Understanding the Problem: Evaluating a Complex Fraction Expression

The question asks us to find the value of a mathematical expression involving fractions, mixed numbers, and multiple operations grouped by brackets, braces, and parentheses. To solve this accurately, we must follow the order of operations, commonly known as BODMAS or PEMDAS.

Applying the BODMAS Rule for Expression Evaluation

The BODMAS rule helps us determine the correct sequence for performing operations in an expression:

• Brackets (or Parentheses) - Solve the expressions inside the brackets first. Start with the innermost ones.
• Of (or Orders/Exponents) - Next, solve any powers, roots, or 'of' operations (which typically means multiplication).
• Division and Multiplication - Perform division and multiplication from left to right.
• Addition and Subtraction - Perform addition and subtraction from left to right.

Let's break down the given expression step by step:

The expression is: \( \frac{2}{7} - \frac{3}{8} - \left[ {2\frac{1}{4} \div 3\frac{1}{2}\,\,{\rm{of}}\,{\rm{1}}\frac{1}{3} + \left\{ {1\frac{{17}}{{40}}\, - \,\left( {3\, - \,1\frac{1}{5}\, - \,\frac{3}{8}} \right)} \right\}} \right] \)

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert all mixed numbers in the expression into improper fractions to make calculations easier.

• \( 2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4} \)
• \( 3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2} \)
• \( 1\frac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{4}{3} \)
• \( 1\frac{{17}}{{40}} = \frac{(1 \times 40) + 17}{40} = \frac{57}{40} \)
• \( 1\frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{6}{5} \)

Substitute these improper fractions back into the expression:

\( \frac{2}{7} - \frac{3}{8} - \left[ {\frac{9}{4} \div \frac{7}{2}\,\,{\rm{of}}\,\,\frac{4}{3} + \left\{ {\frac{57}{40}\, - \,\left( {3\, - \,\frac{6}{5}\, - \,\frac{3}{8}} \right)} \right\}} \right] \)

Step 2: Solve the Innermost Parentheses

According to BODMAS, we start with the innermost brackets, which are the parentheses \( \left( {3\, - \,\frac{6}{5}\, - \,\frac{3}{8}} \right) \).

Find a common denominator for 3, \(\frac{6}{5}\), and \(\frac{3}{8}\). The denominators are 1, 5, and 8. The Least Common Multiple (LCM) of 1, 5, and 8 is 40.

• \( 3 = \frac{3 \times 40}{1 \times 40} = \frac{120}{40} \)
• \( \frac{6}{5} = \frac{6 \times 8}{5 \times 8} = \frac{48}{40} \)
• \( \frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40} \)

Now perform the subtraction within the parentheses:

\( \frac{120}{40} - \frac{48}{40} - \frac{15}{40} = \frac{120 - 48 - 15}{40} = \frac{72 - 15}{40} = \frac{57}{40} \)

Substitute this value back into the expression:

\( \frac{2}{7} - \frac{3}{8} - \left[ {\frac{9}{4} \div \frac{7}{2}\,\,{\rm{of}}\,\,\frac{4}{3} + \left\{ {\frac{57}{40}\, - \,\frac{57}{40}} \right\}} \right] \)

Step 3: Solve the Innermost Braces

Next, evaluate the expression inside the braces: \( \left\{ {\frac{57}{40}\, - \,\frac{57}{40}} \right\} \).

\( \frac{57}{40} - \frac{57}{40} = 0 \)

Substitute this value back into the expression:

\( \frac{2}{7} - \frac{3}{8} - \left[ {\frac{9}{4} \div \frac{7}{2}\,\,{\rm{of}}\,\,\frac{4}{3} + 0} \right] \)

Step 4: Perform 'Of' Operation within the Square Brackets

Inside the square brackets, we have division and 'of'. According to BODMAS, 'of' is done before division. The 'of' operation is \( \frac{7}{2}\,\,{\rm{of}}\,\,\frac{4}{3} \).

'Of' means multiplication:

\( \frac{7}{2} \times \frac{4}{3} = \frac{7 \times 4}{2 \times 3} = \frac{28}{6} \)

Simplify the fraction:

\( \frac{28}{6} = \frac{14}{3} \)

Substitute this back into the expression:

\( \frac{2}{7} - \frac{3}{8} - \left[ {\frac{9}{4} \div \frac{14}{3} + 0} \right] \)

Step 5: Perform Division within the Square Brackets

Next, perform the division inside the square brackets: \( \frac{9}{4} \div \frac{14}{3} \).

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \( \frac{14}{3} \) is \( \frac{3}{14} \).

\( \frac{9}{4} \times \frac{3}{14} = \frac{9 \times 3}{4 \times 14} = \frac{27}{56} \)

Substitute this back into the expression:

\( \frac{2}{7} - \frac{3}{8} - \left[ {\frac{27}{56} + 0} \right] \)

Step 6: Perform Addition within the Square Brackets

Now, perform the addition inside the square brackets: \( \left[ {\frac{27}{56} + 0} \right] \).

\( \frac{27}{56} + 0 = \frac{27}{56} \)

The expression is now simplified to:

\( \frac{2}{7} - \frac{3}{8} - \frac{27}{56} \)

Step 7: Perform the Final Subtraction

Finally, perform the subtraction from left to right.

Find a common denominator for 7, 8, and 56. The LCM of 7, 8, and 56 is 56.

• \( \frac{2}{7} = \frac{2 \times 8}{7 \times 8} = \frac{16}{56} \)
• \( \frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56} \)
• \( \frac{27}{56} \) remains as it is.

Now subtract the fractions:

\( \frac{16}{56} - \frac{21}{56} - \frac{27}{56} = \frac{16 - 21 - 27}{56} \)

Perform the subtractions in the numerator:

\( 16 - 21 = -5 \)

\( -5 - 27 = -32 \)

So the result is \( \frac{-32}{56} \).

Step 8: Simplify the Result

The fraction \( \frac{-32}{56} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 8.

\( \frac{-32 \div 8}{56 \div 8} = \frac{-4}{7} \)

The value of the given expression is \( - \frac{4}{7} \).

Summary of Steps

Final Answer

The calculated value of the expression is \( - \frac{4}{7} \).

Revision Table: Key Math Concepts

Additional Information: Importance of Order of Operations

The order of operations (BODMAS/PEMDAS) is fundamental in mathematics. Without a standard order, expressions could have multiple different values depending on which operation is performed first. Following BODMAS ensures consistency and accuracy in calculations, especially in complex expressions involving various operations and grouping symbols like parentheses, braces, and brackets. This rule is crucial for success in algebra and all higher levels of mathematics.

Understanding how to handle fractions and mixed numbers is also a key skill. Converting mixed numbers to improper fractions often simplifies the calculation process significantly when performing multiplication or division. When adding or subtracting fractions, finding a common denominator is essential before combining the numerators. Always simplify the final result to its lowest terms.