Chapter 5 Notes

5-1 Compare/Order Rational Numbers Graph and compare the fractions in each pair: -(1/2), -(1/10) Order -(1/2), 3/4, -1, and 2/5 from least to greatest

Answers Graph and compare the fractions in each pair: -(1/2), -(1/10) -1/2 < -1/10 Order -(1/2), 3/4, -1, and 2/5 from least to greatest -1, -1/2, 2/5, 3/4

Examples (Graph and Compare) 1. -(4/9), -(2/9) 2. -(4/9), 2/9 3.-(2/3), -(1/3)

Examples (Graph and Compare) - Answers 1. -(4/9), -(2/9) < 2. -(4/9), 2/9 < 3.-(2/3), -(1/3) <

Examples (Order from least to greatest) 1. 2/5, 2/3, 2/7, /3, 1/6, 1, 5/12 3.-(3/10), 1/5, -1, 1/2, -(7/12)

Examples (Order from least to greatest) - Answers 1.2/5, 2/3, 2/7, 2 2/7 < 2/5 < 2/3 < /3, 1/6, 1, 5/12 1/6 < 5/12 < 2/3 < 1 3.-(3/10), 1/5, -1, 1/2, -(7/12) -1 < -(7/12) < -(3/10) < 1/5 < 1/2

Comparing Fractions 2/3 � 7/9 8/17 � (-3/8) (-4/5) � (-7/8) 6/8 � 7/9 -(5/18) � -(1/3)

Comparing Fractions - Answers 2/3 � 7/9 < 8/17 � (-3/8) > (-4/5) � (-7/8) > 6/8 � 7/9 < -(5/18) � -(1/3) >

5-2 Fractions and Decimals Terminating decimal = when division ends with a remainder of zero 5/8 1 7/8 3 3/10

Answers Terminating decimal = when division ends with a remainder of zero 5/8 = /8 = /10 = 3.3

Repeating Decimals Repeating decimal = the same block of digits repeats infinitely many times 2/3 Repeating or Terminating decimals: 7/9 21/2211/88/11

Repeating Decimals - Answers Repeating decimal = the same block of digits repeats infinitely many times 2/3 = …… Repeating or Terminating decimals: 7/9 21/2211/88/ …

Ordering fractions and decimals Write the numbers in order from least to greatest 1/4, -0.2, -3/5, , 4/5, 7/10, 0.5

Ordering fractions and decimals - Answers Write the numbers in order from least to greatest 1/4, -0.2, -3/5, /5, -0.2, 1/4, , 4/5, 7/10, , 0.5, 7/10, 4/5

Writing a decimal as a fraction Examples: /100 =1 6/50 =1 3/25 Examples:

Writing a decimal as a fraction - Answers Examples: /100 =1 6/50 =1 3/25 Examples: 2.32 = 2 8/ = 13/20

Writing a repeating decimal as a fraction Examples: __ _ __ ___

Writing a repeating decimal as a fraction - Answers Examples: __ _ __ ___ /11 7/9 6/11 71/333

5-3 Adding and Subtracting Fractions Adding or subtracting fractions with the same denominator: –Add or subtract the numerators and keep the same denominators –Reduce if necessary –Examples: 7/10 - 3/10 11/y + (-5/y)

5-3 Adding and Subtracting Fractions - Answers Adding or subtracting fractions with the same denominator: –Examples: 7/10 - 3/10 = 2/5 11/y + (-5/y) = 6/y

Add/Subtract Fractions with Different Denominators Find a common denominator Add or subtract the numerators Keep the same denominator Reduce if necessary Examples:1/8 - 5/6 1/8 - 5x/6 3/7 - 2/m

Add/Subtract Fractions with Different Denominators - Answers Examples:1/8 - 5/6 = - 17/24 1/8 - 5x/6 = 6-40x/48 3/7 - 2/m = 3m - 14/7m

Add/Subtract Mixed Numbers Examples: 2 2/ /4 = -5 3/ /8 = 5 2/ /6 = 2 1/ /4 = 1 1/ /8 = 1 ½ - 2 4/5 = 3 2/9 – 5 2/3 =

Add/Subtract Mixed Numbers- answers Examples: 2 2/ /4 = 4 5/ / /8 = -4 1/8 5 2/ /6 = 1 1/2 2 1/ /4 = 4 1/4 1 1/ /8 = 1 ½ 1 ½ - 2 4/5 = -1 3/10 3 2/9 – 5 2/3 = -2 2/9

5-4 Multiplying and Dividing Fractions Multiplying Fractions = multiply straight across -- multiply the numerators together and multiply the denominators together Simplify before multiplying if possible Change any mixed numbers to improper fractions before multiplying!!! Reduce if necessary

Examples 3/7 x 4/5 9/15 x -(5/9) y/4 x 8/11 -(5/15) x 21/ /5 * 2 2/7

Examples - Answers 3/7 x 4/5 = 12/35 9/15 x -(5/9) = - 1/3 y/4 x 8/11 = 2y/11 -(5/15) x 21/25 = - 7/ /5 * -2 2/7 = 3 1/5

Dividing Fractions To divide fractions = keep the 1st fraction the same, change the division sign to a multiplication sign, flip the 2nd fraction, and multiply across Reciprocal To divide mixed numbers = change all mixed numbers to improper fractions, then follow the same rules as when dividing fractions

Examples 2/9 ÷ 2/5 x/3 ÷ x/4 1 3/4 ÷ (-2 5/8) 1 4/5 ÷ -1 ½

Examples 2/9 ÷ 2/5 = 2/9 x 5/2 = 10/18 = 5/9 x/3 ÷ x/4 = 1 1/3 1 3/4 ÷ (-2 5/8) = -2/3 1 4/5 ÷ -1 ½ = -1 1/5

5-7 Solving Equations by Adding or Subtracting Fractions You solve equations with fractions the same way you solve equations with integers and decimals by doing the opposite operation you already have. Example: 1/4 + n = 1/3 -1/4 -1/4 n = 1/3 - 1/4 4/12 - 3/12 n = 1/12

Examples y + 8/9 = 5/9 2/3 = x + 3/5 C + 3/10 = 11/15 6/7 = x - 2/7 3 7/18 = a + 1 1/3 Y + 4 7/8 = 2 A - 2 1/12 = 3 1/12

Examples - Answers y + 8/9 = 5/9 y = -1/3 2/3 = x + 3/5x = 1/15 C + 3/10 = 11/15 c = 13/30 6/7 = x - 2/7x = 1 1/7 3 7/18 = a + 1 1/3a = 2 1/18 Y + 4 7/8 = 2y = -2 7/8 A - 2 1/12 = 3 1/12 a = 5 1/6

5-8 Solving Equations by Multiplying Fractions You can undo multiplication by dividing each side of an equation by the same number. You can also multiply each side of an equation by the reciprocal to undo multiplication.

Multiplying by a reciprocal You can get a variable by itself by multiplying by a reciprocal Example:5a = 1/7

Multiplying by a reciprocal - Answers You can get a variable by itself by multiplying by a reciprocal Example:5a = 1/7 (1/5) 5a = (1/5) (1/7) a = 1/35

Multiplying by the negative reciprocal Example:-(14/25)k = 8/15 Try these:-(6/7)r = (3/4) -(10/13)b = -(2/3)

Multiplying by the negative reciprocal - Answers Example:-(14/25)k = 8/15 (-25/14)(-14/25)k = (8/15)(-25/14) k = -20/21 Try these:-(6/7)r = (3/4) r = -7/8 -(10/13)b = -(2/3) b = 13/15

Solving Equations with Mixed Numbers Change mixed numbers to improper fractions before multiplying Example:1 5/8n = 25

Solving Equations with Mixed Numbers - Answers Change mixed numbers to improper fractions before multiplying Example:1 5/8n = 25 13/8n = 25 (8/13)(13/8)n = 25(8/13) n = 15 5/13

Solving Equations with Mixed Numbers Examples: 3 1x = 28 (-7/20) = 1 1y h = (-12 1/2) 4

Solving Equations with Mixed Numbers - Answers Examples: 3 1x = 28 x = 8 2 (-7/20) = 1 1y y = -3/ h = (-12 1/2) h = 4 6/11 4