Using Properties with Rational Numbers

Warm Up

Identify the property represented.

1. 4 + (9 + 3) = (4 + 9) + 3

2. 10(5 - 6) = 10 . 5 - 10 . 6

3. 17 . 1 = 17

Associative Property

Distributive Property

Identity Property

Using Properties with Rational Numbers

Problem of the DayFran made 18 three-point shots and 12 one-point shots. She had a total of 102 points for the basketball season. How many two-point shots did Fran make for the season?18 two-point shots

Using Properties with Rational Numbers

Learn to use properties of rational numbers to write equivalent expressions and equations.

Using Properties with Rational Numbers

You can use the Distributive Property to calculate Orlando’s total earnings two different ways.

The Distributive Property can be used to help you perform calculations more easily by writing equivalent expressions.

Orlando works part-time at amoving company to earn money for a car. He earns $12.75 per hour.

Using Properties with Rational Numbers

The Distributive Property states:a(b + c) = ab + aca(b - c) = ab - ac

Remember

Using Properties with Rational Numbers

Additional Example 1: Writing Equivalent Expressions

An art teacher pays $13.89 for one box of watercolor brushes. She buys 6 boxes in March and 5 boxes in April. Use the Distributive Property to write equivalent expressions showing two ways to calculate the total cost of the watercolor boxes.

Write an expression to show how much the teacher pays for a box and how many boxes purchased. Then use the Distributive Property to write an equivalent expression

Using Properties with Rational Numbers

Method 1$13.89(6 + 5) $13.89(11)$152.79

Method 2$13.89(6) + $13.89(5)$83.34 + $69.45 $152.79

Both methods result in a calculation of $152.79 for the amount of money spent of watercolor brushes.

Using Properties with Rational Numbers

Check It Out : Example 1

Jamie earns $8.75 per hour. Last week she worked 15 hours and next week she will work 20 hours. Use the Distributive Property to write equivalent expressions showing two ways to calculate how much money she earned.

Write an expression to show how much Jamie earns and the number of hours she works. Then use the Distributive Property to write an equivalent expression.

Using Properties with Rational Numbers

Method 1$8.75(15 + 20) $8.75(35)$306.25

Method 2$8.75(15) + $8.75(20)$131.25 + $175 $306.25

Both methods result in a calculation of $306.25 for Jamie’s salary.

Continued: Check It Out Example 1

Using Properties with Rational Numbers

Additional Example 2 : Writing Equivalent Expressions

34

x + 7 = 56

Write an equivalent equation for that does

not contain fractions. Then solve the equation.

34

X + 7 = 56

The LCM of denominators is 12.

34

x+ 7 = 56

12 12 Multiply both sides by 12.

31 4

x + 12 (7) = 51 6

3 12 12 2 Simplify.

Using Properties with Rational Numbers

Additional Example 2 : Continued

9x + 84 = 10 9x + 84 = 10 is an equivalent expression

9x + 84 = 10-84 -84

9x = -74

Subtract 84 from both sides.

Divide both side by 99 9

x = -829

An equivalent equation is 9x + 84 = 10 and the

solution is x = -829

Using Properties with Rational Numbers

Write an equivalent equation for

that does not contain fractions. Then solve

the equation.

Check It Out: Example 2

12

X + 9 = 46

12

X + 9 = 46

The LCM of denominators is 6

12

x+ 9 = 46

6 6 Multiply both sides by 6.

11 2

x + 6 (9) = 41 6

3 6 6 1 Simplify.

Using Properties with Rational Numbers

3x + 54 = 4 3x + 54 = 4 is an equivalent expression

3x + 54 = 4

-54 -54

3x = -50

3 3

Subtract 54 from both sides.

Divide both side by 3

x = -1623

An equivalent equation is 3x + 34 = 4 and the

solution is x = -1623

Continued: Check It Out Example 2

Using Properties with Rational Numbers

0.75 can also be

written as

Helpful Hint

75100

Using Properties with Rational Numbers

The soccer team uses a 36.75-liter container to take water to games. The team manager fills 0.75 liter bottles from this. He has used 22.5 liters. How many more 0.75 liter bottles can he fill before he runs out of water? Write and solve an equivalent equation without decimals.

Additional Example 3: Construction Application

Write an equation to represent the situation.0.75x + 22.5 = 36.75

Using Properties with Rational Numbers

Write an equivalent equation without decimals.

100(0.75x + 22.5) = (36.75)100

The equation has decimals to the hundredths, so multiply both sides by 100.

Use the Distributive Property

100(0.75x + 100(22.5) = (36.75)100

75x + 2,250 = 3,675

Simplify to get an equivalent equation without decimals

Continued: Example 3

Using Properties with Rational Numbers

75x + 2,250 = 3,675

-2250 -2250

75x = 1,425

75 75

x = 19

The number of 0.75 liter bottles that he can fill before he runs out of water is 19.

Continued: Example 3

Using Properties with Rational Numbers

Check It Out: Example 3…If the soccer team uses a 42.5-liter container, about how many 0.75 liter bottles can the manager fill before he runs out of water?

Write an equation to represent the situation.0.75x + 22.5 = 42.5

Write an equivalent equation without decimals.

100(0.75x + 22.5) = (42.5)100

The equation has decimals to the hundredths, so multiply both sides by 100.

Using Properties with Rational Numbers

Use the Distributive Property

100(0.75x + 100(22.5) = (42.5)100

75x + 2,250 = 4,250

Simplify to get an equivalent equation without decimals

Continued: Check It Out Example 3

75x + 2,250 = 4,250

-2250 -2250

75x = 200075 75

Using Properties with Rational Numbers

Continued: Check It Out Example 3

The number of 0.75 liter bottles that he can fill before he runs out of water is 19.

x ≈ 26.6

Using Properties with Rational Numbers

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Using Properties with Rational Numbers

Lesson Quiz

1. Jai earns $9.75 per hour. Jai works 3 hours one day and then works 7 hours the next day. Use the Distributive Property to write equivalent expressions showing two ways to calculate Jai’s total earnings.

9.75(3) + 9.75(7);9.75(3 + 7); $97.50

Write an equivalent equation that does not contain fractions. Then solve the equation.

45

x + 4 = 12

2. 8x + 40 = 5; x = -438

Using Properties with Rational Numbers

Lesson Quiz

23

x - 4 = 14

3. 8x - 48 = 3; x = 638

4. Joy has $67.85. She buys several pairs of earrings at $9.98 per pair and has $17.95 left. How many pairs of earrings did she buy? Write and solve an equivalent equation without decimals.

9.98x + 17.95 = 67.85; 998x + 1795 = 6785; x = 5;Joy bought 5 pairs of earrings.

Using Properties with Rational Numbers

1. Write an equivalent equation that does not contain fractions.

A. 2x + 6 = 4

B. 2x + 36 = 4

C. 3x + 6 = 4

D. 3x + 36 = 4

Lesson Quiz for Student Response Systems

13

x + 6 = 46

Using Properties with Rational Numbers

2. Write an equivalent equation that does not contain fractions.

A. 8 = 2x + 16

B. 8 = 2x + 2

C. 7 = 2x + 2

D. 7 = 2x + 16

Lesson Quiz for Student Response Systems

78

= 14

x + 2

Using Properties with Rational Numbers

3. Solve the equation

A. x = -4

B. x = -5

C. x = 3

D. x = 5

Lesson Quiz for Student Response Systems

68

= 14

x + 2