Evaluating Mathematical Expressions with Fractions

We are asked to find the value of the mathematical expression: \(\frac{{11}}{5} - \left( {\frac{2}{3}of\frac{3}{5} - \frac{1}{5}} \right) + \left( {\frac{6}{5} \div \frac{4}{5}} \right)\). To solve this, we must follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.

• Brackets first
• Orders (powers, roots) or Of (multiplication related)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Step-by-Step Evaluation

Let's evaluate the expression step by step, following the order of operations.

Step 1: Evaluate the first bracket \(\left( {\frac{2}{3}of\frac{3}{5} - \frac{1}{5}} \right)\)

Inside the first bracket, we have "of" and subtraction. "Of" means multiplication, and it should be performed before subtraction.

First, calculate \(\frac{2}{3}of\frac{3}{5}\):

\(\frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5} = \frac{6}{15}\)

We can simplify the fraction \(\frac{6}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

\(\frac{6 \div 3}{15 \div 3} = \frac{2}{5}\)

Now, substitute this back into the bracket and perform the subtraction:

\(\frac{2}{5} - \frac{1}{5}\)

Since the denominators are the same, we can subtract the numerators directly:

\(\frac{2 - 1}{5} = \frac{1}{5}\)

So, the value of the first bracket is \(\frac{1}{5}\).

Step 2: Evaluate the second bracket \(\left( {\frac{6}{5} \div \frac{4}{5}} \right)\)

Inside the second bracket, we have division. To divide by a fraction, we multiply by its reciprocal.

The reciprocal of \(\frac{4}{5}\) is \(\frac{5}{4}\).

So, \(\frac{6}{5} \div \frac{4}{5} = \frac{6}{5} \times \frac{5}{4}\)

Now, multiply the numerators and the denominators:

\(\frac{6 \times 5}{5 \times 4} = \frac{30}{20}\)

We can simplify the fraction \(\frac{30}{20}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 10.

\(\frac{30 \div 10}{20 \div 10} = \frac{3}{2}\)

So, the value of the second bracket is \(\frac{3}{2}\).

Step 3: Substitute the bracket values back into the original expression

The original expression was \(\frac{{11}}{5} - \left( {\frac{2}{3}of\frac{3}{5} - \frac{1}{5}} \right) + \left( {\frac{6}{5} \div \frac{4}{5}} \right)\). Now it becomes:

\(\frac{11}{5} - \frac{1}{5} + \frac{3}{2}\)

Step 4: Perform subtraction and addition from left to right

First, perform the subtraction:

\(\frac{11}{5} - \frac{1}{5} = \frac{11 - 1}{5} = \frac{10}{5}\)

Simplify \(\frac{10}{5}\):

\(\frac{10}{5} = 2\)

Now, perform the addition:

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

To add a whole number and a fraction, convert the whole number into a fraction with the same denominator as the other fraction. The denominator is 2, so \(2 = \frac{2 \times 2}{2} = \frac{4}{2}\).

Now, add the fractions:

\(\frac{4}{2} + \frac{3}{2} = \frac{4 + 3}{2} = \frac{7}{2}\)

The value of the expression is \(\frac{7}{2}\).

The final calculated value matches option 4.

Revision Table: Order of Operations

Additional Information: Operations with Fractions

Understanding how to perform basic arithmetic operations with fractions is crucial for evaluating expressions like this.

• Multiplication of Fractions: Multiply the numerators together and the denominators together. Simplify the resulting fraction if possible. Example: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
• Division of Fractions: To divide by a fraction, multiply by the reciprocal of the divisor. The reciprocal of a fraction \(\frac{c}{d}\) is \(\frac{d}{c}\). Example: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}\).
• Addition/Subtraction of Fractions:
  • If the denominators are the same, add or subtract the numerators and keep the common denominator. Example: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\).
  • If the denominators are different, find a common denominator (usually the least common multiple, LCM, of the denominators). Convert each fraction to an equivalent fraction with the common denominator, then add or subtract the numerators. Example: \(\frac{a}{b} + \frac{c}{d}\). Find LCM of \(b\) and \(d\), say \(m\). Convert to \(\frac{a'}{m} + \frac{c'}{m} = \frac{a'+c'}{m}\).
• Simplifying Fractions: Divide both the numerator and the denominator by their greatest common divisor (GCD) to reduce the fraction to its lowest terms.