To solve this problem, we will use the distributive property to write two equivalent expressions representing the number of registered dogs belonging to the four most popular breeds.
Step 1: Identify the Percentages
Labrador Retriever = 15.1%
Chihuahua = 10.5%
German Shepherd = 6.9%
Shih Tzu = 5.9%
Let x = the total number of registered dogs.
Step 2: Write the First Expression
Convert each percentage to a decimal:
15.1% = 0.151
10.5% = 0.105
6.9% = 0.069
5.9% = 0.059
Multiply each decimal by x and add:
Expression 1:
0.151x + 0.105x + 0.069x + 0.059x
This represents the number of registered dogs in each of the four breeds added together.
Step 3: Apply the Distributive Property
The distributive property states:
a(b + c) = ab + ac
We can use it in reverse by factoring out the common variable x:
0.151x + 0.105x + 0.069x + 0.059x
= x(0.151 + 0.105 + 0.069 + 0.059)
Expression 2:
x(0.151 + 0.105 + 0.069 + 0.059)
Step 4: Simplify
Add the decimals:
0.151 + 0.105 + 0.069 + 0.059 = 0.384
Therefore:
x(0.384) = 0.384x
This means the four breeds together account for 38.4% of all registered dogs.
Final Answer:
The two expressions are:
1. 0.151x + 0.105x + 0.069x + 0.059x
2. x(0.151 + 0.105 + 0.069 + 0.059)
Both simplify to:
0.384x
Helpful Tip: The distributive property works in both directions! You can multiply a number through parentheses or factor out a common number or variable to create an equivalent expression.
