are doing algebra and are working with unknown quantities, it is nicer if most if not all of the operations you can write down are do-able, and negative numbers allow you to subtract bigger numbers from smaller numbers. This comes up in the real world too, but it isn't a very pleasant subject. If you have 5 dollars and you spend 7 dollars then you are in debt 2 dollars, which could be represented as having -2 dollars.
We can represent negative numbers on the number line by writing them to the left of 0, since on the number line smaller numbers are written to the left of larger numbers. When you are comparing negative numbers with each other you must be careful, because the ones that look bigger are really smaller. If you are 7 dollars in debt you have less money than if you are 5 dollars in debt.

Opposites
Another good reason for having negative numbers is that it allows addition to have something that is called an inverse, which is also helpful when you study algebra. Every number has a number that can be added to it to get zero, and this number is called its opposite. To find the opposite of a number you simply change its sign, so the opposite of a positive number is a negative number, and the opposite of a negative number is a positive number. The symbol for opposite is the same as that for negative so this could be a little confusing. But it doesn't really matter that much to determine which meaning of the minus sign is appropriate, because they sort of are the same thing anyway. If you wanted to, you could simply think of all of the minuses as meaning opposite, because when it is being used to indicate a negative number, that number is the opposite of the positive number that the minus sign is on anyway, so it comes to the same thing. But all you really have to do is interpret the minus sign as indication of negativeness whenever that makes sense, and otherwise take it as meaning opposite. This basically means that the first minus sign on a number means negative and any further ones mean opposite.

Addition
What adding signed numbers is really about is combining adding and subtraction into one operation just as multiplying of fractions is combining multiplication and division into one operation. Adding positive numbers is adding, and adding negative numbers is subtracting. If you want to think of it on the number line you start from 0 and when you add a positive number you go that much to the right, and when you add a negative number you go that much to the left.
Rules for Adding
If you work this out case by case, you can come up with the rules for adding plus and minus numbers. Looking at the possibilities for combinations of signs, you can see that there are 4 possibilities.
+ +
Nothing new here, just add as usual.
+ -
You are going to the right and then to the left, so they are fighting with each other, and you don't get very far. Ignore the signs and subtract the bigger one minus the smaller one. The bigger one wins out as far as which direction you are going, so it determines the sign of your answer.
- +
You are going to the left and then to the right, so again they are fighting, so just like in the last case, you ignore the signs and subtract the bigger one minus the smaller one and the bigger one wins and determines the sign of your answer.
- -
Now you are going to the left and then going to the left again, so to determine how far you are going you ignore the signs and add, and since you will end up to the left of 0, the sign of your answer will be negative.
Actually you can collapse these rules into two cases, and this gives you a simpler statement of the rules.

Like Signs - Add and the sign is the common sign.
Different Signs - Subtract and use the sign of the bigger one.
Absolute Value
You might have noticed that I referred a couple of times to ignoring the signs in the above explanation. To avoid having to talk about ignoring the signs in such explanations and other times when it is called for, it is convenient to have a name for the number stripped of its sign, and the name given for this is absolute value. The absolute value of a number is simply the number without its sign. This means that if the number is positive, the absolute value of it is just itself, but if it is negative it is the number you get when you strip it of its minus sign, that is, the corresponding positive number. On the number line you can think of absolute value also as the distance from 0. The notation for the absolute value of a number x is |x|. So for example |7|=7, but |-4|=4. The answer to an absolute value evaluation is always positive.
Subtraction
With signed numbers we really don't need subtraction, but sometimes real world or intuitive reasons lead us to thinking about a problem in terms of subtraction, so we need a definition that would fit this for subtraction of signed numbers. Our original subtraction of positive numbers like 7-5 can be take care of by adding a negative, since 7+-5 give the same answer as 7-5, so we can generalize this to a definition for subtracting any two signed numbers. So to subtract two signed numbers you change it into an addition problem by changing the second number to its opposite. If you know about division of fractions, you might notice that this is very much like the case there where you divide my multiplying by the reciprocal. To subtract, change the sign of the second number and add. And that's really all there is to it. You just have to remember to do it, and that takes practice.
One thing that I have noticed can be confusing with subtraction is that the negative sign is the same symbol as the subtraction sign, so you have to make sure you don't get them confused or make one do double duty. One way to keep this straight is to draw a circle the two number and a different colored circle around the subtraction sign. To do this, first circle the first number. Then the next minus sign you see is the subtraction sign, so circle that in a different color. The any further minus sign must be a negative sign attached to a number, so circle whatever is left with the first color. Then to change the subtraction to addition, do this. First copy the first number, the one in the first circle. Then write an addition sign in place of the original subtraction sign, the one in the different colored circle. Then write down the opposite of the next number, the one that is circled in the same color as the first number.

Example 1:
3-7
Solution:

Explanation:
The two numbers are circles in red and the operation sign in blue. Write down the first number, the 2. Then change the operation to addition. Then change the sign of the second number. It was a positive 7, so now it becomes a -7. Now for the addition problem we have different signs, so they are fighting with each other, one number telling us to go forward and the other number telling us to go backward, so we subtract and get 4, but since the negative was the bigger one, it is a -4.
Example 2
-4-9
Solution:

Explanation:
This time the first number is -4 and the second number is 9, circled in gold with the minus sign circled in green. First write down the -4. Then change the operation to addition and write the addition plus sign down. Then change the sign of the second number the 9, which will make it a -9. Now for the addition problem we have two negative numbers to add. Since they have the same sign, they both want us to go in the same direction, so we add them to get 13. But since they are both negative numbers, it is a negative 13.
Example 3:
6-(-7)
Solution:

Explanation:
This time the first number is 6, and the second number is -7. The parentheses aren't really necessary here. They are just here to make it easier to read by keeping the minus signs from running together. Then we change the problem to an addition problem. First write down the 7. Then change the subtraction to addition, so write down a plus sign for that. That change the sign of the second number from negative to positive to make it a positive 7 instead of a negative 7. Now for the addition, it is just adding to positive numbers, which we already knew how to do before learning about negative numbers.
Example 4:
-5-(-6)
Solution:

Explanation:
Again the parentheses aren't really necessary. The two numbers are -5 and -6, so to change it to an addition problem we are adding -5 and 6. For that addition problem we have different signs, but this time the positive one is bigger, so we subtract and get a positive answer.
Multiplication
Now lets look at the various combinations of plus and minus for multiplication of signed numbers.
+ +
No negative numbers here, so nothing new.
+ -
Multiplying by positive numbers means repeated addition, so the same thing should be true when you multiply it by a negative number. The repeated addition of a negative number gives a negative number and the absolute value, that is the size without the minus sign, is simply the product of the two absolute values. This means when you multiply a positive number times a negative number, you multiply the two numbers ignoring the signs, and the sign of the answer is negative.
- +
We want multiplication to be commutative, so this should be done the same way as +-.
- -
This one is slightly trickier to understand, and I've never seen a convincing physical interpretation of it. Mainly you have to accept this on the basis that it is the only mathematically consistent way to define it given the other definitions. There are a number of ways to think about it. If -+ is - then -- somehow has to be something different, so it must be +. Or you can think of it as, since -+ is - then multiplying by a minus must change the sign, so in a minus times a minus the first minus must change the sign of the second one to plus. An interesting more formal way of seeing it is to use the distributive property. Take an example with numbers to make it friendlier.

-5(6+-6)=(-5)(0),
but also by the distributive property
-5(6+-6)=(-5)(6)+(-5)(-6)=-30+(-5)(-6).

So whatever (-5)(-6) is, when you add it to -30 you have to get 0, and the only thing you can add to -30 and get 0 is 30. Anyway, however you see it, it seems that the only possible thing for the product of two negatives to be is a positive. I know two wrongs don't make a right, but strange as it seems, in mathematics the product of two negatives is indeed a positive.

I think one important thing to think about if you get bothered by the idea of negative times negative being positive is that multiplication is really a much more complicated operation than addition, and it is definitively not the same as addition. If you are thinking this can't be true because two wrongs don't make a right, you need to realize that combining two wrongs is adding them, not multiplying. Multiplying two negative numbers is something different. If multiplication is repeated addition, what does it mean to repeatedly add something a negative number of times? That doesn't really literally make a lot of sense. Before you can decide what a negative times a negative should be, you have to first decide what is meant by multiplying by a negative number. To some extent we define negative times negative without really thinking about this, and just defining it the only way it would make sense given the above considerations, but if we were to give some thought to what multiplication by a negative means possibly the best way to think about it would be as repeated subtraction. Since multiplying by a positive number is repeated addition, it would make sense to think of multiplication by a negative number as repeated subtraction, and that indeed would make the product of two negatives a positive, since subtracting a negative is the same as adding a positive.

Extra for Experts: If you are a more advanced student or instructor or parent, who has learned about quadratic equations, and you would like to learn about another reason that minus times minus is plus, read my article A Geometrical Approach to Completing the Square.

Again just like with addition we can make this easier to remember by collapsing it down to just two cases, and here it is really much simpler that with addition, because with multiplication you always multiply, so all you have to worry about is what the sign of the answer will be.

Always multiply
Like signs, sign is +
Different signs, sign is -
Big Products
For multiplying it is also interesting to see what happens when you multiply several different numbers. What happens then is every time you have two minus signs the get together and make a plus. so for each minus sign the answer flip flops between - and +, so to determine the sign of your answer you just need to count up the minuses and see whether it is even or odd. If it is even the answer is +, and if it is odd the answer is -.
Multiplication and Addition
It is interesting to compare the rules for multiplication with those for additions so that you don't get them confused.
For multiplication you always multiply, but for addition sometimes you add and sometimes you subtract.
For both multiplication and addition you do different things depending on whether the signs are like or different.
For multiplication like signs mean the answer is +, and for addition like signs mean you add and the sign is the common sign.
For multiplication different signs mean the answer is -, and for addition different signs means you subtract and the sign is the sign of the larger.
Multiplication ++=+, Addition ++=+.
Multiplication +-=-, Addition +-=the sign of the larger.
Multiplication -+=-, Addition -+=the sign of the larger.
Multiplication --=+, Addition --=-.
Powers
Powers mean repeated multiplication, so from the rules for multiplying you should be able to raise numbers to powers. When you raise negative numbers to powers you can also use our rule about counting the minus signs. When you raise a negative number to an even power, then you have an even number of minus signs, so the answer is +. when you raise a negative number to an odd power there are an odd number of minus signs, so the answer is -.
A little Notational Matter
There is a little thing you have to be careful with in the notation of negative numbers raised to powers. When you wish to denote a negative number raised to a power you always enclose it in parentheses. So when you wish to write -2 raised to the 5th power, you write it

the reason for this is that because multiplying by -1 puts a minus sign on a number we think of the minus signs on numbers as equal to multiplying in the order of operations. That means that powers get done before minus signs get attached to numbers, so if you write

this doesn't mean -2 is being raised to the 5th power, instead because the power gets done first, it means that 2 is getting raised to the 5th, and then the minus sign is attached to that number.
Division
The same rules about signs hold for division as for multiplication. Also if you have a problem that is all multiplication and division, you can just ignore the signs and then figure out what the sign of the answer is by counting up the minuses and if it is even the answer is plus and if it is odd the answer is minus. But be careful, you can only do this if the problem is all multiplication and division.