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6-3 Using Properties with Rational Numbers
Warm Up
Identify the property represented.
1. 4 + (9 + 3) = (4 + 9) + 3
Associative Property
2. 10(5 - 6) = 10 . 5 - 10 . 6
Distributive Property
3. 17 . 1 = 17
Identity Property
6-3 Using Properties with Rational Numbers
Learn to use properties of rational
numbers to write equivalent expressions
and equations.
6-3 Using Properties with Rational Numbers
Remember
The Distributive Property states:
a(b + c) = ab + ac
a(b - c) = ab - ac
6-3 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
6-3 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.
6-3 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.
6-3 Using Properties with Rational Numbers
Continued: Check It Out Example 1
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.
6-3 Using Properties with Rational Numbers
Check It Out: Example 2
1
4
Write an equivalent equation for X + 9 =
2
6
that does not contain fractions. Then solve the
equation.
1
X + 9 =4
2
6
6
36
1
x+ 9 = 4 6
2
6
1
1
x
2
The LCM of denominators is 6
Multiply both sides by 6.
+ 6 (9) =
1
4
6
6
1
Simplify.
6-3 Using Properties with Rational Numbers
Continued: Check It Out Example 2
3x + 54 = 4
3x + 54 = 4 is an equivalent
expression
3x + 54 = 4
-54
3x
3
-54
= -50
3
Subtract 54 from both sides.
Divide both side by 3
2
x = -16
3
An equivalent equation is 3x + 34 = 4 and the
2
solution is x = -16
3
6-3 Using Properties with Rational Numbers
Additional Example 3: Construction Application
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.
Write an equation to represent the situation.
0.75x + 22.5 = 36.75
6-3 Using Properties with Rational Numbers
Continued: Example 3
Write an equivalent equation without decimals.
The equation has decimals to the
hundredths, so multiply both sides by 100.
100(0.75x + 22.5) = (36.75)100
Use the Distributive Property
100(0.75x + 100(22.5) = (36.75)100
Simplify to get an equivalent equation
without decimals
75x + 2,250 = 3,675
6-3 Using Properties with Rational Numbers
Continued: Example 3
75x + 2,250 = 3,675
-2250
-2250
75x
75
= 1,425
75
x = 19
The number of 0.75 liter bottles that he can fill
before he runs out of water is 19.
6-3 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.
The equation has decimals to the
hundredths, so multiply both sides by 100.
100(0.75x + 22.5) = (42.5)100
6-3 Using Properties with Rational Numbers
Continued: Check It Out Example 3
Use the Distributive Property
100(0.75x + 100(22.5) = (42.5)100
Simplify to get an equivalent equation
without decimals
75x + 2,250 = 4,250
75x + 2,250 = 4,250
-2250
75x
75
-2250
= 2000
75
6-3 Using Properties with Rational Numbers
Continued: Check It Out Example 3
x ≈ 26.6
The number of 0.75 liter bottles that he can fill
before he runs out of water is 19.
6-3 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.
2. 4 x + 4 = 1
5
2
3
8x + 40 = 5;x = -4
8
6-3 Using Properties with Rational Numbers
Lesson Quiz
3. 2 x - 4 = 1
3
4
8x - 48 = 3; x = 6
3
8
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.