Solving the Mathematical Expression: Step-by-Step Guide

The problem asks us to evaluate the value of the given mathematical expression:

\((0.3) \left[\frac{(200 - 146)}{(3 \times 3 \times 3)} - 3\right]\)

To solve this, we need to follow the order of operations (BODMAS/PEMDAS): Brackets first, then Orders (powers and roots), Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).

Step 1: Calculate the value inside the innermost parenthesis

The innermost parenthesis contains a subtraction:

\(200 - 146\)

Calculating this gives:

\(200 - 146 = 54\)

So the expression becomes:

\((0.3) \left[\frac{54}{(3 \times 3 \times 3)} - 3\right]\)

Step 2: Calculate the value in the denominator

The denominator inside the square bracket is a multiplication:

\(3 \times 3 \times 3\)

Calculating this gives:

\(3 \times 3 = 9\)

\(9 \times 3 = 27\)

So the expression becomes:

\((0.3) \left[\frac{54}{27} - 3\right]\)

Step 3: Perform the division inside the square bracket

Now we perform the division:

\(\frac{54}{27}\)

Calculating this gives:

\(\frac{54}{27} = 2\)

So the expression becomes:

\((0.3) \left[2 - 3\right]\)

Step 4: Perform the subtraction inside the square bracket

Next, we perform the subtraction inside the square bracket:

\(2 - 3\)

Calculating this gives:

\(2 - 3 = -1\)

So the expression becomes:

\((0.3) \times (-1)\)

Step 5: Perform the final multiplication

Finally, we multiply \(0.3\) by \(-1\). Remember that \(0.3\) can be written as a fraction \(\frac{3}{10}\).

\(\frac{3}{10} \times (-1)\)

Calculating this gives:

\(\frac{3}{10} \times (-1) = -\frac{3}{10}\)

Let's re-check the options provided as the calculated value is negative, and options are positive fractions.

The calculation steps were:

• \(200 - 146 = 54\)
• \(3 \times 3 \times 3 = 27\)
• \(\frac{54}{27} = 2\)
• \(2 - 3 = -1\)
• \(0.3 \times (-1) = -0.3 = -\frac{3}{10}\)

Let's carefully re-examine the problem statement and calculations. Assuming the steps and original expression are correct, the result is indeed -\(\frac{3}{10}\).

However, since we are asked to provide a solution based on the given options and correct answer, let's assume there might be a typographical error in the question or options provided, and proceed by showing the steps clearly based on the calculation performed.

The final calculated value is \(-\frac{3}{10}\).

Summary of Calculation Steps

Based on standard mathematical evaluation, the value of the expression \((0.3) [\{(200 - 146)/(3 \times 3 \times 3)\} - 3]\) is \(-\frac{3}{10}\).

Let's assume there was a potential error in transcribing the problem or options and provide the step-by-step process that leads to one of the positive options, *if* a small change were made. For instance, if the subtraction inside the bracket was \(3 - \frac{54}{27}\) instead of \(\frac{54}{27} - 3\), the steps would differ:

• Step 1-3 remain the same, yielding \(\frac{54}{27} = 2\).
• The expression would be \((0.3) [3 - 2]\).
• Step 4 (modified): \(3 - 2 = 1\).
• Step 5 (modified): \(0.3 \times 1 = 0.3 = \frac{3}{10}\).

This modified calculation gives \(\frac{3}{10}\), which is not among the options \(\frac{10}{3}, \frac{5}{3}, \frac{7}{3}, \frac{8}{3}\). The provided correct answer is \(\frac{10}{3}\), which is approximately \(3.33\). Our original calculation gives \(-\frac{3}{10}\) or \(-0.3\).

Let's consider another hypothetical change that might lead to \(\frac{10}{3}\). If the expression was somehow evaluated differently or involved different numbers, the result could change. Since the problem asks for the value based on the given expression, we adhere strictly to the provided formula and standard order of operations.

Our calculation of the expression \((0.3) [\{(200 - 146)/(3 \times 3 \times 3)\} - 3]\) results in \(-\frac{3}{10}\).

Revision Table: Order of Operations

Additional Information: Decimals and Fractions

The number \(0.3\) is a decimal. It can be easily converted into a fraction. The digit '3' is in the tenths place, so \(0.3\) is equivalent to three tenths, which is written as \(\frac{3}{10}\). Understanding the relationship between decimals and fractions is crucial for solving many mathematical problems.

Converting \(0.3\) to \(\frac{3}{10}\) allowed us to perform the final multiplication with a fraction, which is often helpful when options are given as fractions.

The number \(\frac{10}{3}\) is an improper fraction because the numerator (10) is greater than the denominator (3). As a mixed number, it is \(3 \frac{1}{3}\), and as a decimal, it is a repeating decimal \(3.333...\).

Our careful step-by-step evaluation of the expression \((0.3) [\{(200 - 146)/(3 \times 3 \times 3)\} - 3]\) using the correct order of operations yielded the result \(-\frac{3}{10}\).