The Correct Answer and Explanation is:
The correct factored form of the expression is 0.3(90 + 30y + z).
Explanation:
The goal is to factor the expression 27 + 9y + 0.3z by finding the Greatest Common Factor (GCF). Factoring involves pulling out the largest term that divides evenly into all parts of the expression. The problem includes a specific and important rule: after factoring, all the coefficients remaining inside the parentheses must be integers.
First, we identify the coefficients of each term: 27, 9, and 0.3. A common mistake would be to find the GCF of only the integers 27 and 9, which is 9. However, if we factor out 9, the expression becomes 9(3 + y + 0.0333…z). The coefficient for z is not an integer, so this does not meet the problem’s requirement.
To satisfy the integer coefficient rule, we must find a factor that works for all three terms, including the decimal. The GCF of 27, 9, and 0.3 is actually 0.3. We can see this by dividing each term in the original expression by 0.3:
- 27 ÷ 0.3 = 90
- 9y ÷ 0.3 = 30y
- 0.3z ÷ 0.3 = z (which has a coefficient of 1)
Now, we can rewrite the expression with 0.3 factored out. The terms we just calculated (90, 30y, and z) become the new expression inside the parentheses. This gives us the final answer: 0.3(90 + 30y + z). This result is correct because all the coefficients inside the parentheses, which are 90, 30, and 1, are integers. The area model shown in the picture is a simplified visual for the first two terms, but the algebraic principle of finding the GCF applies to the entire expression.
