Problem 73 - Algebra 1

Question

Which of the following has the least value? (A) \(\left[\frac{3}{8}(8-6)+\frac{1}{4}\right] \cdot(-12)\) (B) \(\frac{3}{8} \cdot 8-6+\frac{1}{4} \cdot(-12)\) (c) \(-\frac{3}{8} \cdot 8-6+\frac{1}{4} \cdot 12\) (D) \(-\frac{3}{8} \cdot\left(8-6+\frac{1}{4}\right) \cdot(-12)\)

Step-by-Step Solution

Answer

Option (A), which is -12, has the least value.

Step 1: Calculate option (A)

Solve the expression inside the square brackets first, \(\left[\frac{3}{8}(8-6)+\frac{1}{4}\right] = \left[\frac{3}{8}\cdot2+\frac{1}{4}\right]= \[\frac{3}{4}+\frac{1}{4}\] = 1. Finally, multiply by -12: \(1 \cdot (-12) = -12\).

Step 2: Calculate option (B)

Applying order of operations, perform multiplication and division from left to right first: \(\frac{3}{8} \cdot 8-6+\frac{1}{4} \cdot(-12) = 3 - 6 - 3 = -6\).

Step 3: Calculate option (C)

Perform multiplication as follows: \(-\frac{3}{8} \cdot 8-6+\frac{1}{4} \cdot 12 = -3 - 6 + 3 = -6\).

Step 4: Calculate option (D)

For this expression, the operations inside the parentheses should be performed first: \(-\frac{3}{8} \cdot\left(8-6+\frac{1}{4}\right) \cdot(-12) = -\frac{3}{8} \cdot\left(2+\frac{1}{4}\right) \cdot(-12) = -\frac{3}{8} \cdot\frac{9}{4} \cdot(-12) = -\frac{27}{8} \cdot(-12) = 40.5\).

Key Concepts

Simplifying Expressions

At the very heart of algebra is the concept of simplifying expressions. Simplifying an expression is the process of making it as straightforward as possible. This often involves combining like terms, performing arithmetic operations, and reducing fractions.

Applying this concept to our exercise, the problem provides different algebraic options to determine which has the least value. To simplify each option, it is essential to follow the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), and execute each operation step by step.

For example, in option (A), we simplify what’s inside the square brackets first, which is essentially a smaller expression that needs to be fully simplified before it can be used in further calculations. By systematically working through each expression and simplifying them step by step, we can find and compare their resulting values to identify the least one.

Parentheses in Algebra

Parentheses play a crucial role in algebra as they indicate which operations should be performed first. Without proper attention to parentheses, the entire meaning of an expression can change, leading to incorrect results.

In order to simplify expressions accurately, it's important to evaluate and simplify the contents within the parentheses before moving on to other operations. In our exercise, the parentheses within options (A) and (D) guide us to perform certain calculations before others.

For instance, option (D) involves an operation inside parentheses, followed by multiplication with a negative fraction, and then by another multiplication with a negative number. Careful evaluation of expressions within parentheses is essential for obtaining the correct result and thus, determining which of the options yields the least value.

Negative Numbers

Negative numbers, those less than zero, come with their own set of rules when it comes to calculations. It’s important to remember that multiplying or dividing two negative numbers results in a positive number, while performing these operations with one negative and one positive number results in a negative number.

In the context of our exercise, understanding how to work with negative numbers is key to correctly solving each option. For instance, both options (A) and (D) involve multiplying by negative numbers, which changes the sign of the result.

Further, option (C) starts with a multiplication of a negative and a positive number, leading to an initial negative result, which impacts the subsequent subtraction and addition operations. Mastery of the rules governing negative numbers is essential in simplifying algebraic expressions and solving for the correct values.