Math Skills for Business- Full Chapters 1 U1-Full Chapter- Algebra Chapter3 Introduction to Algebra 3.1 What is Algebra? Algebra is generalized arithmetic operations that use letters of the alphabet to represent known or unknown quantities. We can use y to represent a company's profit or the costs of labour. The letters used to hold the places for unknown quantities are called variables, while known quantities are called constants. Variables denote a number or quantity that may vary in some circumstances. Algebra now occupies the centre of mathematics, because it could be used to solve a variety of complex problems much faster than using arithmetic methods. Many problems that mathematicians could not solve previously with arithmetic methods can now be solved with algebraic methods. As well, algebra has made it possible to apply mathematics in other areas of human endeavour such as economic planning, pharmacology, medicine, and public health. Consider this, 12 + b = 20. What is the value of b? The only quantity that can take the place of b is 8, because 12 + 8 =20. So 8 is the true replacement value for b. What about y + y = 15? The replacement value for the first y could be any number not more than 15. However, the replacement value we pick for the first y will determine the value for the second y. If we say, for example, that the first y is 10, then the second y must be 5. As well, if the first y is 12, the second y must be 3. The reverse is also true. Try it for yourself by picking a replacement value for the second, and determine the value for first y. As we will see later, this simple example is very important for understanding the solution to equations involving two similar variables. Consider another example, 4x + 8 = 40. In this example, we are looking for a number when multiplied by 4 and added 8 to it will give 40. We can try to figure out this number through guessing and checking. Eventually we will find that 8 is the replacement for x. However, with a systematic procedure of solving equations, we can easily solve that problem without going through the throes of guessing and Math Skills for Business- Full Chapters 2 checking. Before we start learning that procedure for solving equations, let us try to understand the meanings of like terms and unlike terms. 3. 2 Like Terms and Unlike Terms Consider again, 12 + b = 20. A number or letter separated by the operation sign + is called a term. So 12 is a term; b too is a term. Letters of the same kind in an algebraic statement or expression are called like terms. For example, b + b + c = 2b + c. The two bs are like terms, so we can carry out the operations of addition on them. Look at more examples below. eg2 2b + 3b = 5b, whatever the value of b is. eg3 3y + 5x cannot be simplified two terms are different. eg4 10t - 4t + 12y + 6t = 6t + 6t + 12y (since 10t 4t =6t) = 12t + 12 y (since 6t + 6t = 12t) This is now fully simplified, since the two remaining terms are not like terms. eg 5 3t - 2t = t. Note: traditionally, mathematicians do not write 1t. They just write t. So, t understandably means 1t (one t). The general rule, however, for adding and subtracting like terms does not apply to multiplication or division. In fact, multiplication and division are done as is shown in the following examples. Multiplication/Division examples: eg 1 y y y = y eg2 x2 x3 = x x x x x = x5 eg 3 t t = t eg 4 4t 6t = (4 6) (t x t) = 24t. Note: 1. 26t is the same as 26 t. 2. 30yz = 30 y z 3. 48x = 48 x x Math Skills for Business- Full Chapters 3 4. 2t t = = 2, because the two t cancel themselves out. 5. 14 x 7 = = 2 x , because 7 goes into 14 two times. 6. 20xy, is the same as 20 x y. Again, there is no need to put the multiplication sign between 20 and x or x and y. 7. y y y x x = yx 8. 5nt = 5 n n t t t 9. To simplify 3a 4 a , first multiply the numbers together to get 12 from (3 4) 2 3 2a and then add the exponents to get a 2 + 3 = a 5 This is the same as a a a a a = a . That is, keep the base a and add the exponents. We now have 12a 2a. Divide the numbers, 12 2 = 6. Then subtract the exponents when dividing, so a 5 a1 = a 5 - 4 This equals a . This is the same thing as a x a x a x a x a a = a . The final answer is 6a 10. = = note: We multiplied the top numbers to get 4 ( 2 x 2) and the number letters to get t. 11. = = note that the exponent 3 tells us the number of times the fraction should be multiplied. We multiplied the top numbers to get 1 (1 x 1 x 1) and then the bottom numbers to get 27 (3 x 3 x 3). 13. 5 = 5 x 5 x 5 = 125 3.3 Substitution and Formula When we put quantities in place of variables it is called substitution. Consider this, what is b + c + d, where b = 4, c= 2, d= 3? We simply substitute each variable with its corresponding quantity. b+ c+ d =4 + 2 + 3 = 9 Math Skills for Business- Full Chapters 4 Example 1 Evaluate, 6x + 8y + 3z, where x = 2, y = 3, z = 4 =6(2) + 8(3) + 3(4) = 12 + 24 + 12 = 48 Example 2 Evaluate 12a 14b, where a =3, b= 1 =12(3) 14 (1) =36 14 = = (dividing by 2, which is the common factor) Example 3 Evaluate 2(x - 3) = 10, if x=8 Remember the distributive property! 2(x-3) = 10 2(x) - 2(3) = 10 (Distribute 2 over x and -3 in order to remove the bracket) 2x - 6 = 10 2(8) - 6 =10 (substituting 8 for x) 16 - 6 = 10 A formula describes how one quantity relates to one or more other quantities. A formula is a shorthand form of procedure for doing calculations. Al-Khwarazmi gave the world algebra by using letters of the alphabet to represent unknown numbers. And, then, he gave these symbols all the properties of numbers. Before this, the only way to know a procedure is to see examples of the procedure for specific numbers. For example, procedure for finding the area of a rectangle was taught by showing the procedure for calculating the area for specific rectangles. In a sense, arithmetic reasoning was more of experience rather than deductive reasoning. However, once a formula is constructed, it can be used to calculate unknown quantities. Formulas too are used to calculate quantities in businesses. A common formula is P = R -E, where P= profit, R= revenue (sales), and E = Expenses. Math Skills for Business- Full Chapters 5 The formula says that profit is equal to revenue subtract expenses. Another common formula is E = A - L, where E = owner's equity (the capital the owner has invested in the business), A = assets (Resources such as machines, furniture, buildings, etc.), and L = liabilities are amounts owed to others. As well, when S = VC + FC, this is called the break-even point, where S =sales, VC= variable cost, and FC = fixed cost. The following are other formulas used in businesses: Net sales = Gross sales sales returns and allowances NS = GS SRA Cost of goods sold = Cost of beginning inventory + cost of purchases cost of ending inventory VC = Cost per unit x Quantities produced VC = CPU x Q Total cost = VC + FC, or TC = VQ + FC The Construction of a formula is a straight-forward process. First, you have to know the purpose for which you want to use the formula. Second, you have to understand the information you will be using to construct the formula. Third, with little knowledge of algebra, particularly substitution, you should have no problem constructing a formula. Lastly, test your formula to make sure that it works. Example 1 A technician charges a basic fee of $40.00 for a house visit plus $15 per hour, when repairing central heating system. Construct a formula for calculating three hours of the technician's charge. Solution Let C = charge, and n = number of hours worked. The charge is made up of a fixed cost of $40 and $15 times the number of hours. This translates into the following: Charge = $30 plus $15 times the number of hours worked. C = 30 + 15nDownload Link: