Question

Which of the following is equivalent to \(\frac{2 m+6}{4} \times \frac{6 m-36}{3 m+9} ?\) A) \(\frac{12 m^2-216}{12 m+36}\) B) \(\frac{8 m-30}{3 m+13}\) c) \(\frac{m-6}{4}\) D) \(m-6\)

Short answer

None of the given options match the simplified expression, so the correct answer is not listed among the options.

Step-by-step solution

Identify the given fractions

First, let's identify the given fractions we need to multiply: \[\frac{2m+6}{4}\] and \[\frac{6m-36}{3m+9}\]

Multiply the fractions

Now, we multiply the two fractions as follows: \[\frac{2m+6}{4} \times \frac{6m-36}{3m+9}\] To do this, we multiply the numerators together and multiply the denominators together: \[\frac{(2m+6)(6m-36)}{(4)(3m+9)}\]

Expand and simplify the expression

Let's expand and simplify the expression: \[\frac{2m(6m) + 6(6m) - 2m(36) - 6(36)}{4(3m+9)}\] \[\frac{12m^2 + 36m - 72m - 216}{12m+36}\] \[\frac{12m^2 - 36m - 216}{12m+36}\] Now, let's factorize the common factor from both the numerator and denominator which is 12: \[\frac{12(m^2 - 3m - 18)}{12(m+3)}\] After that, we can cancel out the common factor 12: \[\frac{m^2 - 3m - 18}{m+3}\]

Check which option matches the simplified expression

Now, let's see which of the given options matches our simplified expression: A) \[\frac{12m^2-216}{12m+36}\] (Not equal to our simplified expression) B) \[\frac{8m-30}{3m+13}\] (Not equal to our simplified expression) C) \[\frac{m-6}{4}\] (Not equal to our simplified expression) D) \(m-6\) (Not equal to our simplified expression) None of the given options matches our simplified expression, so the correct answer is not listed among the options. However, if the correct option was included, it would be: E) \[\frac{m^2 - 3m - 18}{m+3}\]

Key concepts

Fractions Multiplication

Understanding how to multiply fractions is vital for solving many algebra problems, including those on the SAT. To multiply fractions, simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For instance, \( \frac{a}{b} \times \frac{c}{d} \) equals \( \frac{a \times c}{b \times d} \). However, before multiplying, it's important to check if you can simplify the fractions by canceling any common factors. This technique not only makes your calculations easier but also helps in reducing the final result to the simplest form. Remember, numbers and variables can both act as factors which might be simplified or canceled out.

When confronted with complex expressions, like in SAT math problem solving, understanding this concept is key. In our exercise, the multiplication of fractions involves both numerical and algebraic components, which must be tackled systematically to arrive at the correct solution.

Algebraic Expression Simplification

Simplifying an algebraic expression is like cleaning up a cluttered room; you're making it more manageable and easier to understand. This process often involves combining like terms (terms with the same variables raised to the same power), expanding parentheses using the distributive property, and canceling common factors. It's vital to perform operations in the correct order and to always be on the lookout for opportunities to make the expression neater and more streamlined.

Take our SAT math problem as an example. Once the fractions are multiplied, we are left with an expanded form filled with multiple terms. To simplify, we combine like terms and look for common factors that can be factored out. Simplification can sometimes lead to a solution that doesn't match immediately recognizable options, which means extra care is needed to compare the reduced form of the expression with the given answer choices.

Factorization in Algebra

Factorization is the process of breaking down a complex expression into products of simpler factors. It allows simplification of algebraic expressions and solving of equations. This technique is particularly useful when dealing with polynomial expressions. In factorization, look for common factors in terms as well as special patterns such as the difference of squares, perfect square trinomials, and sum and difference of cubes.

In the context of the SAT question we're tackling, factorization comes into play after expanding and simplifying the initial algebraic fraction. By identifying and factoring out the greatest common factor from both the numerator and the denominator, we further simplify the expression, which then allows us to potentially cancel out common terms. Proper factorization is a crucial step in simplifying the algebraic expression to its most reduced form.

SAT Practice Questions

SAT math practice questions are designed to test a range of mathematical skills, including the ability to solve problems involving algebra and geometry. These questions often require a combination of procedural knowledge and conceptual understanding. To excel in SAT math problem solving, it's essential to practice different types of questions, from simple calculations to more complex problems that involve multiple steps and concepts such as fractions multiplication, algebraic expression simplification, and factorization.

When dealing with SAT problems, always read the question carefully, identify which mathematical concepts are being tested, and plan your approach accordingly. As exemplified by the exercise provided, sometimes none of the answer choices match the solution you've calculated. In such cases, double-check your work, ensuring each step adheres to mathematical principles. If the solution still doesn't match the given options, it's possible there could be an error in the question itself, or an additional choice might be the correct one.