### Instruction: Find $53\cdot\left(3\frac{1}{5} - 4\frac{1}{2}\right) \div \left(2\frac{3}{4} + 1\frac{2}{3} \right)$. Express your answer as a mixed number.### Response: To simplify this expression, I need to use the order of operations: parentheses, exponents, multiplication and division, and addition and subtraction.
I also need to convert any mixed numbers to improper fractions, so I can use the common denominator method for adding, subtracting, multiplying, and dividing fractions.
Let me start with the parentheses.
Inside the first one, I have $3\frac{1}{5} - 4\frac{1}{2}$.
To convert these mixed numbers to improper fractions, I multiply the whole number by the denominator and add the numerator.
For example, $3\frac{1}{5} = \frac{3\cdot 5 + 1}{5} = \frac{16}{5}$.
Similarly, $4\frac{1}{2} = \frac{4\cdot 2 + 1}{2} = \frac{9}{2}$.
Now, to subtract these fractions, I need to find a common denominator.
The least common multiple of 5 and 2 is 10, so I multiply the first fraction by $\frac{2}{2}$ and the second fraction by $\frac{5}{5}$ to get equivalent fractions with the same denominator.
This gives me $\frac{16}{5}\cdot\frac{2}{2} - \frac{9}{2}\cdot\frac{5}{5} = \frac{32}{10} - \frac{45}{10} = \frac{-13}{10}$.
So, the first parenthesis simplifies to $\frac{-13}{10}$.
Next, I look at the second parenthesis.
I have $2\frac{3}{4} + 1\frac{2}{3}$.
I convert these mixed numbers to improper fractions as well.
I get $2\frac{3}{4} = \frac{2\cdot 4 + 3}{4} = \frac{11}{4}$ and $1\frac{2}{3} = \frac{1\cdot 3 + 2}{3} = \frac{5}{3}$.
To add these fractions, I need a common denominator again.
The least common multiple of 4 and 3 is 12, so I multiply the first fraction by $\frac{3}{3}$ and the second fraction by $\frac{4}{4}$ to get equivalent fractions with the same denominator.
This gives me $\frac{11}{4}\cdot\frac{3}{3} + \frac{5}{3}\cdot\frac{4}{4} = \frac{33}{12} + \frac{20}{12} = \frac{53}{12}$.
So, the second parenthesis simplifies to $\frac{53}{12}$.
Now, I have $53\cdot\frac{-13}{10} \div \frac{53}{12}$.
To multiply fractions, I just multiply the numerators and the denominators.
This gives me $\frac{53\cdot(-13)}{10\cdot 1} = \frac{-689}{10}$.
To divide fractions, I multiply by the reciprocal of the second fraction.
This gives me $\frac{-689}{10} \cdot \frac{12}{53} = \frac{-689\cdot 12}{10\cdot 53}$.
I can simplify this fraction by canceling out any common factors in the numerator and the denominator.
I notice that both 689 and 53 are divisible by 53, so I divide them by 53 to get $\frac{-13\cdot 12}{10\cdot 1} = \frac{-156}{10}$.
I can simplify this fraction further by dividing both the numerator and the denominator by 2, the greatest common factor.
This gives me $\frac{-78}{5}$.
Finally, I can convert this improper fraction to a mixed number by dividing the numerator by the denominator and writing the remainder as the numerator of a new fraction.
This gives me $\frac{-78}{5} = -15 \frac{3}{5}$.
So, the final answer is -15 and 3/5.