Simplifying Complex Mathematical Expressions

This problem requires us to simplify a complex mathematical expression involving fractions, multiplication, division, and grouping symbols (brackets and braces). We must follow the order of operations, commonly known as BODMAS or PEMDAS, to solve this correctly.

• BODMAS/PEMDAS:
• B/P: Brackets/Parentheses first.
• O/E: Orders/Exponents next.
• D/M: Division and Multiplication (from left to right).
• A/S: Addition and Subtraction (from left to right).

The given expression is:

\([\frac{85}{34}\times \frac{1}{18}- \{(\frac{46}{69}\div\frac{27}{135})-(\frac{86}{129}\div\frac{14}{91})\}\ of \frac{112}{36}]\

Let's break down the expression step-by-step.

Step 1: Simplify the expressions within the innermost parentheses/brackets

We have two division operations inside the curly braces:

First part: \((\frac{46}{69}\div\frac{27}{135})\)

Division of fractions is multiplication by the reciprocal of the second fraction. Also, simplify fractions where possible before multiplying.

\(\frac{46}{69} = \frac{2 \times 23}{3 \times 23} = \frac{2}{3}\)

\(\frac{27}{135} = \frac{27}{5 \times 27} = \frac{1}{5}\)

So, \((\frac{46}{69}\div\frac{27}{135}) = (\frac{2}{3}\div\frac{1}{5}) = \frac{2}{3} \times \frac{5}{1} = \frac{10}{3}\)

Second part: \((\frac{86}{129}\div\frac{14}{91})\)

Simplify fractions:

\(\frac{86}{129} = \frac{2 \times 43}{3 \times 43} = \frac{2}{3}\)

\(\frac{14}{91} = \frac{2 \times 7}{13 \times 7} = \frac{2}{13}\)

So, \((\frac{86}{129}\div\frac{14}{91}) = (\frac{2}{3}\div\frac{2}{13}) = \frac{2}{3} \times \frac{13}{2} = \frac{13}{3}\)

Step 2: Simplify the expression within the curly braces { }

Now we subtract the second part from the first part calculated in Step 1:

\(\{(\frac{46}{69}\div\frac{27}{135})-(\frac{86}{129}\div\frac{14}{91})\} = \{\frac{10}{3} - \frac{13}{3}\}\)

\(= \frac{10 - 13}{3} = \frac{-3}{3} = -1\)

Step 3: Evaluate the 'of' operation

The term 'of' indicates multiplication. So, we multiply the result from Step 2 by \(\frac{112}{36}\).

\(\{-1\}\ of\ \frac{112}{36} = -1 \times \frac{112}{36}\)

Simplify the fraction \(\frac{112}{36}\) by dividing both numerator and denominator by their greatest common divisor, which is 4 (or first by 4, then by 9 as shown below):

\(\frac{112}{36} = \frac{112 \div 4}{36 \div 4} = \frac{28}{9}\)

So, \(-1 \times \frac{28}{9} = -\frac{28}{9}\)

Step 4: Evaluate the multiplication outside the braces

Now consider the first term in the main expression: \(\frac{85}{34}\times \frac{1}{18}\)

Simplify the fraction \(\frac{85}{34}\):

\(\frac{85}{34} = \frac{5 \times 17}{2 \times 17} = \frac{5}{2}\)

So, \(\frac{85}{34}\times \frac{1}{18} = \frac{5}{2} \times \frac{1}{18} = \frac{5 \times 1}{2 \times 18} = \frac{5}{36}\)

Step 5: Perform the final subtraction

The original expression simplifies to the result of Step 4 minus the result of Step 3:

\(\frac{5}{36} - (-\frac{28}{9})\)

\(= \frac{5}{36} + \frac{28}{9}\)

To add these fractions, we need a common denominator. The least common multiple of 36 and 9 is 36.

Convert \(\frac{28}{9}\) to an equivalent fraction with denominator 36:

\(\frac{28}{9} = \frac{28 \times 4}{9 \times 4} = \frac{112}{36}\)

Now, add the fractions:

\(\frac{5}{36} + \frac{112}{36} = \frac{5 + 112}{36} = \frac{117}{36}\)

Step 6: Simplify the final fraction and convert to a mixed number

The fraction \(\frac{117}{36}\) can be simplified by dividing the numerator and denominator by their greatest common divisor. Both 117 and 36 are divisible by 9.

\(\frac{117 \div 9}{36 \div 9} = \frac{13}{4}\)

Now, convert the improper fraction \(\frac{13}{4}\) to a mixed number. Divide 13 by 4.

\(13 = 4 \times 3 + 1\)

So, \(\frac{13}{4} = 3 \frac{1}{4}\)

The simplified value of the expression is \(3 \frac{1}{4}\).

Summary of Steps and Results

Revision Table: Key Concepts for Expression Simplification

Additional Information on Fraction Operations

Understanding fraction operations is crucial for simplifying expressions like this one. Always simplify fractions early if possible, as it makes calculations easier with smaller numbers. Remember that 'of' usually implies multiplication, especially in the context of fractions or percentages. Be careful with signs, particularly when subtracting negative numbers, as \(- (-\text{number}) = +\text{number}\).

For example, in Step 5, we had \(\frac{5}{36} - (-\frac{28}{9})\). This correctly became \(\frac{5}{36} + \frac{28}{9}\).

Converting improper fractions (where the numerator is greater than or equal to the denominator) to mixed numbers is often required for final answers, especially if the options are given in mixed number format. To convert \(\frac{13}{4}\), we divide 13 by 4. The quotient (3) is the whole number part, the remainder (1) is the new numerator, and the denominator (4) stays the same, giving \(3 \frac{1}{4}\).