1. 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

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

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

4. 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

5. 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

6. ÷ 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

7. 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

8. 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

9. Additional Example 2: Dividing Decimals
Find ÷ 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.

10. Find 0.65 ÷ 0.25. Estimate the reasonableness of your answer.
Check It Out! Example 2
Find 0.65 ÷ 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
– n 0
The answer is reasonable.
0.65 ÷ 0.25 = 2.6
Estimate: 60 ÷ 20 = 3

11. 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

12. 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

13. 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

14. Additional Example 4 Continued2
Make a Plan
Set up an equation.

15. 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

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