Study Guide For use with pages

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1 . GOAL For use with pages Write fractions as decimals and vice versa. VOCABULARY A rational number is a number that can be written as a quotient of two integers. In a terminating decimal, the division ends because you obtain a final remainder of zero. In a repeating decimal, a digit or block of digits in the quotient repeats without end. EXAMPLE Identifying Rational Numbers Show that the number is rational by writing it as a quotient of two integers. a. b. c. Lesson. a. Write the integer as. b. Write the mixed number as the improper fraction. c. Think of as the opposite of. First write as. Then you can write as. To write as a quotient of two integers, you can assign the negative sign to either the numerator or the denominator. You can write or. Exercises for Example Show that the number is rational by writing it as a quotient of two integers..... EXAMPLE Writing Fractions as Decimals a. Write as a decimal. b. Write as a decimal. a. 0. b The remainder is 0, so 0 the decimal is a 0 terminating decimal: Use a bar to show the repeating digits in the repeating decimal: 0. ww. Chapter Pre-Algebra Resource Book

2 . Continued For use with pages Exercises for Example Write the fraction or mixed number as a decimal..... Lesson. EXAMPLE Writing Terminating Decimals as Fractions a Place value of is tenths, so denominator is 0. b Place value of is hundredths, so denominator is 00. EXAMPLE Writing a Repeating Decimal as a Fraction To write 0. ww as a fraction, let x 0. ww. () Because 0. ww has repeating digits, multiply each side of x 0. ww by 0, or 00. Then 00x. ww. 00x. ww () Subtract x from 00x. (x 0. ww) x () Solve for x and simplify. x x Answer: The decimal 0. ww is equivalent to the fraction. Exercises for Examples and Write the decimal as a fraction or mixed number ww..w.. Pre-Algebra Chapter Resource Book

3 . GOAL For use with pages Add and subtract like fractions. EXAMPLE Adding Like Fractions You use of a tank of gasoline to travel to your aunt s house. You use of a tank of gasoline to travel to the store. What fraction of a tank of gasoline did you use? Write sum of numerators over denominator. Add. Answer: You used of a tank of gasoline. EXAMPLE Subtracting Like Fractions a. Write difference of numerators over denominator. Subtract. b. To subtract, add. Write sum of numerators over denominator. Lesson. Add. Exercises for Examples and Find the sum or difference Chapter Pre-Algebra Resource Book

4 . Continued For use with pages EXAMPLE Adding and Subtracting Mixed Numbers a. Write mixed numbers as improper fractions. Write sum of numerators over denominator. 0 b. Add. Then write fraction as a mixed number. Write mixed numbers as improper fractions. Write difference of numerators over denominator. Subtract. Then write fraction as a mixed number. Lesson. Exercises for Example Find the sum or difference EXAMPLE Simplifying Variable Expressions c c a. 0 0 c c 0 c c 0 b. 0x 0x 0x 0 x x Exercises for Example Write sum of numerators over denominator. Add. Divide out common factor. Then simplify. Write difference of numerators over denominator. Subtract. Divide out common factor. Then simplify. Find the sum or difference.. x 0x. y y.. a a f f Pre-Algebra Chapter Resource Book

5 . GOAL For use with pages 0 Add and subtract unlike fractions. EXAMPLE Adding and Subtracting Fractions a. Write using LCD. Write sum of numerators over denominator. Add. b. 0 0 Write fractions using LCD. 0 Write difference of numerators over denominator. 0 Subtract. Exercises for Example Find the sum or difference..... EXAMPLE Adding Mixed Numbers Write mixed numbers as improper fractions. Lesson. Write fractions using LCD. Write sum of numerators over denominator. 0 Add. Then write fraction as a mixed number. Exercises for Example Find the sum Pre-Algebra Chapter Resource Book

6 . Continued For use with pages 0 EXAMPLE Subtracting Mixed Numbers Write mixed numbers as improper fractions. Exercises for Example Write fractions using LCD. Write difference of numerators over denominator. Subtract. Write fraction as a mixed number. Find the difference EXAMPLE Simplifying an Expression Simplify the expression d d. d d d p d p Write fractions using LCD. d d 0 0 d d 0 d 0 Exercises for Example Multiply. Write sum of numerators over denominator. Add. Lesson. Simplify the expression.. d d. a a. x x. a a Chapter Pre-Algebra Resource Book

7 . GOAL For use with pages Multiply fractions and mixed numbers. EXAMPLE Multiplying Fractions p p p () p p ( ) p Assign negative signs to numerators. Divide out common factors. Multiply. Exercises for Example Find the product.. p. p 0. p. p EXAMPLE Multiplying a Mixed Number and an Integer Every day for a week you run miles. What is the total distance you run for the week? Total distance Daily distance p Number of days p Substitute values. p p p Write numbers as improper fractions. Multiply. Write fraction as a mixed number. Lesson. Answer: You run a total of miles during the week. Pre-Algebra Chapter Resource Book

8 . Continued For use with pages EXAMPLE Multiplying Mixed Numbers p p 0 Write mixed numbers as improper fractions. p 0 p Divide out common factors. Multiply. Exercises for Examples and Find the product.. p. p 0. p (). p (). p 0. p. p. p EXAMPLE Simplifying an Expression x x Simplify the expression p. p x p x x p x x Divide out common factors. Product of powers rule x Add exponents. Exercises for Example Simplify the expression.. a p a. x x p. y p y. 0z p z 0 Lesson. Chapter Pre-Algebra Resource Book

9 Lesson. LESSON. For use with pages GOAL Divide fractions and mixed numbers. VOCABULARY Two nonzero numbers whose product is are reciprocals. EXAMPLE Dividing a Fraction by a Fraction 0 p 0 p p 0 Multiply by reciprocal. Divide out common factors. Multiply. Write fraction as a mixed number. Check To check, multiply the quotient by the divisor to see if you get the dividend: p 0 p p 0 checks. Exercises for Example Find the quotient..... EXAMPLE Dividing a Mixed Number by a Mixed Number Write mixed numbers as improper fractions. p Multiply by reciprocal. p ( ) p Divide out common factors. Multiply. Write fraction as a mixed number. Pre-Algebra Chapter Resource Book

10 . Continued For use with pages Lesson. Exercises for Example Find the quotient EXAMPLE Dividing a Whole Number by a Mixed Number You are cutting strips of cloth out of a -inch by -inch piece of cloth. You want each strip to be inches long and inches wide. How many strips can you cut out of the cloth? Divide to find how many strips you can cut out of the cloth. in. in. Number of strips Cloth width Width of each strip Substitute values. in. p p p Write numbers as improper fractions. Multiply by reciprocal. Divide out common factor. Multiply. 0 Write fraction as a mixed number. Answer: Because you cannot use fractions of a strip, you can cut 0 strips out of the cloth. Exercises for Example Find the quotient (). 0 0 Chapter Pre-Algebra Resource Book

11 . GOAL For use with pages Use multiplicative inverses to solve equations. VOCABULARY The multiplicative inverse of a nonzero number is the number s reciprocal. Lesson. EXAMPLE Solving a One-Step Equation x 0 Original equation x 0 Multiply each side by multiplicative inverse of. x 0 Multiplicative inverse property x p 0 p Divide out common factors. Multiply. Write fraction as a mixed number. Answer: The solution is. EXAMPLE Solving a Two-Step Equation x x 0 0 Original equation Subtract from each side x 0 0 x 0 Subtract. x x 0 Multiply each side by multiplicative inverse of. Divide out common factors. Pre-Algebra Chapter Resource Book

12 . Continued For use with pages Exercises for Examples and EXAMPLE Solve the equation. Check your solution.. x. x 0. x. x. x. x. x Writing and Solving a Two-Step Equation You are training for a marathon. Currently, you can run miles. You increase the distance you run on your long-run day by mile each week. In how many 0 more weeks will you be able to run miles? Write a verbal model. Let w represent the number of weeks you need to reach your goal. Lesson. Current distance you can run Weekly Number of increase p weeks Distance goal w 0 w 0 w 0 Substitute values. Subtract from each side. 0 w 0 0 () Multiply each side by multiplicative inverse of. 0 w 0 Multiply. Answer: You will be able to run miles in 0 weeks. Exercise for Example. A baby with a length of inches at birth grows at a rate of inch per month. How many months old will the baby be when his length is inches? Chapter Pre-Algebra Resource Book

13 . GOAL For use with pages Use the LCD to solve equations and inequalities. EXAMPLE Solving an Equation by Clearing Fractions x 0 0 x x x 0 x x Original equation Multiply each side by LCD of fractions. Use distributive property. Add 0 to each side. x Divide each side by. x Lesson. EXAMPLE Solving an Equation by Clearing Decimals Solve the equation. 0.x. Because the greatest number of decimal places in any of the terms with decimals is, multiply each side of the equation by 0, or x Write original equation. 000(.) 000( 0.x) Multiply each side by x Use distributive property x 000 Subtract 000 from each side. 0x 0x 0 0 Divide each side by 0.. x Exercises for Examples and Solve the equation by first clearing the fractions.. x. x x Solve the equation by first clearing the decimals...a... 0.b....m 0.. Pre-Algebra Chapter Resource Book

14 . Continued EXAMPLE For use with pages Solving an Inequality with Fractions You have cups of raisins. You need cup of raisins to make one loaf of raisin bread and cup of raisins to make one dozen oatmeal raisin cookies. You want to make dozen oatmeal raisin cookies. How many loaves of raisin bread can you make? Write a verbal model. Let n represent the loaves of raisin bread you can make. Cups of raisins per dozen Number of Cups of Number of p dozens of raisins per p loaves of cookies loaf bread Cups of raisins you have p n Substitute. n n n () Subtract from each side and simplify. Multiply each side by the multiplicative inverse of. Lesson. n Answer: You can make at most loaves of raisin bread. EXAMPLE Solving an Inequality by Clearing Fractions x > 0 0 x 0 0 x 0 0 x > 0 x > x < > 0 > 0 Exercises for Examples and Original inequality Multiply each side by LCD of fractions. Use distributive property. Subtract from each side and simplify. Divide each side by, reverse inequality symbol, and simplify. Solve the inequality.. x <. x. y Chapter Pre-Algebra Resource Book