In order to master mathematics, you definitely need to master fractions. These appear in every single aspect of this discipline, from algebra to calculus to engineering to related fields like physics. Fractions present a lot of trouble to students, yet most of these problems can be easily resolved if the right approach is taken. Here in Part II, we examine some other techniques necessary to master this area.
Equivalent fractions are nothing more than fractions that have the same value. Thus 1/2, 2/4, and 3/6 are all equivalent fractions. Equivalent fractions have the same value but have different numerators and denominators from the other fractions to which they are equivalent. There are infinitely many equivalent fractions to 1/2, let us say. Each of these can be derived by multiplying 1/2 by 1 in disguised form. What we mean by 1 in disguised form is a fraction which has the same numerator as denominator. Thus 2/2, 3/3, 4/4...etc. are all 1 in disguised form. Remember: 1 is the multiplicative identity, and thus no matter what we multiply by 1 does not have its value changed.
Equivalent fractions come in very handy when we add or subtract fractions, because this operation requires that we have the same denominator. Thus if adding 1/2 and 3/8, we need to convert the 1/2 into an equivalent fraction with 8 as its denominator. We simply ask ourselves what we need to multiply 2 by to get 8. The answer is easy and is 4. Thus we use 4/4, 1 in disguised form to multiply and convert 1/2 into the equivalent fraction 4/8. We can then add 4/8 and 3/8 to get 7/8.
Reducing fractions is another important aspect to mastering these numbers. Reducing fractions allows us to deal with fractions in lowest terms. This is important because in mathematics we always want our answers in the most simplified form and reducing fractions permits this. Mathematics is complicated as it is, thus providing the most simple form is always important. Could you imagine how much more complicated this field would be if we did not do this? At any rate, simplifying fractions simply requires that we factor out the GCF (Greatest Common Factor) from both numerator and denominator and canceling. The GCF is the largest factor common to both numerator and denominator. For example, 20/25 can be reduced to 4/5 because the GCF of both 20 and 25 is 5. Thus we write 20/25 = (4x5)/(5x5) and cancel the 5 to get 4/5. Similarly for 38/57, we can express this as (19x2)/(19x3) and cancel the 19 to get 2/3. Obviously, it is easier to work with smaller numbers than larger ones, as many times we use the results of one operation for further operations. Thus 2/3 is easier to work with than 38/57, and thus the reason for reducing fractions becomes evident.
Another important aspect of fractions is multiplying and dividing them. This is probably one of the easiest operations involving fractions because we need not concern ourselves with common denominators. To multiply two fractions, we simply multiply the numerators and then the denominators. It should be pointed out that we should first try to reduce the fractions so that our end result is in lowest terms. Doing this first, also simplifies the multiplications. For example, (38/57)x(20/25) is easier to do if we first reduce each fraction as mentioned above to 2/3 and 4/5, respectively. We then multiply 2x4 and 3x5 to get 8/15 as our answer, and this is in lowest terms. If you do not simplify first, you are looking at multiplying 38x20 and 57x25, which are harder multiplications than the ones we did.
Dividing fractions is really no different than multiplying them, with one exception. Before we do the multiplication, we invert the numerator and denominator of the second fraction. We then simply multiply. Thus (9/15)/(8/16) is the same as (9/15)x 16/8). Let's reduce and multiply. We have (3/5)x(2/1) = 6/5.
Mastering these techniques will give you the edge in conquering fractions. Use these articles and the techniques laid out therein to overcome any problems you might have had with these stubborn mathematical entities. You will soon realize that fractions are actually quite fun to work with.
Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey.
Joe propagates his Wiz Kid Teaching Philosophy through his writings and lectures and loves to turn "math-haters" into "math-lovers."
Equivalent fractions are nothing more than fractions that have the same value. Thus 1/2, 2/4, and 3/6 are all equivalent fractions. Equivalent fractions have the same value but have different numerators and denominators from the other fractions to which they are equivalent. There are infinitely many equivalent fractions to 1/2, let us say. Each of these can be derived by multiplying 1/2 by 1 in disguised form. What we mean by 1 in disguised form is a fraction which has the same numerator as denominator. Thus 2/2, 3/3, 4/4...etc. are all 1 in disguised form. Remember: 1 is the multiplicative identity, and thus no matter what we multiply by 1 does not have its value changed.
Equivalent fractions come in very handy when we add or subtract fractions, because this operation requires that we have the same denominator. Thus if adding 1/2 and 3/8, we need to convert the 1/2 into an equivalent fraction with 8 as its denominator. We simply ask ourselves what we need to multiply 2 by to get 8. The answer is easy and is 4. Thus we use 4/4, 1 in disguised form to multiply and convert 1/2 into the equivalent fraction 4/8. We can then add 4/8 and 3/8 to get 7/8.
Reducing fractions is another important aspect to mastering these numbers. Reducing fractions allows us to deal with fractions in lowest terms. This is important because in mathematics we always want our answers in the most simplified form and reducing fractions permits this. Mathematics is complicated as it is, thus providing the most simple form is always important. Could you imagine how much more complicated this field would be if we did not do this? At any rate, simplifying fractions simply requires that we factor out the GCF (Greatest Common Factor) from both numerator and denominator and canceling. The GCF is the largest factor common to both numerator and denominator. For example, 20/25 can be reduced to 4/5 because the GCF of both 20 and 25 is 5. Thus we write 20/25 = (4x5)/(5x5) and cancel the 5 to get 4/5. Similarly for 38/57, we can express this as (19x2)/(19x3) and cancel the 19 to get 2/3. Obviously, it is easier to work with smaller numbers than larger ones, as many times we use the results of one operation for further operations. Thus 2/3 is easier to work with than 38/57, and thus the reason for reducing fractions becomes evident.
Another important aspect of fractions is multiplying and dividing them. This is probably one of the easiest operations involving fractions because we need not concern ourselves with common denominators. To multiply two fractions, we simply multiply the numerators and then the denominators. It should be pointed out that we should first try to reduce the fractions so that our end result is in lowest terms. Doing this first, also simplifies the multiplications. For example, (38/57)x(20/25) is easier to do if we first reduce each fraction as mentioned above to 2/3 and 4/5, respectively. We then multiply 2x4 and 3x5 to get 8/15 as our answer, and this is in lowest terms. If you do not simplify first, you are looking at multiplying 38x20 and 57x25, which are harder multiplications than the ones we did.
Dividing fractions is really no different than multiplying them, with one exception. Before we do the multiplication, we invert the numerator and denominator of the second fraction. We then simply multiply. Thus (9/15)/(8/16) is the same as (9/15)x 16/8). Let's reduce and multiply. We have (3/5)x(2/1) = 6/5.
Mastering these techniques will give you the edge in conquering fractions. Use these articles and the techniques laid out therein to overcome any problems you might have had with these stubborn mathematical entities. You will soon realize that fractions are actually quite fun to work with.
Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey.
Joe propagates his Wiz Kid Teaching Philosophy through his writings and lectures and loves to turn "math-haters" into "math-lovers."
