Solving Complex Fraction Expressions with BODMAS

To find the value of the given expression, we need to follow the order of operations, commonly known as BODMAS or PEMDAS. This rule dictates the sequence in which operations should be performed:

• Brackets (or Parentheses)
• Orders (or Exponents/Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In the given expression, \(\left( {\frac{7}{5} \div \frac{7}{{10}}of\frac{3}{4}} \right) \div \frac{4}{9} + \left( {\frac{7}{{16}} \div 10\frac{1}{2} \times 7\frac{1}{5}} \right) \times \frac{5}{{12}}\), we have brackets, 'of', division, multiplication, and addition.

Step 1: Evaluate the First Bracket

The first bracket is \(\left( {\frac{7}{5} \div \frac{7}{{10}}of\frac{3}{4}} \right)\). Inside this bracket, we first perform the 'of' operation, which is multiplication:

\(\frac{7}{{10}}of\frac{3}{4} = \frac{7}{{10}} \times \frac{3}{4} = \frac{7 \times 3}{10 \times 4} = \frac{21}{40}\)

Now, the expression inside the first bracket becomes \(\frac{7}{5} \div \frac{21}{40}\). We perform the division by multiplying by the reciprocal of the second fraction:

\(\frac{7}{5} \div \frac{21}{40} = \frac{7}{5} \times \frac{40}{21}\)

Simplify before multiplying:

\(= \frac{\cancel{7}^1}{5} \times \frac{40}{\cancel{21}^3} = \frac{1}{5} \times \frac{40}{3}\)

\(= \frac{1}{\cancel{5}^1} \times \frac{\cancel{40}^8}{3} = \frac{1 \times 8}{1 \times 3} = \frac{8}{3}\)

So, the value of the first bracket is \(\frac{8}{3}\).

Step 2: Evaluate the Second Bracket

The second bracket is \(\left( {\frac{7}{{16}} \div 10\frac{1}{2} \times 7\frac{1}{5}} \right)\). First, convert the mixed numbers to improper fractions:

\(10\frac{1}{2} = \frac{(10 \times 2) + 1}{2} = \frac{21}{2}\)

\(7\frac{1}{5} = \frac{(7 \times 5) + 1}{5} = \frac{36}{5}\)

The expression inside the second bracket becomes \(\left( {\frac{7}{{16}} \div \frac{21}{2} \times \frac{36}{5}} \right)\). Within the bracket, perform division and multiplication from left to right.

First, perform the division:

\(\frac{7}{{16}} \div \frac{21}{2} = \frac{7}{{16}} \times \frac{2}{{21}}\)

Simplify before multiplying:

\(= \frac{\cancel{7}^1}{16} \times \frac{2}{\cancel{21}^3} = \frac{1}{16} \times \frac{2}{3}\)

\(= \frac{1}{\cancel{16}^8} \times \frac{\cancel{2}^1}{3} = \frac{1 \times 1}{8 \times 3} = \frac{1}{24}\)

Now, perform the multiplication with the result:

\(\frac{1}{24} \times \frac{36}{5}\)

Simplify before multiplying:

\(= \frac{1}{\cancel{24}^2} \times \frac{\cancel{36}^3}{5} = \frac{1 \times 3}{2 \times 5} = \frac{3}{10}\)

So, the value of the second bracket is \(\frac{3}{10}\).

Step 3: Perform Division and Multiplication Outside Brackets

Substitute the simplified bracket values back into the original expression. The expression now looks like:

\(\frac{8}{3} \div \frac{4}{9} + \frac{3}{10} \times \frac{5}{{12}}\)

According to BODMAS, we perform division and multiplication next, from left to right.

First, the division:

\(\frac{8}{3} \div \frac{4}{9} = \frac{8}{3} \times \frac{9}{4}\)

Simplify before multiplying:

\(= \frac{\cancel{8}^2}{\cancel{3}^1} \times \frac{\cancel{9}^3}{\cancel{4}^1} = \frac{2 \times 3}{1 \times 1} = 6\)

Next, the multiplication:

\(\frac{3}{10} \times \frac{5}{{12}}\)

Simplify before multiplying:

\(= \frac{\cancel{3}^1}{\cancel{10}^2} \times \frac{\cancel{5}^1}{\cancel{12}^4} = \frac{1 \times 1}{2 \times 4} = \frac{1}{8}\)

The expression is now simplified to:

\(6 + \frac{1}{8}\)

Step 4: Perform Addition

Finally, perform the addition:

\(6 + \frac{1}{8} = \frac{6 \times 8}{8} + \frac{1}{8} = \frac{48}{8} + \frac{1}{8} = \frac{48 + 1}{8} = \frac{49}{8}\)

Final Result

The value of the expression is \(\frac{49}{8}\).

Comparing this with the given options, we find that the value matches option 1.

Revision Table: BODMAS Rules & Operations

Additional Information on Fraction Operations

Working with fractions requires understanding basic operations:

• Addition and Subtraction: Fractions must have a common denominator. Find the least common multiple (LCM) of the denominators, convert fractions to equivalent fractions with the LCM as the new denominator, and then add or subtract the numerators.
• Multiplication: Multiply the numerators together and multiply the denominators together. Simplify the resulting fraction if possible. \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
• Division: To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{c}{d}\) is \(\frac{d}{c}\). So, \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).
• Mixed Numbers: Convert mixed numbers (like \(10\frac{1}{2}\)) to improper fractions before performing calculations. An improper fraction has a numerator greater than or equal to its denominator. \(a\frac{b}{c} = \frac{(a \times c) + b}{c}\).
• 'Of' Operation: In mathematical expressions, 'of' signifies multiplication and is typically performed after operations within brackets but before other multiplication and division.

Applying these rules systematically, along with the BODMAS order, is crucial for correctly solving complex expressions involving fractions.