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COURSE :Math conceptsDEPARTMENT : ChemistryPROFESSOR : LARRYBYRDLecture 1 : Part_1View the Power Point Presentation of the Lecture
The following lesson is one lecture in a series of chemistry programs developed by Professor Larry Byrd Department of Chemistry Western Kentucky University. Excellent assistance has been provided by Dr. Robert Wyatt, Ms. Elizabeth Romero, and Ms. Kathy Barnes. Todays lesson is part one of mathematical concepts. Often in the sciences we are involved in using mathematical concepts. In this section we will review all of the fundamental mathematical processes that you will use in a basic chemistry course. Fractions are commonly used to indicate the division of one number by another number and they are written as one over ten notice the normal way or however one over ten with a slash or four twelfths or four over twelve. When we read the first above fraction we may say that in this fraction one is divided by ten or equally correct we could say that there is one part per total of ten parts. Notice that the term per means division, the other fraction could be read as four is divided by twelve or four parts per twelve parts. The number above the line notice the number above the line in a fraction is the numerator and the number below the line is the denominator, a whole number can also be considered a fraction in which the denominator is equal to one so three is the same as three over one, a hundred eighty is the same as a hundred eighty divided by one. If the numerator and the denominator of the fraction are both multiplied by the same number a fraction equal to the first is formed. The fraction two thirds is equivalent to four sixths and it is also equivalent to ten fifteenths. Notice the very simplest form two thirds however if we would like to have it where it has a denominator of six we multiply the numerator and denominator by two so two thirds becomes four over six thus two thirds is equivalent to four sixths however notice this is the lowest it can be, lowest common form. Two thirds can also be in terms of fifteenths multiply the numerator denominator by five so two thirds becomes ten fifths. Observe that the term times means to multiply. A fraction, notice ratio, a fraction is nothing more than a representation of how many parts we have out of a total amount so a ratio. If a chemistry class is composed of twenty students and eleven of these are girls we say that eleven of the young ladies out of the twenty total are the number of young ladies in the class. Notice that the fractions are simply ratios, eleven girls in a class of twenty students is the same as saying eleven members of that class per twenty members, eleven members numerator denominator twenty members. When adding and subtracting fractions they must have the same common denominator, if they do not have a common denominator we must first convert both of those so they have the same common denominator or we cannot add or subtract these. Addition of fractions is very easy, if both of them have the same common denominator, simply add the numerator so notice the denominator is the same so two fifths plus one fifth is equal to two plus one over five or the correct answer is three fifths. If the denominators are not the same then one or both of the fractions must be changed to develop a common denominator. For example if we need to add eleven twentieths plus one tenth we must first find a common denominator. We must pick the lowest number that both twenty and ten will divide into a whole number of times, so we see a twenty and a ten and quite easily you see that twenty is going to be the lowest common denominator. So eleven twentieths plus one tenth is eleven over twenty plus so many twenties we find then that eleven twentieths plus two twentieths that was the same as the one tenth up here, eleven plus two over twenty or the correct answer is thirteen twentieths. So eleven twentieths plus one tenth, once we have a common denominator we can add the numerators eleven plus two thirteen twentieths, notice that the new numerator as we said before is always the sum of the numerators of the fractions with the same common denominator. If we want to subtract one half from three fourths we must write it as follows, so we want to take one half from three fourths so three fourths minus one half. We see that the lowest common denominator is four so we have to convert one half into four so that would be one half is equal to so many fourths and we could easily see that one half is the same as two over four, we can now do the mathematics three fourths minus two fourths is going to be three minus two, simply subtract the numerator. Three minus two would be one fourth. If we want to subtract one sixteenth from two thirds we need to find a common denominator that both sixteen and three will divide into a whole number of times. In this case it is found by simply multiplying the two given denominators so we have a sixteen denominator, a three denominator, sixteen times three the lowest common denominator they both will divide into a whole number of times is forty eight thus two thirds becomes three thirty two over forty eight and one sixteenth becomes three over forty eight. Now we can do the needed mathematics, in this case subtraction, two thirds minus one sixteenth we are going to have forty eighths and we are going to convert the two thirds to thirty two forty eighths and the one sixteen becomes three forty eighths. We now can do the math, thirty two over forty eight minus three over forty eight, got the same common denominator simply do the subtraction thirty two minus three is twenty nine over forty eight. Some other examples, very important to always reduce to the lowest form, if we got eight ninths plus one half we are going to see that the eighteen is going to be our lowest common denominator so in the first case we are going to multiply the numerator and the denominator eight and nine times two and that is going to give us sixteen over eighteen, the common denominator we are looking for. Then in the case of one half we want an eighteen here so we are going to have to multiply the numerator and denominator by nine so nine times one nine times two and that is going to give our eighteen. We now have a common denominator so we simply say sixteen plus nine is twenty five, common denominator eighteen, reduce to the lowest form we will get one and seven eighteenths. Some other examples, if we have two fifths plus three the number three may also be written as three over one, two fifths plus three over one that means we need a common denominator of five since five times one would be the common denominator so two fifths plus so many fifths, three over one would be how many fifths and as we already figured out three over one is the same as fifteen over five so in the since five will divide into fifteen three times three over one is the same as fifteen over five. We now have a common denominator, fives two fifths plus fifteen fifths add the two and the fifteen we get seventeen fifths, reduce it to the lowest form in this case and this will be five into this will be fifteen and two fifths so three and two fifths. Notice that when we converted one tenth to twentieths then we found that one tenth was the same as two twentieths then one tenth is the same as two twentieths or in other words we are saying that two twentieths will reduce back to the one tenth. However we can often do these a lot quicker if we remember the means and extreme rule. So if we use a mathematical rule, algebraic rule known as the means and extreme rule we can always be sure our work is correct. In terms of the means and extreme rule it states that if we have a fraction and notice a fraction equal to another fraction so we must have a fraction on the left hand side equal to a fraction on the right hand side then the numerator and this case a of the left fraction times the denominator of the right fraction in this case will always be equal to the numerator in this case c, of the right fraction times the denominator of the left fraction. Here is our fraction, we have a fraction equal to a fraction so a times d, lets always put these multiplications in parentheses, is going to equal c in parentheses times the value of b. For example, we want to convert our one tenth into twentieths we would do as follows, one over ten equal so many twentieths. We start out and put our unknown in this case we are looking for c, one tenth equals c over twenty plus one tenth c over twenty we have the algebra now, a fraction equal to a fraction. We can start with the c, multiply c, put it into parentheses times ten put it in parentheses, one times twenty put both of these in parentheses so c times ten is the same as ten c, remember if you have two things written together ten c is the same as ten c or c times ten. One times twenty is twenty, now we are going to bribe both sides with ten so we have ten c over ten, observe they cancel out or actually simply divide out, same word, we now have twenty over ten and notice twenty into ten gives us two and thus as we already found one tenth is the same as two twentieths. Some other examples of means and extremes, we have two divided by seventeen is equal to so many two hundred and fourths, two zero four, notice this one will be a little bit harder to do. So here we are going to put unknown c, we are going to multiply c in parentheses times seventeen in parentheses, two in parentheses times two zero four, seventeen c is going to equal to times two zero four or four zero eight. Then we are going to divide both sides, we want to find out what one c is what that value is up here and we are going to divide both sides by seventeen so seventeen and seventeen is the same as one c, seventeen four hundred and eight gives us twenty four or two over seventeen is equal to twenty four over two zero four. Now here is where we want to check our work, we can always check our work by multiplying across the equals sign to make sure our new fraction the one with the new denominator in this case is equal to the original fraction. In out example, two times two zero four must be equal to twenty four times seventeen or we have made an error. Notice we did alright, notice we got it, two seventeenths is equal to twenty four over two zero four. Our proof is two times two zero four, four zero eight, twenty four times seventeen, four zero eight. If these values were not correct then we had actually missed a problem. Always do this if you are converting fractions. Practice test number one questions, number one what are fractions, number two for the fraction one fifth and two sevenths what is the lowest possible common denominator between these two. Number three, we have twenty four divided by seventeen is equal to forty eight over b, we are going to find the value of b and number four we have six divided by a is equal to a hundred and twenty over twenty five and we are looking for the value of eight. First question, what are fractions, as we said before fractions are nothing more than a ratio, we will have some value as a numerator and the denominator will be the other value. So we have a ratio between the numerator and denominator, for the fraction one fifth actually the fraction is one fifth and two sevenths, the lowest common denominator that both five and seven will divide into a whole number times simply was found by multiplying five times seven so the thirty five is the lowest common denominator. Number three we got twenty four over seventeen equals forty eight over b it kind of looks like well I am not really sure of this but however using the means and extremes we can immediately get it, multiply it twenty four times b so always get our unknown on the left hand side so we multiply a numerator times a denominator twenty four times b, numerator forty eight times seventeen thus b is equal to thirty four. And number four we have six over A equals a hundred and twenty over twenty five and we are going to multiply the hundred and twenty times A so we want to keep A on our left hand side, six times twenty five and if we do the math we will find that A kind of odd looking here one and one fourth. Thank you for watching another tutor bird lesson. If you would like to continue on there is then part two and three also.
