Solving Complex Mathematical Expressions with BODMAS

The question asks for the value of a given mathematical expression involving various operations like division, multiplication, and 'of' with fractions and whole numbers. To solve this, we must follow the order of operations, commonly known as BODMAS or PEMDAS.

BODMAS stands for:

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

The 'of' operator in mathematics represents multiplication and is typically evaluated after brackets but before standard multiplication and division. In the context of BODMAS, 'of' is often grouped with multiplication/division but is usually performed immediately after resolving brackets and orders.

Let's break down the given expression into three parts based on the main operations:

Expression: \( \left( {18 \div 2\;of\frac{1}{4}} \right)\; \times \;\left( {\frac{2}{3} \div \frac{3}{4}\; \times \;\frac{5}{8}} \right) \div \left( {\frac{2}{3} \div \frac{3}{4}of\frac{3}{4}} \right) \)

Let's evaluate each part separately.

Evaluating the First Part: \( \left( {18 \div 2\;of\frac{1}{4}} \right) \)

Inside the bracket, we have 'of' and division. According to the rule, 'of' is performed before division.

1. Calculate \(2\;of\frac{1}{4}\): $$ 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} $$
2. Now, perform the division: $$ 18 \div \frac{1}{2} $$ Dividing by a fraction is the same as multiplying by its reciprocal. $$ 18 \times \frac{2}{1} = 18 \times 2 = 36 $$

So, the value of the first part is 36.

Evaluating the Second Part: \( \left( {\frac{2}{3} \div \frac{3}{4}\; \times \;\frac{5}{8}} \right) \)

Inside the bracket, we have division and multiplication. These operations have the same precedence and should be performed from left to right.

1. Perform the division first: \( \frac{2}{3} \div \frac{3}{4} \) $$ \frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9} $$
2. Now, perform the multiplication: \( \frac{8}{9} \times \frac{5}{8} \) $$ \frac{8}{9} \times \frac{5}{8} = \frac{8 \times 5}{9 \times 8} = \frac{40}{72} $$ We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 8. $$ \frac{40 \div 8}{72 \div 8} = \frac{5}{9} $$

So, the value of the second part is \( \frac{5}{9} \).

Evaluating the Third Part: \( \left( {\frac{2}{3} \div \frac{3}{4}of\frac{3}{4}} \right) \)

Inside the bracket, we have 'of' and division. The 'of' operation is performed before division.

1. Calculate \( \frac{3}{4}of\frac{3}{4} \): $$ \frac{3}{4} \times \frac{3}{4} = \frac{3 \times 3}{4 \times 4} = \frac{9}{16} $$
2. Now, perform the division: \( \frac{2}{3} \div \frac{9}{16} \) $$ \frac{2}{3} \div \frac{9}{16} = \frac{2}{3} \times \frac{16}{9} = \frac{2 \times 16}{3 \times 9} = \frac{32}{27} $$

So, the value of the third part is \( \frac{32}{27} \).

Combining the Results

Now, substitute the values of the three parts back into the original expression:

\( \left( {18 \div 2\;of\frac{1}{4}} \right)\; \times \;\left( {\frac{2}{3} \div \frac{3}{4}\; \times \;\frac{5}{8}} \right) \div \left( {\frac{2}{3} \div \frac{3}{4}of\frac{3}{4}} \right) \)

becomes

\( 36 \times \frac{5}{9} \div \frac{32}{27} \)

Now we have multiplication and division. We perform these operations from left to right.

1. Perform the multiplication: \( 36 \times \frac{5}{9} \) $$ 36 \times \frac{5}{9} = \frac{36 \times 5}{9} = \frac{180}{9} = 20 $$
2. Perform the division: \( 20 \div \frac{32}{27} \) $$ 20 \div \frac{32}{27} = 20 \times \frac{27}{32} $$ Now, simplify before multiplying. Both 20 and 32 are divisible by 4. $$ 20 \div 4 = 5 $$ $$ 32 \div 4 = 8 $$ So, the expression becomes: $$ 5 \times \frac{27}{8} = \frac{5 \times 27}{8} = \frac{135}{8} $$

The result is the improper fraction \( \frac{135}{8} \). Let's convert this to a mixed number.

To convert \( \frac{135}{8} \) to a mixed number, divide 135 by 8:

\( 135 \div 8 = 16 \) with a remainder of \( 135 - (16 \times 8) = 135 - 128 = 7 \).

So, \( \frac{135}{8} = 16 \frac{7}{8} \).

The final value of the expression is \( 16 \frac{7}{8} \).

Revision Table: Key Concepts

Additional Information: Importance of Order of Operations

Understanding and applying the correct order of operations is crucial in mathematics to ensure that calculations are performed consistently and correctly. Without a standard order, the same expression could yield multiple different results, leading to ambiguity and errors. The BODMAS rule provides this standard framework, making mathematical expressions unambiguous and their solutions unique.

For expressions involving fractions, it's often helpful to convert division into multiplication by the reciprocal. Simplifying fractions at intermediate steps can also make calculations easier and reduce the chance of errors with large numbers.

Always remember that multiplication and division, as well as addition and subtraction, are performed from left to right when they appear consecutively in an expression after higher priority operations have been resolved.