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An improper fraction is a fraction where the numerator (the top number) is larger than the denominator (the bottom number). In other words, it represents a value greater than one. For example, in the problem, when we converted the mixed number \[3 \frac{3}{4}\] to an improper fraction \[\frac{15}{4}\], we ended up with a larger numerator (15) than the denominator (4). Improper fractions are useful in mathematical operations because they simplify calculations. Here’s how you convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator.
• Add that product to the numerator.
• Place the result over the original denominator.

Using this method, you can handle more complex problems efficiently.

A mixed number has both a whole number and a fraction part. When solving problems involving multiplication or division with mixed numbers, it’s easier to convert them to improper fractions first. As shown, the mixed number \[3 \frac{3}{4}\] was converted to the improper fraction \[\frac{15}{4}\].
To do this conversion:

• Multiply the whole part (3) by the denominator (4) to get 12.
• Add the numerator (3) to get 15.
• Place the sum (15) over the original denominator (4), resulting in \[\frac{15}{4}\].

Once you've solved the problem, you may need to convert an improper fraction back to a mixed number at the end. Let's rehash this with our result \[\frac{75}{4}\]:

• Divide 75 by 4, which is 18 with a remainder of 3.
• The quotient becomes the whole number part (18).
• The remainder becomes the numerator of the fractional part (\[\frac{3}{4}\]).
• Therefore, \[\frac{75}{4} = 18 \frac{3}{4}\].

Multiplying fractions involves a straightforward process: multiply the numerators together and the denominators together. When multiplying a whole number by a fraction, convert the whole number to a fraction by placing it over 1 (since any number divided by 1 is the number itself).
For example, multiplying 5 by \[\frac{15}{4}\] looks like this:

• Convert 5 to \[\frac{5}{1}\].
• The multiplication now looks like \[\frac{5}{1} \times \frac{15}{4}\].
• Multiply the numerators: \[5 \times 15 = 75\].
• Multiply the denominators: \[1 \times 4 = 4\].
• The result is \[\frac{75}{4}\].

Multiplying fractions is a fundamental skill, and understanding it eases the process of more complex operations.

Breaking down mathematical problems into manageable steps can significantly improve understanding and accuracy. In the provided exercise, three main steps were used to solve the problem:
1. **Convert the Mixed Number to an Improper Fraction**
This simplifies the multiplication process. Here, the mixed number \[3 \frac{3}{4}\] was converted to \[\frac{15}{4}\].
2. **Perform the Multiplication**
With the mixed number now an improper fraction, you multiply it by the whole number. This was done by converting 5 to \[\frac{5}{1}\] and then multiplying \[\frac{5}{1} \times \frac{15}{4} = \frac{75}{4}\].
3. **Convert the Result to a Mixed Number**
Finally, the improper fraction \[\frac{75}{4}\] was converted back to a mixed number: 18 with a remainder of \[\frac{3}{4}\], resulting in \[18 \frac{3}{4}\].
Following these steps ensures clarity and precision, making complex problems easier to solve.