The value of \(\left( {2\frac{6}{7}of4\frac{1}{5} \div \frac{2}{3}} \right) \times 5\frac{1}{9} \div \left( {\frac{3}{4} \times 2\frac{2}{3}of\frac{1}{2} \div \frac{1}{4}} \right)\) is:
23
Simplifying Complex Fraction Expressions using BODMAS
The problem asks us to find the value of a given mathematical expression involving fractions, mixed numbers, and different operations. To correctly solve this, we need to follow the order of operations, often remembered by the acronym BODMAS or PEMDAS.
BODMAS stands for:
- Brackets first
- Orders (powers, square roots, etc.) or Of
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
In this expression, we have brackets, 'of', division, and multiplication. The 'of' operator acts like multiplication but is performed before standard multiplication and division.
The expression is: \(\left( {2\frac{6}{7}of4\frac{1}{5} \div \frac{2}{3}} \right) \times 5\frac{1}{9} \div \left( {\frac{3}{4} \times 2\frac{2}{3}of\frac{1}{2} \div \frac{1}{4}} \right)\)
Let's break this down into parts and simplify them step-by-step.
Step 1: Simplify the First Bracket
The first bracket is \(\left( {2\frac{6}{7}of4\frac{1}{5} \div \frac{2}{3}} \right)\).
First, convert the mixed numbers to improper fractions:
- \(2\frac{6}{7} = \frac{(2 \times 7) + 6}{7} = \frac{14 + 6}{7} = \frac{20}{7}\)
- \(4\frac{1}{5} = \frac{(4 \times 5) + 1}{5} = \frac{20 + 1}{5} = \frac{21}{5}\)
Now the bracket is \(\left( {\frac{20}{7}of\frac{21}{5} \div \frac{2}{3}} \right)\).
According to BODMAS, perform the 'of' operation first:
\(\frac{20}{7} \text{ of } \frac{21}{5} = \frac{20}{7} \times \frac{21}{5}\)
We can cancel common factors before multiplying:
\(\frac{4 \times 5}{7} \times \frac{3 \times 7}{5} = \frac{4 \times \cancel{5}}{\cancel{7}} \times \frac{3 \times \cancel{7}}{\cancel{5}} = 4 \times 3 = 12\)
Now the bracket is \(\left( {12 \div \frac{2}{3}} \right)\).
Perform the division. Dividing by a fraction is the same as multiplying by its reciprocal:
\(12 \div \frac{2}{3} = 12 \times \frac{3}{2}\)
Cancel common factors:
\(\frac{12}{1} \times \frac{3}{2} = \frac{6 \times \cancel{2}}{1} \times \frac{3}{\cancel{2}} = 6 \times 3 = 18\)
So, the value of the first bracket is 18.
Step 2: Simplify the Second Bracket
The second bracket is \(\left( {\frac{3}{4} \times 2\frac{2}{3}of\frac{1}{2} \div \frac{1}{4}} \right)\).
Convert the mixed number to an improper fraction:
- \(2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}\)
Now the bracket is \(\left( {\frac{3}{4} \times \frac{8}{3}of\frac{1}{2} \div \frac{1}{4}} \right)\).
Inside the bracket, perform the 'of' operation first:
\(\frac{8}{3} \text{ of } \frac{1}{2} = \frac{8}{3} \times \frac{1}{2}\)
Cancel common factors:
\(\frac{4 \times \cancel{2}}{3} \times \frac{1}{\cancel{2}} = \frac{4}{3}\)
Now the bracket is \(\left( {\frac{3}{4} \times \frac{4}{3} \div \frac{1}{4}} \right)\).
Inside the bracket, perform multiplication and division from left to right.
First, multiplication:
\(\frac{3}{4} \times \frac{4}{3}\)
Cancel common factors:
\(\frac{\cancel{3}}{\cancel{4}} \times \frac{\cancel{4}}{\cancel{3}} = 1\)
Now the bracket is \(\left( {1 \div \frac{1}{4}} \right)\).
Perform the division:
\(1 \div \frac{1}{4} = 1 \times \frac{4}{1} = 4\)
So, the value of the second bracket is 4.
Step 3: Simplify the Remaining Expression
The original expression simplifies to the value of the first bracket times the value of the second part, where the second part is \(5\frac{1}{9} \div (\text{value of second bracket})\).
The expression is now: \(18 \times 5\frac{1}{9} \div 4\).
First, convert the mixed number \(5\frac{1}{9}\) to an improper fraction:
\(5\frac{1}{9} = \frac{(5 \times 9) + 1}{9} = \frac{45 + 1}{9} = \frac{46}{9}\)
The expression becomes: \(18 \times \frac{46}{9} \div 4\).
According to BODMAS, perform multiplication and division from left to right.
First, multiplication:
\(18 \times \frac{46}{9}\)
Cancel common factors:
\(\frac{\cancel{18}^2}{1} \times \frac{46}{\cancel{9}^1} = 2 \times 46 = 92\)
The expression becomes: \(92 \div 4\).
Perform the final division:
\(92 \div 4 = \frac{92}{4} = 23\)
The final value of the expression is 23.
| Part of Expression | Calculation | Result |
|---|---|---|
| First Bracket (\(2\frac{6}{7}of4\frac{1}{5} \div \frac{2}{3}\)) | \(\left( {\frac{20}{7} \times \frac{21}{5}} \right) \div \frac{2}{3} = 12 \div \frac{2}{3} = 12 \times \frac{3}{2} = 18\) | 18 |
| 'of' in Second Bracket (\(2\frac{2}{3}of\frac{1}{2}\)) | \(\frac{8}{3} \times \frac{1}{2} = \frac{4}{3}\) | \(\frac{4}{3}\) |
| Second Bracket (\(\frac{3}{4} \times \frac{4}{3} \div \frac{1}{4}\)) | \(1 \div \frac{1}{4} = 4\) | 4 |
| Main Expression (\(18 \times 5\frac{1}{9} \div 4\)) | \(18 \times \frac{46}{9} \div 4 = 92 \div 4 = 23\) | 23 |
Revision Table: Key Concepts
| Concept | Description | Example |
|---|---|---|
| BODMAS/PEMDAS | Rule for the order of operations in mathematical expressions: Brackets, Orders/Of, Division/Multiplication, Addition/Subtraction. | Solve \(3 + 4 \times 2\) as \(3 + (4 \times 2) = 3 + 8 = 11\), not \((3+4) \times 2 = 7 \times 2 = 14\). |
| 'Of' operator | Means multiplication, but is evaluated before division and standard multiplication in the D/M step. | \(5 \text{ of } 2 = 5 \times 2 = 10\). In \(3 + 4 \text{ of } 2\), calculate \(4 \text{ of } 2\) first: \(3 + 8 = 11\). |
| Mixed Numbers | A number consisting of a whole number and a fraction. | \(2\frac{1}{2}\) |
| Improper Fractions | A fraction where the numerator is greater than or equal to the denominator. | \(\frac{5}{2}\) |
| Converting Mixed to Improper | Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. | \(2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}\) |
| Dividing by a Fraction | Multiply by the reciprocal of the divisor fraction. | \(a \div \frac{b}{c} = a \times \frac{c}{b}\) |
Additional Information on Order of Operations
Understanding the correct order of operations is crucial for solving mathematical expressions accurately. The BODMAS rule ensures that everyone gets the same answer when evaluating an expression. It's a standard convention followed globally.
Let's look at the hierarchy of operations:
- Parentheses/Brackets: Any calculation inside brackets is performed first. If there are nested brackets, work from the innermost one outwards.
- Exponents/Orders/Of: Next come powers, roots, and the 'of' operator. The 'of' operator is essentially multiplication but is typically evaluated immediately after brackets and before division/multiplication in the same step.
- Division and Multiplication: These are performed next, working from left to right. Neither operation has priority over the other.
- Addition and Subtraction: These are performed last, also working from left to right. Neither has priority over the other.
In the given problem, the presence of 'of' is a key detail. It's important to remember that 'of' takes precedence within the multiplication/division group, evaluated right after brackets and orders.
Important Questions from Simplification
Simplify the following expression.
\(\left(\frac{7}{16} \div \frac{1}{2}\:of\: \frac{1}{5}\right)\times \frac{4}{5}-\frac{1}{3}\times\frac{5}{8}\div \frac{1}{2}+\frac{3}{4}\)
The value of \(\left[ {\frac{4}{7}\rm \;of\;2\frac{4}{5} \times 1\frac{2}{3} - \left( {3\frac{1}{2} - 2\frac{1}{6}} \right)} \right] \div \left( {3\frac{1}{5} \div 4\frac{1}{2}\;\rm of\;\;5\frac{1}{3}} \right)\) is:
The value of \(\frac{{0.0203 \times 2.92}}{{0.7 \times 0.0365 \times 2.9}} \div \frac{{{{\left( {12.12} \right)}^2} - {{\left( {8.12} \right)}^2}}}{{{{\left( {0.25} \right)}^2} + \left( {0.25} \right)\left( {19.99} \right)}}\) is:
The value of 4 ÷ 12 of [3 ÷ 4 of {(4 - 2) × 6 ÷ 2}] - 2 × 6 ÷ 8 + 3 is:
The value of \(\frac{7+3\sqrt5}{3+\sqrt5}-\frac{7-3\sqrt5}{3-\sqrt{5}}\) lies between:
