Let's solve the expression step by step:

$\left( 1 \frac{4}{5} + \left( -\frac{2}{3} \right) \right) \times \frac{15}{4}$

Step 1: Convert the mixed number to an improper fraction

First, convert $1 \frac{4}{5}$ to an improper fraction: $1 \frac{4}{5} = \frac{5}{5} + \frac{4}{5} = \frac{9}{5}$

Now the expression becomes: $\left( \frac{9}{5} + \left( -\frac{2}{3} \right) \right) \times \frac{15}{4}$

Step 2: Add the fractions

To add $\frac{9}{5}$ and $-\frac{2}{3}$, we need a common denominator. The least common denominator (LCD) of 5 and 3 is 15. Let's rewrite both fractions with the denominator 15:

$\frac{9}{5} = \frac{9 \times 3}{5 \times 3} = \frac{27}{15}$ $-\frac{2}{3} = \frac{-2 \times 5}{3 \times 5} = \frac{-10}{15}$

Now add the fractions: $\frac{27}{15} + \frac{-10}{15} = \frac{27 - 10}{15} = \frac{17}{15}$

Step 3: Multiply by $\frac{15}{4}$

Now the expression is: $\frac{17}{15} \times \frac{15}{4}$

Multiply the numerators and denominators: $\frac{17 \times 15}{15 \times 4} = \frac{255}{60}$

Step 4: Simplify the fraction

To simplify $\frac{255}{60}$, divide both the numerator and denominator by their greatest common divisor (GCD), which is 15: $\frac{255 \div 15}{60 \div 15} = \frac{17}{4}$

Final Answer:

The simplified result is: $\frac{17}{4} \quad \text{or} \quad 4 \frac{1}{4}$

Would you like a more detailed explanation, or do you have any other questions?

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Here are 5 related questions:

1. How do you convert a mixed number into an improper fraction?
2. How can you find the least common denominator of two fractions?
3. What is the process for multiplying two fractions?
4. How do you simplify a fraction by finding the greatest common divisor (GCD)?
5. How do you convert an improper fraction to a mixed number?

Tip: When adding or subtracting fractions, always find a common denominator to ensure you're working with equivalent fractions.