Simplifying Complex Fraction Expressions

To simplify the given expression, we need to follow the order of operations, often remembered by the acronym BODMAS or PEMDAS. This order is:

• B/P: Brackets or Parentheses
• O/E: Order or Exponents
• D/M: Division and Multiplication (from left to right)
• A/S: Addition and Subtraction (from left to right)

The expression is:

\[\rm\left(\frac{3}{4}-\frac{1}{4}\div\frac{1}{4}\ of\ \frac{2}{5}\right)\div\left(\frac{3}{4}\div\frac{2}{3}\ of\ \frac{3}{5}\right)\]

We will simplify the two parts within the main parentheses first, one by one.

Step 1: Simplify the first parenthesis

The first part is \(\rm\left(\frac{3}{4}-\frac{1}{4}\div\frac{1}{4}\ of\ \frac{2}{5}\right)\).

Inside this parenthesis, we have subtraction, division, and 'of'. According to BODMAS, 'of' (which means multiplication) is done before division.

First, calculate \(\frac{1}{4}\ of\ \frac{2}{5}\):

\[ \frac{1}{4} \times \frac{2}{5} = \frac{1 \times 2}{4 \times 5} = \frac{2}{20} = \frac{1}{10} \]

Now, the expression inside the first parenthesis becomes:

\[ \left(\frac{3}{4} - \frac{1}{4} \div \frac{1}{10}\right) \]

Next, perform the division: \(\frac{1}{4} \div \frac{1}{10}\). Dividing by a fraction is the same as multiplying by its reciprocal.

\[ \frac{1}{4} \div \frac{1}{10} = \frac{1}{4} \times \frac{10}{1} = \frac{1 \times 10}{4 \times 1} = \frac{10}{4} = \frac{5}{2} \]

The expression inside the first parenthesis is now:

\[ \left(\frac{3}{4} - \frac{5}{2}\right) \]

Finally, perform the subtraction. To subtract fractions, they must have a common denominator. The least common multiple of 4 and 2 is 4.

\[ \frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4} \]

So, the subtraction is:

\[ \frac{3}{4} - \frac{10}{4} = \frac{3 - 10}{4} = \frac{-7}{4} \]

The first parenthesis simplifies to \(\frac{-7}{4}\) or \(-\frac{7}{4}\).

Step 2: Simplify the second parenthesis

The second part is \(\rm\left(\frac{3}{4}\div\frac{2}{3}\ of\ \frac{3}{5}\right)\).

Inside this parenthesis, we have division and 'of'. According to BODMAS, 'of' is done before division.

First, calculate \(\frac{2}{3}\ of\ \frac{3}{5}\):

\[ \frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5} = \frac{6}{15} = \frac{2}{5} \]

Now, the expression inside the second parenthesis becomes:

\[ \left(\frac{3}{4} \div \frac{2}{5}\right) \]

Next, perform the division: \(\frac{3}{4} \div \frac{2}{5}\). Multiply by the reciprocal of \(\frac{2}{5}\).

\[ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} \]

The second parenthesis simplifies to \(\frac{15}{8}\).

Step 3: Perform the final division

Now the original expression simplifies to the division of the results from Step 1 and Step 2:

\[ \left(-\frac{7}{4}\right) \div \left(\frac{15}{8}\right) \]

Divide the first fraction by the second fraction by multiplying the first fraction by the reciprocal of the second fraction:

\[ -\frac{7}{4} \div \frac{15}{8} = -\frac{7}{4} \times \frac{8}{15} \]

Multiply the numerators and the denominators:

\[ -\frac{7 \times 8}{4 \times 15} = -\frac{56}{60} \]

Finally, simplify the resulting fraction by dividing the numerator and the denominator by their greatest common divisor. Both 56 and 60 are divisible by 4.

\[ 56 \div 4 = 14 \] \[ 60 \div 4 = 15 \]

So, the simplified fraction is:

\[ -\frac{14}{15} \]

Summary of Simplification Steps

• Simplify inside the first parenthesis: \(\rm\left(\frac{3}{4}-\frac{1}{4}\div\frac{1}{4}\ of\ \frac{2}{5}\right)\) becomes \(\frac{-7}{4}\).
• Simplify inside the second parenthesis: \(\rm\left(\frac{3}{4}\div\frac{2}{3}\ of\ \frac{3}{5}\right)\) becomes \(\frac{15}{8}\).
• Divide the result of the first parenthesis by the result of the second parenthesis: \(\frac{-7}{4} \div \frac{15}{8}\).
• Perform the division: \(\frac{-7}{4} \times \frac{8}{15} = \frac{-56}{60}\).
• Simplify the final fraction: \(\frac{-56}{60} = -\frac{14}{15}\).

The final simplified expression is \(-\frac{14}{15}\).

Revision Table: Key Fraction Operations

Additional Information: Understanding BODMAS/PEMDAS

The BODMAS (Brackets, Order, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) rule is crucial for solving mathematical expressions correctly.

• Operations within brackets or parentheses are always performed first.
• 'Of' represents multiplication, but it's often treated as part of the 'Order' or evaluated immediately after brackets, before other multiplication or division. In cases like 'a of b', it means \(a \times b\).
• Division and Multiplication have equal priority and are performed from left to right.
• Addition and Subtraction have equal priority and are performed from left to right.

Following this specific order ensures a unique and correct result for any given mathematical expression.