Simplifying Complex Mathematical Expressions with BODMAS

To simplify the given mathematical expression, we need to follow the order of operations, commonly known as BODMAS or PEMDAS.

Understanding the BODMAS Rule

BODMAS stands for:

• B: Brackets (solve operations inside brackets first, starting from the innermost)
• O: Orders (powers, square roots, etc. - not present in this expression) / O: Of (multiplication related to 'of')
• D: Division
• M: Multiplication
• A: Addition
• S: Subtraction

Operations for Division and Multiplication are performed from left to right. Similarly, Addition and Subtraction are performed from left to right.

In the given expression, the bar over \(\overline{36\div9}\) acts like a grouping symbol (like parentheses), indicating that this division should be performed first.

Step-by-Step Simplification of the Expression

The expression is: \(\rm \frac{3}{5}\ of\ \left\{\left(48\div\overline{36\div9}\times\frac{1}{6}\right)+\frac{1}{3}\right\}\)

Step 1: Simplify the innermost grouped operation (under the bar).

\(\overline{36\div9} = 4\)

Now substitute this back into the expression:

\(\rm \frac{3}{5}\ of\ \left\{\left(48\div4\times\frac{1}{6}\right)+\frac{1}{3}\right\}\)

Step 2: Simplify the operations inside the parentheses \((\ )\) from left to right (Division then Multiplication).

First, perform the division: \(48\div4\)

\(48\div4 = 12\)

Now, perform the multiplication: \(12\times\frac{1}{6}\)

\(12\times\frac{1}{6} = \frac{12}{6} = 2\)

Substitute this result back into the expression:

\(\rm \frac{3}{5}\ of\ \left\{2+\frac{1}{3}\right\}\)

Step 3: Simplify the operation inside the braces \(\{\ \}\) (Addition).

\(2+\frac{1}{3}\)

To add an integer and a fraction, we find a common denominator. \(2 = \frac{6}{3}\).

So, \(2+\frac{1}{3} = \frac{6}{3}+\frac{1}{3} = \frac{6+1}{3} = \frac{7}{3}\)

Substitute this result back into the expression:

\(\rm \frac{3}{5}\ of\ \frac{7}{3}\)

Step 4: Perform the 'of' operation.

The term 'of' means multiplication. So, we multiply the two fractions:

\(\frac{3}{5}\times\frac{7}{3}\)

Multiply the numerators and the denominators:

\(\frac{3\times7}{5\times3} = \frac{21}{15}\)

Step 5: Simplify the resulting fraction.

The fraction \(\frac{21}{15}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

\(\frac{21\div3}{15\div3} = \frac{7}{5}\)

Therefore, the simplified value of the expression is \(\frac{7}{5}\).

Revision Table: Key Mathematical Operations

Additional Information: Fractions and Order of Operations

Simplifying expressions involving fractions requires careful application of the order of operations. When dealing with fractions, remember:

• Multiplying fractions: Multiply numerators together and denominators together. \(\frac{a}{b}\times\frac{c}{d} = \frac{a\times c}{b\times d}\).
• Dividing by a fraction: Multiply by the reciprocal of the second fraction. \(a\div\frac{b}{c} = a\times\frac{c}{b}\).
• Adding/Subtracting fractions: Find a common denominator before adding/subtracting the numerators. \(\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}\).
• The bar notation \(\overline{a\div b}\) means perform the operation \(a\div b\) first, just like if it were in parentheses \((a\div b)\).

Practicing different types of expressions helps solidify the understanding of the BODMAS rule and fraction arithmetic.