CH 0 MORE FRACTIONS Introduction I n this chapter we tie up some loose ends. First, we split a single fraction into two fractions, followed by performing our standard math operations on positive and negative fractions. Then we solve equations and number problems which contain fractions. Splitting Up a Fraction Before this course is over we will see the need to split up a fraction in a couple of ways, one using multiplication and the other using addition. EXAMPLE : A. Express x as the product of two quantities: Solution: The easiest way to see this process is to just do it and then check that it s right. Here s the claim: x x and here s the reason: x x x x Ch 0 More Fractions
6 B. Express x as the sum of two fractions. Solution: What two fractions have a sum of x? The tells us that we could use fractions each with a denominator of. Since the numerator is x +, we can make one of the numerators x and the other one. That is x x + (check by adding the two fractions) C. Combine the ideas of parts A and B to split up x. Solution: x x x D. Express Solution: x 6 6 as the sum of two quantities. x 6 x 6 x x 6 6 6 Homework. Express each fraction as the product of two quantities, using part A of Example as a guide: a. x x y 6 a n x 6 g. 6x m x j. z Ch 0 More Fractions
. Express each fraction as the sum or difference of two quantities, using parts C and D of Example as a guide: a. x x 6 y n w x g. x x 0 Signed Fractions Adding and Subtracting 6 6 6 0 6 6 Multiplying and Dividing 0 0 Ch 0 More Fractions
Powers and Square Roots An exponent still means what it always has, so these next examples should be clear. 6 6 0 6 6 6 6 As for the square root sign, we still ask the question: What number (that s not negative) times itself gives the number in the radical sign? This is true because. This is because. does not exist, because is a negative number, and square roots of negative numbers don t exist in this cours Ch 0 More Fractions
number: does exist, because the fraction is actually a positive. Homework Perform the indicated operation:. a. 6 g. ( ). a. 6 6 6 g.. a. 0 0 0 0 g. 6 Ch 0 More Fractions
0 6. a. 00 6 6 g. 6 0 ( abc ) Equations and Word Problems EXAMPLE : Solve for x: x 6 Solution: equation: To isolate the x, we will first add to each side of the x 6 Now we simplify each side of the equation: Left side: Right side: x x 0 6 6 The LCD is So now the equation reads x The last step is to divide each side of the equation by order to isolate the x): x (in Ch 0 More Fractions
x (multiply by the reciprocal) x (factor and cross-cancel) x 6 EXAMPLE : I m thinking of a number. If is added to times the number, the result is. Find the number. Solution: We ve seen this type of word problem befor The only new issue is what to do with those fractions. Suppose we let n represent the unknown number. The equation which results from the wording of the problem is n The first step in isolating the n is to subtract the equation: from each side of n The left side of the equation is obviously n. The right side is found by using a common denominator:. The equation now reads n We isolate the n with the usual step of dividing each side of the equation by : Ch 0 More Fractions
n n Therefore, the number I was thinking of is Homework. Solve each equation: a. x x n t n z g. x 6 a 0 y. I m thinking of a number. If is added to times the number, the result is. Find the number. 0. I m thinking of a number. If is subtracted from times the number, the result is. Find the number. 0. I m thinking of a number. If is added to twice the number, the result is. Find the number. 0 Ch 0 More Fractions
. I m thinking of a number. If is subtracted from times the number, the result is. Find the number.. If times a number is increased by, the result is. What is the number?. If times a number is decreased by, the result is. What is the number?. If times a number is increased by, the result is. What is the number?. If 6 times a number is decreased by, the result is. What is the number? Review Problems 6. Express y 0 as a product.. Express 6 x as the sum of two quantities.. Evaluate each expression: a. Ch 0 More Fractions
j. g. k. 0 0 l. 6 6. Solve each equation: a. x y z 0. I m thinking of a number. If is added to times the number, the result is. Find the number.. If times a number is decreased by, the result is. What is the number? Solutions. a. x x 6 y a n x g. x m x j. z. a. x x w x g. y x n x 0 0 Ch 0 More Fractions
. a. 0 0 g.. a. g. 6 0. a. g. 6 6. a. 0 Does not exist g. 6 0. a. g.. n 0 n 0 0 0 0 0 n n 0 0 0 0 0. n n 0. 6. 0 0.... 60 6. 0 y. x Ch 0 More Fractions
6. a. k. 0 0 g. l. Does not exist 6 6 j. 0. a. 0. n ; n. n ; n 0 You can know the name of a bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bir.. So let's look at the bird and see what it's doing -- that's what counts. I learned very early the difference between knowing the name of something and knowing something. Richard Feynman Ch 0 More Fractions