Solving the Mathematical Expression Using Order of Operations

To find the value of the given mathematical expression, we need to follow the correct order of operations. A commonly used rule for the order of operations is BODMAS or BIDMAS.

BODMAS stands for:

• Brackets first
• Orders (powers, square roots, etc.)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

BIDMAS stands for:

• Brackets first
• Indices (powers, square roots, etc.)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In the given expression: \(25 \div 15 \text{ of } 4 \times [4 \div 5 \times (9 - 7)] - (20 \div 5 \text{ of } 9)\)

Let's break down the calculation step-by-step.

Step 1: Solve the Brackets ()

First, we evaluate the operations inside the innermost brackets.

• The first set of brackets is \((9 - 7)\).
• \((9 - 7) = 2\)

The expression becomes: \(25 \div 15 \text{ of } 4 \times [4 \div 5 \times 2] - (20 \div 5 \text{ of } 9)\)

Step 2: Evaluate the 'of' operation

The 'of' operation is performed after brackets but before division and multiplication. 'of' means multiplication.

• Evaluate \(15 \text{ of } 4\): \(15 \times 4 = 60\)
• Evaluate \(5 \text{ of } 9\) in the second bracket: \(5 \times 9 = 45\)

The expression is now: \(25 \div 60 \times [4 \div 5 \times 2] - (20 \div 45)\)

Step 3: Solve the Square Brackets []

Next, evaluate the expression inside the square brackets \([4 \div 5 \times 2]\). We perform division and multiplication from left to right.

• First, \(4 \div 5 = \frac{4}{5}\).
• Then, \(\frac{4}{5} \times 2 = \frac{4 \times 2}{5} = \frac{8}{5}\).

The expression becomes: \(25 \div 60 \times \frac{8}{5} - (20 \div 45)\)

Step 4: Simplify the second Bracket ()

Now, let's simplify the expression in the second bracket \((20 \div 45)\).

• \(20 \div 45 = \frac{20}{45}\).
• We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor, which is 5.
• \(\frac{20 \div 5}{45 \div 5} = \frac{4}{9}\).

The expression is now: \(25 \div 60 \times \frac{8}{5} - \frac{4}{9}\)

Step 5: Perform Division and Multiplication (from left to right)

We have \(25 \div 60 \times \frac{8}{5} - \frac{4}{9}\). Perform division first, then multiplication.

• First, \(25 \div 60 = \frac{25}{60}\).
• Simplify the fraction \(\frac{25}{60}\) by dividing by 5: \(\frac{25 \div 5}{60 \div 5} = \frac{5}{12}\).
• The expression becomes: \(\frac{5}{12} \times \frac{8}{5} - \frac{4}{9}\).
• Now, perform the multiplication: \(\frac{5}{12} \times \frac{8}{5}\). We can cancel out the 5 from the numerator and denominator.
• \(\frac{\cancel{5}}{12} \times \frac{8}{\cancel{5}} = \frac{8}{12}\).
• Simplify the fraction \(\frac{8}{12}\) by dividing by 4: \(\frac{8 \div 4}{12 \div 4} = \frac{2}{3}\).

The expression is now: \(\frac{2}{3} - \frac{4}{9}\)

Step 6: Perform Subtraction

Finally, we perform the subtraction: \(\frac{2}{3} - \frac{4}{9}\). To subtract fractions, they must have a common denominator. The least common multiple of 3 and 9 is 9.

• Convert \(\frac{2}{3}\) to a fraction with a denominator of 9: \(\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}\).
• Now subtract: \(\frac{6}{9} - \frac{4}{9} = \frac{6 - 4}{9} = \frac{2}{9}\).

The value of the expression is \(\frac{2}{9}\).

Revision Table: Key Mathematical Concepts

Additional Information on Order of Operations

Understanding the correct order of operations is crucial not just for solving expressions like this, but for many areas of mathematics and science. Without a standard order, expressions could have multiple possible values, leading to confusion.

While BODMAS and BIDMAS are common, sometimes PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or other mnemonics are used, particularly in different regions. The core principle remains the same: perform operations in a specific sequence.

A key point to remember is that multiplication and division have equal priority, as do addition and subtraction. When you have a sequence of only multiplication and division (or only addition and subtraction), you perform them from left to right.

For example, \(10 - 5 + 2\) is calculated as \((10 - 5) + 2 = 5 + 2 = 7\), not \(10 - (5 + 2) = 10 - 7 = 3\).

Similarly, \(10 \div 5 \times 2\) is calculated as \((10 \div 5) \times 2 = 2 \times 2 = 4\), not \(10 \div (5 \times 2) = 10 \div 10 = 1\).

In the context of 'of', it acts like multiplication but is given higher priority than standard multiplication/division in the BODMAS/BIDMAS structure when it appears alongside them. This is why '15 of 4' was calculated before the division `25 ÷ 60` and multiplication `× [4 ÷ 5 × 2]`. Similarly, `5 of 9` was calculated before the division `20 ÷ 45`.