Solving Complex Fraction Expressions with BODMAS

This problem requires us to evaluate a mathematical expression involving fractions, multiplication, division, and addition. To solve this correctly, we must follow the standard order of operations, often remembered by acronyms like BODMAS or PEMDAS.

BODMAS stands for:

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

PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Let's evaluate the given expression step-by-step:

The expression is: \(\rm\frac{1}{7} \times \frac{1}{8} \div \frac{72}{51} of \left(\frac{1}{9}+\frac{1}{8}\right)+\left(\frac{63}{56} \times \frac{48}{72}\right)\)

Step 1: Evaluate the expression inside the Brackets

First, we evaluate the sum inside the first set of brackets: \(\left(\frac{1}{9}+\frac{1}{8}\right)\). To add fractions, we need a common denominator. The least common multiple of 9 and 8 is 72.

Convert the fractions to have a denominator of 72:

\(\frac{1}{9} = \frac{1 \times 8}{9 \times 8} = \frac{8}{72}\)

\(\frac{1}{8} = \frac{1 \times 9}{8 \times 9} = \frac{9}{72}\)

Now, add the fractions:

\(\frac{8}{72} + \frac{9}{72} = \frac{8+9}{72} = \frac{17}{72}\)

So, the expression becomes: \(\rm\frac{1}{7} \times \frac{1}{8} \div \frac{72}{51} of \left(\frac{17}{72}\right)+\left(\frac{63}{56} \times \frac{48}{72}\right)\)

Step 2: Evaluate the 'of' operation

Next, we evaluate the 'of' operation, which means multiplication. We have \(\frac{72}{51} of \left(\frac{17}{72}\right)\). This is \(\frac{72}{51} \times \frac{17}{72}\).

Multiply the fractions:

\(\frac{72}{51} \times \frac{17}{72}\)

We can cancel out the 72 in the numerator and denominator:

\(\frac{\cancel{72}}{51} \times \frac{17}{\cancel{72}} = \frac{17}{51}\)

Simplify the fraction \(\frac{17}{51}\). Both 17 and 51 are divisible by 17 (since \(51 = 3 \times 17\)).

\(\frac{17 \div 17}{51 \div 17} = \frac{1}{3}\)

The expression is now: \(\rm\frac{1}{7} \times \frac{1}{8} \div \frac{1}{3}+\left(\frac{63}{56} \times \frac{48}{72}\right)\)

Step 3: Evaluate Division and Multiplication (from left to right)

According to BODMAS/PEMDAS, division and multiplication are performed next, from left to right.

First, the division: \(\frac{1}{8} \div \frac{1}{3}\). Dividing by a fraction is the same as multiplying by its reciprocal:

\(\frac{1}{8} \div \frac{1}{3} = \frac{1}{8} \times \frac{3}{1} = \frac{3}{8}\)

The expression becomes: \(\rm\frac{1}{7} \times \frac{3}{8}+\left(\frac{63}{56} \times \frac{48}{72}\right)\)

Next, the multiplication: \(\frac{1}{7} \times \frac{3}{8}\)

\(\frac{1}{7} \times \frac{3}{8} = \frac{1 \times 3}{7 \times 8} = \frac{3}{56}\)

The expression is now: \(\rm\frac{3}{56}+\left(\frac{63}{56} \times \frac{48}{72}\right)\)

Now, evaluate the multiplication inside the second set of brackets: \(\left(\frac{63}{56} \times \frac{48}{72}\right)\)

Simplify the fractions before multiplying:

• \(\frac{63}{56}\): Both are divisible by 7. \(\frac{63 \div 7}{56 \div 7} = \frac{9}{8}\)
• \(\frac{48}{72}\): Both are divisible by 24. \(\frac{48 \div 24}{72 \div 24} = \frac{2}{3}\)

Multiply the simplified fractions:

\(\frac{9}{8} \times \frac{2}{3}\)

We can cancel common factors (9 and 3, 2 and 8):

\(\frac{\cancel{9}^3}{\cancel{8}^4} \times \frac{\cancel{2}^1}{\cancel{3}^1} = \frac{3 \times 1}{4 \times 1} = \frac{3}{4}\)

The expression is now: \(\rm\frac{3}{56}+\frac{3}{4}\)

Step 4: Evaluate Addition

Finally, perform the addition: \(\frac{3}{56}+\frac{3}{4}\). To add these fractions, we need a common denominator. The least common multiple of 56 and 4 is 56 (since \(56 = 14 \times 4\)).

Convert \(\frac{3}{4}\) to have a denominator of 56:

\(\frac{3}{4} = \frac{3 \times 14}{4 \times 14} = \frac{42}{56}\)

Now, add the fractions:

\(\frac{3}{56} + \frac{42}{56} = \frac{3+42}{56} = \frac{45}{56}\)

Final Result

The value of the expression is \(\frac{45}{56}\).

Comparing with Options

Let's compare our result with the given options:

Our calculated value, \(\frac{45}{56}\), matches Option 4.

Revision Table: BODMAS Steps for Fractions

Additional Information on Order of Operations

The order of operations is crucial in mathematics to ensure that expressions are evaluated consistently, leading to a single correct answer. Without a standard order, different interpretations could lead to different results. BODMAS (or PEMDAS) provides this standard order.

Key points about the order of operations:

• Operations within brackets (or parentheses) are always performed first.
• Powers and roots (Orders or Exponents) are evaluated after brackets.
• Multiplication and Division have the same priority and are performed from left to right as they appear in the expression.
• Addition and Subtraction have the lowest priority among these operations and are also performed from left to right.

Understanding how to apply these rules, especially when working with fractions, decimals, or integers, is fundamental for solving mathematical problems accurately.