Simplifying Complex Mathematical Expressions

The problem asks us to simplify the mathematical expression: \(441 \div \left[270 \div \frac{3}{7}+\left(17\div \frac{1}{3}\right)-\left(8\frac{1}{2}-\frac{5}{2}\right)\right]\). To solve this, we must follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.

This means we first evaluate expressions inside brackets `[ ]`, then parentheses `( )`, followed by division and multiplication (from left to right), and finally addition and subtraction (from left to right).

Step 1: Evaluate the expressions within the inner parentheses

There are three parts inside the main bracket `[ ]` that involve inner parentheses or fractions requiring immediate attention:

1. The first part is \(270 \div \frac{3}{7}\). Dividing by a fraction is the same as multiplying by its reciprocal.

   \(270 \div \frac{3}{7} = 270 \times \frac{7}{3}\)

   We can simplify this: \(270/3 = 90\). So, \(90 \times 7 = 630\).

2. The second part is \(\left(17\div \frac{1}{3}\right)\). Again, divide by a fraction by multiplying by its reciprocal.

   \(17 \div \frac{1}{3} = 17 \times 3 = 51\).

3. The third part is \(\left(8\frac{1}{2}-\frac{5}{2}\right)\). First, convert the mixed number \(8\frac{1}{2}\) to an improper fraction.

   \(8\frac{1}{2} = \frac{(8 \times 2) + 1}{2} = \frac{16 + 1}{2} = \frac{17}{2}\).

   Now perform the subtraction: \(\frac{17}{2} - \frac{5}{2} = \frac{17 - 5}{2} = \frac{12}{2} = 6\).

Step 2: Substitute the results back into the main bracket

Now, replace the calculated values back into the original expression inside the square brackets:

The expression inside the bracket was \(\left[270 \div \frac{3}{7}+\left(17\div \frac{1}{3}\right)-\left(8\frac{1}{2}-\frac{5}{2}\right)\right]\).

Substituting the results from Step 1:

\(\left[630 + 51 - 6\right]\)

Step 3: Evaluate the expression inside the main bracket

Perform the addition and subtraction within the bracket from left to right:

\(630 + 51 = 681\)

\(681 - 6 = 675\)

So, the value inside the square bracket is \(675\).

Step 4: Perform the final division

The original expression is \(441 \div \left[\text{result from bracket}\right]\).

\(441 \div 675\)

We need to express this division as a fraction and simplify it if possible.

\(\frac{441}{675}\)

Let's find common factors for the numerator (441) and the denominator (675). We can check divisibility by small prime numbers.

• Check for divisibility by 3:

  Sum of digits for 441: \(4+4+1 = 9\). 9 is divisible by 3, so 441 is divisible by 3. \(441 \div 3 = 147\).

  Sum of digits for 675: \(6+7+5 = 18\). 18 is divisible by 3, so 675 is divisible by 3. \(675 \div 3 = 225\).

  The fraction becomes \(\frac{147}{225}\).

• Check for divisibility by 3 again:

  Sum of digits for 147: \(1+4+7 = 12\). 12 is divisible by 3, so 147 is divisible by 3. \(147 \div 3 = 49\).

  Sum of digits for 225: \(2+2+5 = 9\). 9 is divisible by 3, so 225 is divisible by 3. \(225 \div 3 = 75\).

  The fraction becomes \(\frac{49}{75}\).

Now check if 49 and 75 have any common factors. The factors of 49 are 1, 7, 49. The factors of 75 are 1, 3, 5, 15, 25, 75. They have no common factors other than 1.

So, the simplified fraction is \(\frac{49}{75}\).

Putting it all together:

Final Answer

The simplified expression is \(\frac{49}{75}\).

Revision Table: Simplifying Expressions

Additional Information: Order of Operations in Math

Understanding the correct order of operations is crucial for simplifying complex mathematical expressions accurately. The acronyms BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) help remember the sequence.

• B/P: Brackets or Parentheses - Evaluate expressions inside grouping symbols first. This includes parentheses \(( )\), square brackets \([ ]\), and curly braces \(\{ \}\).
• O/E: Orders or Exponents - Evaluate powers, roots, and other orders.
• DM: Division and Multiplication - Perform these operations from left to right as they appear in the expression. They have equal precedence.
• AS: Addition and Subtraction - Perform these operations from left to right as they appear in the expression. They have equal precedence.

In the problem solved, we meticulously followed these rules, starting with the innermost parentheses and working outwards, performing divisions, subtractions, and additions in the correct order to arrive at the simplified fraction.