Simplify Fractional Expression using BODMAS Rule

This problem requires us to simplify a mathematical expression involving fractions and different operations. To correctly solve such expressions, we must follow the order of operations, commonly known as BODMAS or PEMDAS.

The expression we need to simplify is: \( \frac{{33}}{{40}} + \frac{1}{5}\left[ {\frac{4}{5} - \frac{1}{5} \times \left( {\frac{7}{8} - \frac{5}{4}} \right)} \right] - \frac{4}{5} \)

Understanding the BODMAS/PEMDAS Rule

The BODMAS rule helps us decide the sequence in which operations should be performed in a mathematical expression. The acronym stands for:

• Brackets (Parentheses)
• Orders (Exponents, Square Roots, etc.)
• Division
• Multiplication
• Addition
• Subtraction

PEMDAS is another version of the rule, where 'P' is Parentheses, 'E' is Exponents, and 'M' and 'D' (Multiplication and Division) and 'A' and 'S' (Addition and Subtraction) are done from left to right.

We solve operations within brackets first, starting from the innermost bracket. Then we proceed with the other operations according to the order.

Step-by-Step Solution for Simplifying the Expression

Step 1: Solve the Innermost Bracket

The innermost bracket contains the expression \( \left( {\frac{7}{8} - \frac{5}{4}} \right) \). To subtract these fractions, we need a common denominator. The least common multiple of 8 and 4 is 8.

Convert \( \frac{5}{4} \) to an equivalent fraction with denominator 8:

\( \frac{5}{4} = \frac{5 \times 2}{4 \times 2} = \frac{10}{8} \)

Now perform the subtraction inside the bracket:

\( \frac{7}{8} - \frac{10}{8} = \frac{7 - 10}{8} = \frac{-3}{8} \)

The expression now becomes:

\( \frac{{33}}{{40}} + \frac{1}{5}\left[ {\frac{4}{5} - \frac{1}{5} \times \left( \frac{-3}{8} \right)} \right] - \frac{4}{5} \)

Step 2: Perform Multiplication inside the Square Bracket

Inside the square bracket, we have \( \frac{4}{5} - \frac{1}{5} \times \left( \frac{-3}{8} \right) \). According to BODMAS, multiplication comes before subtraction. So, we first calculate \( \frac{1}{5} \times \left( \frac{-3}{8} \right) \).

Multiply the numerators and the denominators:

\( \frac{1}{5} \times \frac{-3}{8} = \frac{1 \times (-3)}{5 \times 8} = \frac{-3}{40} \)

The expression inside the square bracket is now:

\( \frac{4}{5} - \left( \frac{-3}{40} \right) \)

The overall expression is now:

\( \frac{{33}}{{40}} + \frac{1}{5}\left[ \frac{4}{5} - \left( \frac{-3}{40} \right) \right] - \frac{4}{5} \)

Step 3: Perform Subtraction (which becomes Addition) inside the Square Bracket

We need to calculate \( \frac{4}{5} - \left( \frac{-3}{40} \right) \). Subtracting a negative number is the same as adding the positive number.

\( \frac{4}{5} - \left( \frac{-3}{40} \right) = \frac{4}{5} + \frac{3}{40} \)

To add these fractions, find a common denominator, which is 40.

Convert \( \frac{4}{5} \) to an equivalent fraction with denominator 40:

\( \frac{4}{5} = \frac{4 \times 8}{5 \times 8} = \frac{32}{40} \)

Now perform the addition:

\( \frac{32}{40} + \frac{3}{40} = \frac{32 + 3}{40} = \frac{35}{40} \)

Simplify the fraction \( \frac{35}{40} \) by dividing the numerator and denominator by their greatest common divisor, which is 5.

\( \frac{35 \div 5}{40 \div 5} = \frac{7}{8} \)

So, the value inside the square bracket is \( \frac{7}{8} \). The expression now is:

\( \frac{{33}}{{40}} + \frac{1}{5}\left[ \frac{7}{8} \right] - \frac{4}{5} \)

Step 4: Perform Multiplication outside the Square Bracket

We need to calculate \( \frac{1}{5} \times \frac{7}{8} \).

\( \frac{1}{5} \times \frac{7}{8} = \frac{1 \times 7}{5 \times 8} = \frac{7}{40} \)

The expression is now:

\( \frac{{33}}{{40}} + \frac{7}{40} - \frac{4}{5} \)

Step 5: Perform Addition and Subtraction from Left to Right

We have \( \frac{{33}}{{40}} + \frac{7}{40} - \frac{4}{5} \).

First, perform the addition \( \frac{{33}}{{40}} + \frac{7}{40} \). Since the denominators are already the same, we just add the numerators:

\( \frac{33}{40} + \frac{7}{40} = \frac{33 + 7}{40} = \frac{40}{40} = 1 \)

The expression simplifies to \( 1 - \frac{4}{5} \).

Now, perform the subtraction \( 1 - \frac{4}{5} \). To subtract these, write 1 as a fraction with denominator 5:

\( 1 = \frac{5}{5} \)

Now subtract:

\( \frac{5}{5} - \frac{4}{5} = \frac{5 - 4}{5} = \frac{1}{5} \)

The final value of the expression is \( \frac{1}{5} \).

Revision Table: Order of Operations (BODMAS/PEMDAS)

Additional Information on Fraction Operations

When simplifying expressions with fractions, it's important to remember the rules for fraction arithmetic:

• Adding/Subtracting Fractions: Fractions must have a common denominator before adding or subtracting. Find the least common multiple (LCM) of the denominators, convert the fractions to equivalent fractions with this LCM as the new denominator, and then add or subtract the numerators.
• Multiplying Fractions: To multiply fractions, multiply the numerators together and multiply the denominators together. \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \). Simplify the resulting fraction if possible.
• Dividing Fractions: To divide by a fraction, multiply by its reciprocal. \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \).
• Simplifying Fractions: Divide the numerator and the denominator by their greatest common divisor (GCD) to express the fraction in its simplest form.

Applying these rules carefully, along with the BODMAS order, is key to accurately solving complex fractional expressions.