What Is The Value Of The Expression?$\left(12 \frac{1}{3} + 5 \frac{1}{3}\right) - \left(7 \frac{4}{8} - 7 \frac{1}{6}\right$\]Enter Your Answer In The Box As A Mixed Number In Simplest Form.

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Understanding the Problem

The given expression involves mixed numbers and fractions, and we need to simplify it to find its value. To start, let's break down the expression into smaller parts and simplify each part separately.

Simplifying the First Part

The first part of the expression is (1213+513)\left(12 \frac{1}{3} + 5 \frac{1}{3}\right). To simplify this, we need to find a common denominator for the two mixed numbers. The least common multiple (LCM) of 3 and 3 is 3, so we can rewrite the mixed numbers with a common denominator.

12 \frac{1}{3} = \frac{(12 \times 3) + 1}{3} = \frac{36 + 1}{3} = \frac{37}{3}
5 \frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3}

Now, we can add the two fractions together.

\frac{37}{3} + \frac{16}{3} = \frac{37 + 16}{3} = \frac{53}{3}

So, the simplified first part of the expression is 533\frac{53}{3}.

Simplifying the Second Part

The second part of the expression is (748−716)\left(7 \frac{4}{8} - 7 \frac{1}{6}\right). To simplify this, we need to find a common denominator for the two mixed numbers. The LCM of 8 and 6 is 24, so we can rewrite the mixed numbers with a common denominator.

7 \frac{4}{8} = \frac{(7 \times 8) + 4}{8} = \frac{56 + 4}{8} = \frac{60}{8} = \frac{15}{2}
7 \frac{1}{6} = \frac{(7 \times 6) + 1}{6} = \frac{42 + 1}{6} = \frac{43}{6}

Now, we can subtract the two fractions.

\frac{15}{2} - \frac{43}{6} = \frac{15 \times 3}{2 \times 3} - \frac{43 \times 1}{6 \times 1} = \frac{45}{6} - \frac{43}{6} = \frac{45 - 43}{6} = \frac{2}{6} = \frac{1}{3}

So, the simplified second part of the expression is 13\frac{1}{3}.

Combining the Parts

Now that we have simplified the two parts of the expression, we can combine them to find the final value.

\left(12 \frac{1}{3} + 5 \frac{1}{3}\right) - \left(7 \frac{4}{8} - 7 \frac{1}{6}\right) = \frac{53}{3} - \frac{1}{3} = \frac{53 - 1}{3} = \frac{52}{3}

So, the final value of the expression is 523\frac{52}{3}.

Converting to a Mixed Number

To convert the improper fraction 523\frac{52}{3} to a mixed number, we need to divide the numerator by the denominator.

\frac{52}{3} = 17 \frac{1}{3}

So, the final value of the expression is 171317 \frac{1}{3}.

The final answer is: 1713\boxed{17 \frac{1}{3}}

Q: What is the value of the expression (1213+513)−(748−716)\left(12 \frac{1}{3} + 5 \frac{1}{3}\right) - \left(7 \frac{4}{8} - 7 \frac{1}{6}\right)?

A: The value of the expression is 171317 \frac{1}{3}.

Q: How do I simplify the first part of the expression, (1213+513)\left(12 \frac{1}{3} + 5 \frac{1}{3}\right)?

A: To simplify the first part of the expression, you need to find a common denominator for the two mixed numbers. The least common multiple (LCM) of 3 and 3 is 3, so you can rewrite the mixed numbers with a common denominator. Then, you can add the two fractions together.

Q: How do I simplify the second part of the expression, (748−716)\left(7 \frac{4}{8} - 7 \frac{1}{6}\right)?

A: To simplify the second part of the expression, you need to find a common denominator for the two mixed numbers. The LCM of 8 and 6 is 24, so you can rewrite the mixed numbers with a common denominator. Then, you can subtract the two fractions.

Q: What is the common denominator for the two mixed numbers in the first part of the expression?

A: The common denominator for the two mixed numbers in the first part of the expression is 3.

Q: What is the common denominator for the two mixed numbers in the second part of the expression?

A: The common denominator for the two mixed numbers in the second part of the expression is 24.

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, you need to divide the numerator by the denominator. The quotient will be the whole number part of the mixed number, and the remainder will be the numerator of the fraction part.

Q: What is the final value of the expression in mixed number form?

A: The final value of the expression in mixed number form is 171317 \frac{1}{3}.

Q: Can I use a calculator to simplify the expression?

A: Yes, you can use a calculator to simplify the expression. However, it's always a good idea to understand the steps involved in simplifying the expression to ensure that you get the correct answer.

Q: What is the least common multiple (LCM) of 3 and 3?

A: The LCM of 3 and 3 is 3.

Q: What is the least common multiple (LCM) of 8 and 6?

A: The LCM of 8 and 6 is 24.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you need to list the multiples of each number and find the smallest multiple that is common to both lists.

Q: What is the numerator of the fraction part of the mixed number 171317 \frac{1}{3}?

A: The numerator of the fraction part of the mixed number 171317 \frac{1}{3} is 1.

Q: What is the denominator of the fraction part of the mixed number 171317 \frac{1}{3}?

A: The denominator of the fraction part of the mixed number 171317 \frac{1}{3} is 3.

Q: Can I simplify the expression further?

A: No, the expression cannot be simplified further. The final value of the expression is 171317 \frac{1}{3}.