The value of \(9 \div [\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}\div(\frac{3}{4}-\frac{1}{3})\;of\;\frac{2}{9}]\) is:
Simplifying Complex Fraction Expressions Using BODMAS
To find the value of the given mathematical expression, we need to follow the order of operations, often remembered by the acronym BODMAS or PEMDAS. This rule dictates the sequence in which operations should be performed:
- Brackets (or Parentheses)
- Orders (or Exponents/Indices)
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
The expression is: \(9 \div [\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}\div(\frac{3}{4}-\frac{1}{3})\;of\;\frac{2}{9}]\)
Let's simplify the expression step by step according to the BODMAS rule.
Step 1: Simplify the innermost bracket
We start with the expression inside the parentheses: \((\frac{3}{4}-\frac{1}{3})\). To subtract fractions, we need a common denominator. The least common multiple (LCM) of 4 and 3 is 12.
\(\frac{3}{4}-\frac{1}{3} = \frac{3 \times 3}{4 \times 3}-\frac{1 \times 4}{3 \times 4} = \frac{9}{12}-\frac{4}{12} = \frac{9-4}{12} = \frac{5}{12}\)
The expression inside the bracket becomes: \( [\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}\div(\frac{5}{12})\;of\;\frac{2}{9}] \)
Step 2: Evaluate the 'of' operation within the bracket
Next, we handle the 'of' operation. Recall that 'of' means multiplication. So, we calculate \((\frac{5}{12})\;of\;\frac{2}{9}\).
\(\frac{5}{12}\;of\;\frac{2}{9} = \frac{5}{12} \times \frac{2}{9} = \frac{5 \times 2}{12 \times 9} = \frac{10}{108}\)
We can simplify \(\frac{10}{108}\) by dividing both numerator and denominator by their greatest common divisor, which is 2.
\(\frac{10}{108} = \frac{10 \div 2}{108 \div 2} = \frac{5}{54}\)
The expression inside the bracket is now: \( [\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}\div\frac{5}{54}] \)
Step 3: Perform the division within the bracket
According to BODMAS, division comes before addition. We need to calculate \(\frac{1}{6}\div\frac{5}{54}\). Dividing by a fraction is the same as multiplying by its reciprocal.
\(\frac{1}{6}\div\frac{5}{54} = \frac{1}{6} \times \frac{54}{5} = \frac{1 \times 54}{6 \times 5} = \frac{54}{30}\)
We can simplify \(\frac{54}{30}\) by dividing both numerator and denominator by their greatest common divisor, which is 6.
\(\frac{54}{30} = \frac{54 \div 6}{30 \div 6} = \frac{9}{5}\)
The expression inside the bracket is now: \( [\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{9}{5}] \)
Step 4: Perform the addition within the bracket
Now we need to add the fractions inside the bracket: \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{9}{5}\). To add fractions, we find the LCM of their denominators (2, 3, 4, 5). The LCM is 60.
Convert each fraction to an equivalent fraction with a denominator of 60:
- \(\frac{1}{2} = \frac{1 \times 30}{2 \times 30} = \frac{30}{60}\)
- \(\frac{1}{3} = \frac{1 \times 20}{3 \times 20} = \frac{20}{60}\)
- \(\frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60}\)
- \(\frac{9}{5} = \frac{9 \times 12}{5 \times 12} = \frac{108}{60}\)
Now add the equivalent fractions:
\(\frac{30}{60}+\frac{20}{60}+\frac{15}{60}+\frac{108}{60} = \frac{30+20+15+108}{60} = \frac{173}{60}\)
The value inside the bracket is \(\frac{173}{60}\).
The original expression is now reduced to: \(9 \div \frac{173}{60}\)
Step 5: Perform the final division
Finally, we perform the division outside the bracket. Dividing by a fraction is the same as multiplying by its reciprocal.
\(9 \div \frac{173}{60} = 9 \times \frac{60}{173} = \frac{9 \times 60}{173} = \frac{540}{173}\)
The value of the expression is \(\frac{540}{173}\).
Let's check the options:
| Option | Value |
|---|---|
| 1 | \(\frac{540}{173}\) |
| 2 | \(\frac{340}{173}\) |
| 3 | \(\frac{480}{173}\) |
| 4 | \(\frac{2540}{173}\) |
Our calculated value matches Option 1.
Revision Table: BODMAS Order
| Step | Operation Type | Details |
|---|---|---|
| 1 | Brackets/Parentheses | Simplify expressions inside () first, then [] |
| 2 | Orders/Exponents | Calculate powers and roots |
| 3 | Division and Multiplication | Perform from left to right |
| 4 | Addition and Subtraction | Perform from left to right |
Additional Information: Fraction Operations
Working with fractions requires understanding basic operations:
- Addition/Subtraction: Find a common denominator (LCM) and add/subtract the numerators.
- Multiplication: Multiply numerators together and denominators together. Simplify the resulting fraction.
- Division: Multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
- 'of' operation: This signifies multiplication.
Always simplify fractions to their lowest terms at the end, or during intermediate steps if it makes calculations easier.
Important Questions from Fractions
Which fraction among the following is the least ?
\(\frac{5}{11}, \frac{7}{12}, \frac{8}{13}, \frac{9}{17}\)
Find the value of the following expression:
\(\frac{{3 \div 1 \times 2 + 5 - 2}}{{3 \times 3 - 2}}\)
Simplify the expression 441 ÷ \(\left[270 \div \frac{3}{7}+\left(17\div \frac{1}{3}\right)-\left(8\frac{1}{2}-\frac{5}{2}\right)\right]\)
If the sum of two positive numbers is 65 and the square root of their product is 26, then the sum of their reciprocals is:
The value of \(3\frac{1}{5} \div 4\frac{1}{2}\;of\;5\frac{1}{3} + \frac{1}{8} \div \frac{1}{2}of\frac{1}{4} - \frac{1}{4}\left( {\frac{1}{2} \div \frac{1}{8} \times \frac{1}{4}} \right)\) is:
