Simplifying Complex Fraction Expressions

To simplify the given mathematical expression, we need to follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS. The expression is:

\(\left(\frac{7}{16} \div \frac{1}{2}\:of\: \frac{1}{5}\right)\times \frac{4}{5}-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}+\frac{3}{4}\)

Let's break down the expression and simplify it step-by-step.

Step 1: Handle the 'of' operation within the brackets

The term \(\frac{1}{2}\:of\: \frac{1}{5}\) means \(\frac{1}{2} \times \frac{1}{5}\).

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

Now the expression inside the brackets becomes \(\frac{7}{16} \div \frac{1}{10}\).

Step 2: Perform the division inside the brackets

Dividing by a fraction is the same as multiplying by its reciprocal.

\(\frac{7}{16} \div \frac{1}{10} = \frac{7}{16} \times \frac{10}{1} = \frac{7 \times 10}{16 \times 1} = \frac{70}{16}\)

This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 2.

\(\frac{70}{16} = \frac{70 \div 2}{16 \div 2} = \frac{35}{8}\)

The expression now looks like:

\(\left(\frac{35}{8}\right)\times \frac{4}{5}-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}+\frac{3}{4}\)

Step 3: Perform the multiplication outside the first bracket

Next, calculate \(\frac{35}{8}\times \frac{4}{5}\).

\(\frac{35}{8}\times \frac{4}{5} = \frac{35 \times 4}{8 \times 5} = \frac{140}{40}\)

Simplify this fraction by dividing numerator and denominator by 10, then by 4 (or directly by 40, or stepwise by common factors).

\(\frac{140}{40} = \frac{140 \div 10}{40 \div 10} = \frac{14}{4}\)

\(\frac{14}{4} = \frac{14 \div 2}{4 \div 2} = \frac{7}{2}\)

The expression is now:

\(\frac{7}{2}-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}+\frac{3}{4}\)

Step 4: Perform multiplication and division from left to right in the middle part

The middle part is \(-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}\). First, multiply:

\(-\frac{1}{3}\times\frac{5}{8} = -\frac{1 \times 5}{3 \times 8} = -\frac{5}{24}\)

Now, divide the result by \(\frac{1}{2}\). Remember to multiply by the reciprocal.

\(-\frac{5}{24} \div \frac{1}{2} = -\frac{5}{24} \times \frac{2}{1} = -\frac{5 \times 2}{24 \times 1} = -\frac{10}{24}\)

Simplify this fraction by dividing numerator and denominator by 2.

\(-\frac{10}{24} = -\frac{10 \div 2}{24 \div 2} = -\frac{5}{12}\)

The expression is now simplified to:

\(\frac{7}{2} - \frac{5}{12} + \frac{3}{4}\)

Step 5: Perform addition and subtraction from left to right

To add and subtract these fractions, we need a common denominator. The denominators are 2, 12, and 4. The least common multiple (LCM) of 2, 12, and 4 is 12.

Convert each fraction to an equivalent fraction with a denominator of 12:

• \(\frac{7}{2} = \frac{7 \times 6}{2 \times 6} = \frac{42}{12}\)
• \(\frac{5}{12}\) remains \(\frac{5}{12}\)
• \(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)

Now substitute these into the expression:

\(\frac{42}{12} - \frac{5}{12} + \frac{9}{12}\)

Combine the numerators over the common denominator:

\(\frac{42 - 5 + 9}{12}\)

Calculate the numerator:

\(42 - 5 = 37\)

\(37 + 9 = 46\)

So the expression simplifies to:

\(\frac{46}{12}\)

Step 6: Simplify the final fraction

The fraction \(\frac{46}{12}\) can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 2.

\(\frac{46}{12} = \frac{46 \div 2}{12 \div 2} = \frac{23}{6}\)

The simplified expression is \(\frac{23}{6}\).

Revision Table: Key Concepts in Fraction Simplification

Additional Information: Understanding BODMAS/PEMDAS

The order of operations ensures that mathematical expressions are evaluated consistently, leading to a single correct answer. Different regions use slightly different acronyms, but the order of operations is the same:

• B/P: Brackets / Parentheses - Evaluate expressions inside brackets or parentheses first.
• O/E: Orders / Exponents - Evaluate powers and roots. This step is not applicable in the given problem.
• D/M: Division and Multiplication - Perform these operations from left to right as they appear in the expression.
• A/S: Addition and Subtraction - Perform these operations from left to right as they appear in the expression.

In the given problem, the 'of' operation (\(\frac{1}{2}\:of\: \frac{1}{5}\)) acts like multiplication but is typically performed before other multiplication and division operations outside of brackets, similar to how you would evaluate an exponent or an operation within a grouping symbol. Within the standard order of operations, 'of' falls under the 'Orders' or 'Exponents' category conceptually in some interpretations when it's used in this context, meaning it takes precedence over standard multiplication/division that is not part of a grouping. However, within the context of brackets as seen here \(\left(\frac{7}{16} \div \frac{1}{2}\:of\: \frac{1}{5}\right)\), you first evaluate the 'of' within the bracket, then the division within the bracket, before moving outside the bracket.

Applying this to our problem ensured we correctly handled the division and 'of' within the brackets first, then the multiplication outside, and finally the remaining multiplication/division before the final addition and subtraction.