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Mixed numbers are a combination of a whole number and a fractional part. They are often seen in everyday life, such as when measuring ingredients or dividing objects. For example, the mixed number \(8 \frac{1}{3}\) comprises a whole number 8 and the fraction \(\frac{1}{3}\).

• To operate with mixed numbers, it's usually easier to convert them into improper fractions first.
• Improper fractions express the amount as "the whole thing" plus a part of it, expressed entirely in fractions.

To convert a mixed number into an improper fraction:

• Multiply the denominator of the fraction by the whole number.
• Add the result to the numerator of the fraction.
• Write this total over the original denominator.

For instance, to convert \(8 \frac{1}{3}\) to an improper fraction, calculate \(8 \times 3 + 1 = 25\), then place it over the original denominator, \(\frac{25}{3}\). This method simplifies calculations especially in operations like multiplication and division.

Improper fractions have a numerator larger than or equal to the denominator. While they might seem a bit strange at first, they are very practical in mathematical calculations.

• Improper fractions can represent numbers equal to or greater than one.
• They are often used as an intermediate step in operations involving mixed numbers.

Consider the fraction \(\frac{12}{5}\), which means "twelve fifths". When this is part of an expression with other fractions, particularly involving multiplication or division, improper fractions help to keep everything in terms of fractions, simplifying processes and allowing for straightforward multiplication and division.

• Improper fractions can always be simplified and if desired, converted back to mixed numbers as a final step.
• For example, after calculations, \(\frac{5}{3}\) can be simplified to \(1 \frac{2}{3}\).

When you multiply and divide fractions, it's useful to remember a few simple steps. **Multiplication:**

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

For example, to multiply \(\frac{25}{3}\) by \(\frac{36}{75}\), you calculate \(25 \times 36\) for the numerator and \(3 \times 75\) for the denominator. **Division:**

• Instead of dividing by a fraction, multiply by its reciprocal.
• To find the reciprocal, swap the numerator and the denominator of the fraction.

In our problem, \(\div \frac{12}{5}\) becomes \(\times \frac{5}{12}\). This makes it a multiplication problem, which is usually simpler to handle.Once the fractions are lined up for multiplication, always look to simplify by canceling common factors across the numerators and denominators before actually doing the multiplication. This reduces the numbers you’re working with and simplifies the final result. Remember, practicing these steps will make these processes automatic and more intuitive.