Adding, Subtracting, Multiplying and Dividing Fractions
Fractions can be included in mathematical expressions such as every other number. However, calculating with fractions can be quite different to calculating with normal numbers. For different types of operations, you have to go through different procedures in order to get the correct solution to the problem.Adding FractionsIf you want to add two or more fractions together, you need to make sure that the denominators of all these fractions are the same. To get common denominators you have to convert one or more of the fractions. Ideally, you should end up with the lowest common multiple or the highest common factor of all the denominators as the common denominator. To properly convert the fraction (not change its value) you have to apply the operation that you used to get to the common denominator to the numerator as well. In the example, the common denominator is 6. To get from 3 to six we have to multiply it by 2. To make sure the new fraction is equivalent we also have multiply the numerator (1) by 2 so that we end up with 2/6. Then you just have to add the two numerators together so that you end up with 5/6. Here are some example.Subtracting FractionsSubtracting Fractions works in the same way as adding fractions. Again we have to find a common denominator for the fractions in the expression. The only difference is that we have to now subtract the second fraction from the first instead of adding both together. Here are a couple of examples.Multiplying FractionsMultiplying fractions is a lot easier than adding and subtracting them. This is because the only thing you have to do is to multiply the numerators and the denominators. To finish, you should simplify your answer. However, multiplication questions can often have very large numbers as solution and simplifying them can be difficult. There is a simpler way though. By cancelling a fraction, you are basically simplifying it before doing the operation. Look at the multiplication problem on the right with the red lines going through the numbers. This is cancelling. When cancelling you must look if there are two numbers, one on the top and one on the bottom (it does not matter in which fraction), that can be divided by the same number. The first pair in this fraction is 5 and 10. Both number can be divided by 5. To cancel, divide both numbers by 5 and put the result as the new numerator or denominator of the fraction. In this case the answers are 1 and 2. The second pair of numbers that we can cancel is 6 and 9 as we can divide them by 3. Do the division operation and you end up with 2 and 3. Now the expression has transformed from 5/6 x 9/10 to 1/2 x 3/2. The from both problems are 45/60 and 3/4. They are equivalent but one is already simplified. You may sometimes only be able to cancel one pair or even no pair but as long as you cancel the fractions as much as you can you will always get the most simple answer. This saves you the pain of simplifying and also makes multiplying a lot easier. Underneath you will find a few examples (there is no cancelling in the examples).Dividing FractionsIf you know how to multiply fraction,dividing fractions will be very easy. To divide fractions, you just have to invert the second fraction in the expression and then multiply as shown in the example on the right. You again have the opportunity to cancel which makes the procedure a lot easier. Here are a few examples just to make sure that you understand.Calculating With Improper Fractions or Mixed NumbersIf you have a mixed number in a mathematical expression, the easiest way you can deal with it is to transform it into an improper fraction, meaning that the numerator is larger than the denominator. This method is especially useful for multiplication and division but can also be used when adding or subtracting. You can also calculate the fractions and whole numbers separately when doing addition and subtraction.Mastering FractionsYou should now know almost everything about numerical fractions. You should try to use one of our learning resources if you want to become even better or look at algebraic fractions. You can also have a look at some other topics on this site by returning to the library. Please share this page if you like it or found it helpful!
Example 1In this example the common dominator is 12 and both original fractions had to be converted. We have to apply the conversion operation to both the numerator and the denominator in order for the new fraction to be equivalent. We could simplify this fraction to make it 5/6.
Example 2In this example, we already have a common denominator. We only have to add the two numerators together to get the solution.
Example 3In this example, we were able to get the common denominator (8) by converting just 1 fraction. We could simplify this fraction to make it 3/4.
Example 4This example is a bit more complicated because we are adding 3 fractions together. However, the only difference is that we have to convert 3 fractions and add 3 numerators together to get the answer.
Example 1In this example, we convert the first fraction so that it has the common denominator (9) before subtracting the second fraction from it.
Example 2In this example, we have the 3 fractions in the expression. Fractions 2 and 3 are subtracted from fraction one after they have been converted to have the common denominator (8).
Example 1In this example, the numerators and denominators were times together to get the answer which was then simplified. (Remember to always try to cancel instead of simplifying as it is easier.)
Example 2In this example, the numerators and denominators were again multiplied. However, this time we are unable to cancel or simplify as the fraction is already i its simplest form.
Example 1In this division example, we invert the second fraction and then multiply the numerators and denominators. Finally we simplify it. (Always try to cancel when doing multiplication questions.)
Example 2In this example, we have again inverted the second fraction and multiplied both fractions together. As a result we have gotten a mixed number (or improper fraction). The solution was already in its simplest form so we are unable to cancel or simplify.
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Chapter 3.2: Learning OutcomesStudents will be able to add and subtract fractions by finding a common denominator!Students will be able to multiply and divide fractions!Students will know how to deal with mixed numbers and improper fractions when calculating with fractions!