Simplifying Mathematical Expressions with BODMAS/PEMDAS

To solve the given mathematical expression, we need to follow the order of operations. A common acronym used for remembering the order is BODMAS or PEMDAS.

• B/P: Brackets or Parentheses (solve operations inside these first)
• O/E: Orders or Exponents (powers, square roots, etc.)
• D/M: Division or Multiplication (from left to right)
• A/S: Addition or Subtraction (from left to right)

The expression is: \(2 + 5 \div [5 + 8 \div\left(1+\frac{1}{3}\right) - 1 ]\)

Step-by-Step Calculation

Let's break down the calculation according to the order of operations.

Step 1: Solve the operation inside the innermost parentheses \(\left(1+\frac{1}{3}\right)\)

First, calculate the sum inside the small brackets:

\(1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{3+1}{3} = \frac{4}{3}\)

The expression now becomes:

\(2 + 5 \div [5 + 8 \div \left(\frac{4}{3}\right) - 1 ]\)

Step 2: Solve the division inside the square brackets \([5 + 8 \div \left(\frac{4}{3}\right) - 1 ]\)

Next, perform the division inside the square brackets:

\(8 \div \left(\frac{4}{3}\right) = 8 \times \frac{3}{4}\)

Multiply 8 by the reciprocal of \(\frac{4}{3}\):

\(8 \times \frac{3}{4} = \frac{8 \times 3}{4} = \frac{24}{4} = 6\)

The expression now becomes:

\(2 + 5 \div [5 + 6 - 1 ]\)

Step 3: Solve the operations inside the square brackets \([5 + 6 - 1 ]\)

Perform the addition and subtraction from left to right inside the square brackets:

\(5 + 6 - 1 = 11 - 1 = 10\)

The expression now becomes:

\(2 + 5 \div [10]\)

or simply

\(2 + 5 \div 10\)

Step 4: Perform the remaining division

Now, perform the division:

\(5 \div 10 = \frac{5}{10} = \frac{1}{2}\)

The expression now becomes:

\(2 + \frac{1}{2}\)

Step 5: Perform the final addition

Finally, add the numbers:

\(2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{4+1}{2} = \frac{5}{2}\)

The value of the expression is \(\frac{5}{2}\).

Summary of Calculation Steps

Thus, the result of the expression is \(\frac{5}{2}\).

Revision Table: Order of Operations

Additional Information: Fractions and Mixed Numbers

In this problem, we encountered a mixed number operation (\(1 + \frac{1}{3}\)). A mixed number combines a whole number and a fraction. To perform arithmetic with mixed numbers, it's often easiest to convert them into improper fractions.

• A number like \(1\) can be written as a fraction with any denominator (e.g., \(\frac{1}{1}\), \(\frac{2}{2}\), \(\frac{3}{3}\)). To add it to \(\frac{1}{3}\), we wrote \(1\) as \(\frac{3}{3}\).
• An improper fraction is one where the numerator is greater than or equal to the denominator (e.g., \(\frac{4}{3}\)).

We also performed division by a fraction. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{4}{3}\) is \(\frac{3}{4}\).