NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g. identity, inverse, distributive, associative, commutative) and justify the process used. California Standards

Two numbers whose product is 1 are multiplicative inverses, or reciprocals.

A division problem can always be rewritten as a multiplication problem by using the reciprocal of the divisor.

Additional Example 1A: Dividing Fractions Divide. Write the answer in simplest form. 5 11 1 2 ÷ 5 11 ÷ 1 2 5 11 • 2 1 = Multiply by the reciprocal. 5 11 • 2 1 = No common factors. 10 11 = Simplest form

Additional Example 1B: Dividing Fractions Divide. Write the answer in simplest form. 3 8 2 ÷ 2 3 8 ÷ 2 = 19 8 2 1 ÷ Write as an improper fraction. = 19 8 1 2 Multiply by the reciprocal. 19 • 1 8 • 2 = No common factors 19 16 = 3 16 = 1

÷ 28 45 Divide. Write the answer in simplest form. 7 15 3 4 7 15 ÷ 3 4 Check It Out! Example1A Divide. Write the answer in simplest form. 7 15 3 4 ÷ 7 15 ÷ 3 4 7 15 • 4 3 = Multiply by the reciprocal. 7 • 4 15 • 3 = No common factors 28 45 = Simplest form

4 ÷ 3 ÷ ÷ 3 4 1 Divide. Write the answer in simplest form. Check It Out! Example1B Divide. Write the answer in simplest form. 2 5 4 ÷ 3 = 22 5 3 1 ÷ Write as an improper fraction. 2 5 ÷ 3 4 = 22 5 1 3 Multiply by the reciprocal. 22 • 1 5 • 3 = No common factors 7 15 = or 1 22 15

When dividing a decimal by a decimal, multiply both numbers by a power of 10 so you can divide by a whole number. To decide which power of 10 to multiply by, look at the denominator. The number of decimal places represents the number of zeros after the 1 in the power of 10. 1.32 0.4 1.32 0.4 10 13.2 4 = = 1 decimal place 1 zero

Additional Example 2: Dividing Decimals Find 0.384 ÷ 0.24. Estimate the reasonableness of your answer. 0.24 )0.384 0.24 has two decimal places so multiple both numbers by 100 to make the divisor an integer. 1 .6 24 )38.4 Then divide as a whole number. - 24 0.384 ÷ 0.24 = 1.6 Estimate: 40 ÷ 20 = 2 14 4 – 144 The answer is reasonable.

Find 0.65 ÷ 0.25. Estimate the reasonableness of your answer. Check It Out! Example 2 Find 0.65 ÷ 0.25. Estimate the reasonableness of your answer. 0.25 )0.65 0.25 has two decimal places so multiple both numbers by 100 to make the divisor an integer. 2 .6 25 )65.0 Then divide as a whole number. - 50 15 –150 n 0 The answer is reasonable. 0.65 ÷ 0.25 = 2.6 Estimate: 60 ÷ 20 = 3

Additional Example 3: Evaluating Expressions with Rational Numbers Evaluate the expression for the given value of the variable. 5.25 for n = 0.15 n 5.25 n 0.15 = Substitute 0.15 for n. 0.15 has two decimal places, so multiply both numbers by 100 to make the divisor an integer. 0.15 )5.25 35 15 )525 When n = 0.15, = 35. 5.25 n - 525

3.60 for n = 0.12 n 3.60 = Substitute 0.12 for n. n 0.12 Check It Out! Example 3 Evaluate the expression for the given value of the variable. 3.60 for n = 0.12 n 3.60 n 0.12 = Substitute 0.12 for n. 0.12 has two decimal places, so multiply both numbers by 100 to make the divisor an integer. 0.12 )3.60 30 12 )360 When n = 0.12, = 30. 3.60 n - 360

Understand the Problem Additional Example 4: Problem Solving Application 1 2 A muffin recipe calls for cup of oats. You have cup of oats. How many batches of muffins can you bake using all of the oats you have? 3 4 1 Understand the Problem The number of batches of muffins you can bake is the number of batches using the oats that you have. List the important information: The amount of oats is cup. One batch of muffins calls for cup of oats. 12 34

Additional Example 4 Continued 2 Make a Plan Set up an equation.

Additional Example 4 Continued Solve 3 Let n = number of batches. 12 34 = n ÷ 34 21 = n • 64 , or 1 = n 12 You can bake 1 batches of the muffins. 12

Additional Example 4 Continued Look Back 4 One cup of oats would make two batches so 1 is a reasonable answer. 12