Simplifying the Expression with Fractions

We need to find the value of the given expression:

$\frac{4}{5} \div \frac{2}{15} \text{ of } \left(\frac{2}{3} + \frac{4}{21}\right) - \left(\frac{7}{4} - \frac{4}{3}\right) = ?$

To solve this, we follow the order of operations (BODMAS/PEMDAS): Brackets, Of (multiplication), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Solving the Brackets

First, let's solve the expressions inside the brackets.

Bracket 1: $\left(\frac{2}{3} + \frac{4}{21}\right)$

To add fractions, we find a common denominator. The least common multiple (LCM) of 3 and 21 is 21.

• Convert $\frac{2}{3}$ to an equivalent fraction with denominator 21: $\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}$.
• Now add: $\frac{14}{21} + \frac{4}{21} = \frac{14 + 4}{21} = \frac{18}{21}$.
• Simplify the fraction $\frac{18}{21}$ by dividing both numerator and denominator by their greatest common divisor, which is 3: $\frac{18 \div 3}{21 \div 3} = \frac{6}{7}$.

So, $\left(\frac{2}{3} + \frac{4}{21}\right) = \frac{6}{7}$.

Bracket 2: $\left(\frac{7}{4} - \frac{4}{3}\right)$

To subtract fractions, we find a common denominator. The LCM of 4 and 3 is 12.

• Convert $\frac{7}{4}$ to an equivalent fraction with denominator 12: $\frac{7}{4} = \frac{7 \times 3}{4 \times 3} = \frac{21}{12}$.
• Convert $\frac{4}{3}$ to an equivalent fraction with denominator 12: $\frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12}$.
• Now subtract: $\frac{21}{12} - \frac{16}{12} = \frac{21 - 16}{12} = \frac{5}{12}$.

So, $\left(\frac{7}{4} - \frac{4}{3}\right) = \frac{5}{12}$.

Solving 'Of' (Multiplication)

Next, we evaluate the 'of' part, which means multiplication: $\frac{2}{15} \text{ of } \left(\frac{2}{3} + \frac{4}{21}\right)$. We found that $\left(\frac{2}{3} + \frac{4}{21}\right) = \frac{6}{7}$.

So, $\frac{2}{15} \text{ of } \frac{6}{7} = \frac{2}{15} \times \frac{6}{7}$.

Multiply the numerators and the denominators:

$\frac{2 \times 6}{15 \times 7} = \frac{12}{105}$.

Simplify the fraction $\frac{12}{105}$ by dividing both numerator and denominator by their greatest common divisor, which is 3: $\frac{12 \div 3}{105 \div 3} = \frac{4}{35}$.

So, $\frac{2}{15} \text{ of } \left(\frac{2}{3} + \frac{4}{21}\right) = \frac{4}{35}$.

Solving Division

Now we perform the division: $\frac{4}{5} \div \left(\frac{2}{15} \text{ of } \left(\frac{2}{3} + \frac{4}{21}\right)\right)$. We found that $\frac{2}{15} \text{ of } \left(\frac{2}{3} + \frac{4}{21}\right) = \frac{4}{35}$.

So, $\frac{4}{5} \div \frac{4}{35}$.

Dividing by a fraction is the same as multiplying by its reciprocal:

$\frac{4}{5} \times \frac{35}{4}$.

Cancel out the common factor 4 in the numerator and denominator, and simplify 35/5:

$\frac{\cancel{4}}{5} \times \frac{35}{\cancel{4}} = \frac{35}{5} = 7$.

So, $\frac{4}{5} \div \frac{4}{35} = 7$.

Solving Final Subtraction

Finally, we perform the subtraction: $\left(\frac{4}{5} \div \frac{4}{35}\right) - \left(\frac{7}{4} - \frac{4}{3}\right)$. We found the first part is 7 and the second bracket is $\frac{5}{12}$.

So, $7 - \frac{5}{12}$.

Convert 7 to a fraction with denominator 12: $7 = \frac{7 \times 12}{1 \times 12} = \frac{84}{12}$.

Now subtract:

$\frac{84}{12} - \frac{5}{12} = \frac{84 - 5}{12} = \frac{79}{12}$.

Thus, the value of the expression is $\frac{79}{12}$.

Final Answer

The value that comes in the place of the question mark is $\frac{79}{12}$.