The question asks us to find the value of a mathematical expression involving mixed fractions, simple fractions, and various types of grouping symbols like brackets \(\left[\right]\) and braces \(\lbrace\rbrace\). To solve this, we need to follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.

Solving the Expression with BODMAS/PEMDAS

BODMAS stands for Brackets, Order (powers/roots), Division, Multiplication, Addition, Subtraction. PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Both convey the same order: solve operations inside grouping symbols first, then powers/roots, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

The given expression is: \(3\frac{5}{6}+\left[3\frac{2}{3}+\lbrace{\frac{15}{4}\left(5\frac{4}{5}\div 14\frac{1}{2}\right)\rbrace}\right]\)

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

Step 1: Convert Mixed Fractions to Improper Fractions

First, convert all mixed fractions in the expression into improper fractions. This makes calculations easier.

• \(3\frac{5}{6} = \frac{(3 \times 6) + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}\)
• \(3\frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}\)
• \(5\frac{4}{5} = \frac{(5 \times 5) + 4}{5} = \frac{25 + 4}{5} = \frac{29}{5}\)
• \(14\frac{1}{2} = \frac{(14 \times 2) + 1}{2} = \frac{28 + 1}{2} = \frac{29}{2}\)

The expression now becomes: \(\frac{23}{6}+\left[\frac{11}{3}+\lbrace{\frac{15}{4}\left(\frac{29}{5}\div \frac{29}{2}\right)\rbrace}\right]\)

Step 2: Solve the Innermost Operation (Division within Parentheses)

The innermost operation is the division inside the parentheses:

\(\frac{29}{5}\div \frac{29}{2}\)

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

\(\frac{29}{5}\times \frac{2}{29} = \frac{\cancel{29}}{5}\times \frac{2}{\cancel{29}} = \frac{2}{5}\)

The expression is now: \(\frac{23}{6}+\left[\frac{11}{3}+\lbrace{\frac{15}{4}\left(\frac{2}{5}\right)\rbrace}\right]\)

Step 3: Solve the Operation within Braces (Multiplication)

Next, solve the multiplication inside the braces:

\(\frac{15}{4}\times \frac{2}{5}\)

Multiply the numerators and the denominators, cancelling common factors if possible:

\(\frac{\cancel{15}^3}{4_{\cancelto{2}{4}}}\times \frac{\cancel{2}^1}{\cancel{5}^1} = \frac{3}{2}\times \frac{1}{1} = \frac{3}{2}\)

The expression becomes: \(\frac{23}{6}+\left[\frac{11}{3}+\frac{3}{2}\right]\)

Step 4: Solve the Operation within Square Brackets (Addition)

Now, solve the addition inside the square brackets:

\(\frac{11}{3}+\frac{3}{2}\)

To add fractions, find a common denominator. The least common multiple (LCM) of 3 and 2 is 6.

\(\frac{11}{3} = \frac{11 \times 2}{3 \times 2} = \frac{22}{6}\)

\(\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}\)

Now add the fractions:

\(\frac{22}{6}+\frac{9}{6} = \frac{22 + 9}{6} = \frac{31}{6}\)

The expression is now: \(\frac{23}{6}+\frac{31}{6}\)

Step 5: Solve the Final Operation (Addition)

Finally, perform the last addition:

\(\frac{23}{6}+\frac{31}{6}\)

Since the fractions already have a common denominator (6), simply add the numerators:

\(\frac{23 + 31}{6} = \frac{54}{6}\)

Simplify the fraction:

\(\frac{54}{6} = 9\)

The value of the expression is 9.

Summary of Calculation Steps

Here's a quick recap of the steps taken to evaluate the expression:

1. Convert all mixed numbers to improper fractions.
2. Solve the division inside the parentheses: \(\frac{29}{5} \div \frac{29}{2} = \frac{2}{5}\).
3. Solve the multiplication inside the braces: \(\frac{15}{4} \times \frac{2}{5} = \frac{3}{2}\).
4. Solve the addition inside the square brackets: \(\frac{11}{3} + \frac{3}{2} = \frac{22}{6} + \frac{9}{6} = \frac{31}{6}\).
5. Solve the final addition: \(\frac{23}{6} + \frac{31}{6} = \frac{54}{6} = 9\).

Final Answer Value

The calculation shows that the value of the given expression \(3\frac{5}{6}+\left[3\frac{2}{3}+\lbrace{\frac{15}{4}\left(5\frac{4}{5}\div 14\frac{1}{2}\right)\rbrace}\right]\) is 9.

Revision Table: Key Concepts for Expression Evaluation

Additional Information: Handling Fractions and Brackets

When dealing with expressions like this, precision at each step is key. Converting to improper fractions initially simplifies the process significantly. Remember that brackets, braces, and parentheses all serve the same purpose – to group operations that must be performed before those outside the grouping symbol.

Working from the inside out of the grouping symbols ensures that the order of operations is correctly applied. If you encounter complex fractions within the expression, simplify them first before combining them with other terms.