Simplifying Fraction Expressions Using BODMAS

To simplify the given expression, we need to follow the order of operations, often remembered using the acronym BODMAS or PEMDAS.

• Brackets
• Orders (powers, roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

In this expression, we have 'of', division, multiplication, addition, and subtraction. 'Of' indicates multiplication and is usually performed before division and multiplication in the standard order.

The expression is: \(\rm \frac{7}{12} \div \frac{1}{10} \ of \ \frac{2}{3} - \frac{5}{3} \times \frac{9}{10} + \frac{5}{8} \div \frac{3}{4} \ of \ \frac{2}{3}\)

Step 1: Evaluate the 'of' operations

First, calculate the terms involving 'of':

• \(\frac{1}{10} \ of \ \frac{2}{3} = \frac{1}{10} \times \frac{2}{3} = \frac{1 \times 2}{10 \times 3} = \frac{2}{30}\)
• Simplify the fraction: \(\frac{2}{30} = \frac{2 \div 2}{30 \div 2} = \frac{1}{15}\)

And the second 'of' term:

• \(\frac{3}{4} \ of \ \frac{2}{3} = \frac{3}{4} \times \frac{2}{3} = \frac{3 \times 2}{4 \times 3} = \frac{6}{12}\)
• Simplify the fraction: \(\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}\)

Now substitute these back into the expression:

\(\rm \frac{7}{12} \div \frac{1}{15} - \frac{5}{3} \times \frac{9}{10} + \frac{5}{8} \div \frac{1}{2}\)

Step 2: Perform Division and Multiplication (from left to right)

Next, we perform the division and multiplication operations:

• First division: \(\frac{7}{12} \div \frac{1}{15}\)
• Dividing by a fraction is the same as multiplying by its reciprocal: \(\frac{7}{12} \times \frac{15}{1} = \frac{7 \times 15}{12 \times 1} = \frac{105}{12}\)
• Simplify the fraction: \(\frac{105}{12} = \frac{105 \div 3}{12 \div 3} = \frac{35}{4}\)

• Multiplication: \(\frac{5}{3} \times \frac{9}{10}\)
• Multiply the numerators and denominators: \(\frac{5 \times 9}{3 \times 10} = \frac{45}{30}\)
• Simplify the fraction: \(\frac{45}{30} = \frac{45 \div 15}{30 \div 15} = \frac{3}{2}\)

• Second division: \(\frac{5}{8} \div \frac{1}{2}\)
• Multiply by the reciprocal: \(\frac{5}{8} \times \frac{2}{1} = \frac{5 \times 2}{8 \times 1} = \frac{10}{8}\)
• Simplify the fraction: \(\frac{10}{8} = \frac{10 \div 2}{8 \div 2} = \frac{5}{4}\)

Substitute these results back into the expression:

\(\rm \frac{35}{4} - \frac{3}{2} + \frac{5}{4}\)

Step 3: Perform Addition and Subtraction (from left to right)

Now, combine the terms. To do this, we need a common denominator, which is 4.

• Convert \(\frac{3}{2}\) to have a denominator of 4: \(\frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}\)

The expression becomes:

\(\rm \frac{35}{4} - \frac{6}{4} + \frac{5}{4}\)

Now, perform the subtraction and addition from left to right:

• Subtract the first two terms: \(\frac{35}{4} - \frac{6}{4} = \frac{35 - 6}{4} = \frac{29}{4}\)
• Add the last term: \(\frac{29}{4} + \frac{5}{4} = \frac{29 + 5}{4} = \frac{34}{4}\)

Step 4: Simplify the final fraction

The resulting fraction is \(\frac{34}{4}\). We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

\(\frac{34}{4} = \frac{34 \div 2}{4 \div 2} = \frac{17}{2}\)

Step 5: Convert the improper fraction to a mixed number

The fraction \(\frac{17}{2}\) is an improper fraction (numerator is greater than the denominator). We can convert it to a mixed number.

Divide 17 by 2:

• \(17 \div 2 = 8\) with a remainder of \(1\).

So, \(\frac{17}{2}\) is equal to \(8\) whole times with \(\frac{1}{2}\) remaining. This gives the mixed number \(8 \frac{1}{2}\).

Thus, the simplified expression is \(8 \frac{1}{2}\).

Revision Table: BODMAS Rules and Fraction Arithmetic

Additional Information: Importance of Order of Operations in Math

Understanding the correct order of operations is crucial for accurately simplifying mathematical expressions. Without a standard order like BODMAS or PEMDAS, the same expression could yield multiple different results, leading to confusion and errors. This standard ensures that everyone interprets and solves expressions consistently, which is fundamental in mathematics and related fields.

The 'of' operation specifically refers to finding a fraction of a quantity, which is why it's a multiplication operation. Its placement in the order (often performed before standard multiplication and division when encountered in written form) is a convention to handle expressions precisely as intended.

Fraction arithmetic is a building block for more advanced mathematical concepts, including algebra, calculus, and statistics. Mastering fraction operations and the order of operations is essential for success in these areas.