Problem 59 - Algebra 1: Concepts and Skills

Question

What does \(-3(6)\left(-\frac{1}{3}\right)\) equal? A) \(-6\) B) \(-2\) C) \(2\) D) \(6\)

Step-by-Step Solution

Answer

The correct answer is D) 6

Step 1: Multiply -3 and 6

The first operation to carry out is the multiplication of -3 and 6. This yields \( -3 \times 6 = -18 \)

Step 2: Multiply the result with -1/3

Next, the result from the previous step should be multiplied with -1/3. This multiplication yields \( -18 \times -\frac{1}{3} = 6 \)

Key Concepts

Order of Operations

When solving mathematical expressions, the order of operations is crucial to find the correct answer. Not following the right sequence can lead to incorrect results.

The common phrase PEMDAS helps remember the order:

• P: Parentheses
• E: Exponents
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

In this exercise, the expression \(-3(6)(-\frac{1}{3})\)\ requires you to first address multiplications since there are no parentheses or exponents to simplify initially. Thus, you perform the multiplication \((-3 \times 6) = -18\) first and then multiply the result by \(-\frac{1}{3}\).Applying PEMDAS ensures that calculations are done in the proper order, reducing errors in more complex arithmetic computations.

Negative Numbers

Negative numbers are numbers below zero and are represented with a minus sign. Understanding how negative numbers interact with other integers is important when performing operations like multiplication.

Here's a quick overview of rules involving negative numbers:

• Multiplying two positive numbers results in a positive number.
• Multiplying two negative numbers also results in a positive number. For example, \((-3) \times (-2) = 6\).
• Multiplying a positive number and a negative number results in a negative number. For instance, \(3 \times (-2) = -6\).

In our specific problem, first, we multiplied \(-3\), a negative number, with \(6\), a positive number, giving us \(-18\)\. Then, multiplying \(-18\)\ by another negative number \(-\frac{1}{3}\) reversed the sign to positive, leading to \(6\)\ as the final answer.

Fractions in Arithmetic

Fractions represent parts of a whole and are used in arithmetic to express numbers that aren't whole. When multiplying fractions, especially involving whole numbers and negative numbers, it helps to break down the process.

Multiplying a fraction by a whole number involves multiplying the numerator by the whole number while keeping its denominator the same. For example, multiplying \(-18\) by \(-\frac{1}{3}\) involves:

• First, multiply the numerators: \((-18)\ \times (-1) = 18\).
• Second, keep the denominator: \(3\).
• The division \(18 \div 3 = 6\) gives the result.

This simplifies to \(6\)\, which is our final answer. Remember that when multiplying fractions, if both numbers are negative, the product will be positive.