Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{64}{15}=\) ______

Short answer

The mixed number is 4\(\frac{4}{15}\).

Step-by-step solution

Divide the Numerator by the Denominator

Start by dividing the numerator (64) by the denominator (15). When you divide 64 by 15, you get a quotient of 4 with a remainder.

Determine the Whole Number

The whole number part of the mixed number is the quotient from the division, which is 4.

Find the Remainder

Calculate the remainder of the division. Multiply the whole number (4) by the denominator (15) to get 60. Subtract 60 from the numerator (64) to find the remainder: 64 - 60 = 4.

Form the Fraction Part

The remainder of the division will form the numerator of the fraction part, and the denominator remains 15. So the fraction part is \(\frac{4}{15}\).

Combine Whole Number and Fraction

Combine the whole number (4) and the fraction part \(\frac{4}{15}\) to form the mixed number: 4\(\frac{4}{15}\).

Check for Simplification

Check if \(\frac{4}{15}\) can be simplified. The greatest common divisor (GCD) of 4 and 15 is 1, so \(\frac{4}{15}\) is already in its simplest form.

Key concepts

Improper Fractions

Improper fractions can sometimes be a bit confusing, but they are just fractions where the numerator is larger than the denominator. This means that the fraction is greater than one. For example, \(\frac{64}{15}\) is an improper fraction because 64 (the numerator) is bigger than 15 (the denominator).

To better understand improper fractions, think of them as ways to express whole numbers along with extra parts. To make them easier to work with or visualize, we often change them into mixed numbers. This involves dividing the numerator by the denominator to identify the whole number and the remaining fraction that represents the leftover part.

Division Process

The division process is a crucial step when converting an improper fraction to a mixed number. This is where you divide the numerator by the denominator.

In our example, we divide 64 by 15:

• The quotient is the whole number part of the mixed number. For \(\frac{64}{15}\), the quotient is 4.
• The remainder is what you get after dividing completely. It represents the numerator of the fractional part of the mixed number.

After dividing, subtract the product of the quotient and the denominator from the original numerator (as shown in \(64 - 60 = 4\))to get the remainder, which becomes the new numerator of your fraction part.

Simplification

Simplification ensures that a fraction is in its simplest form. This means the numerator and denominator have no common factors other than 1.

After converting an improper fraction into a mixed number, check if the fractional part can be simplified.
For instance, in the fraction \(\frac{4}{15}\), the greatest common divisor (GCD) of 4 and 15 is 1.
This means that \(\frac{4}{15}\) is already simplified, so no further reduction is necessary.

Numerator and Denominator

Understanding the roles of the numerator and denominator is key in fraction math. The numerator is the top number and signifies how many parts we have, whereas the denominator, the bottom number, indicates the total parts that make up a whole.

When working with improper fractions such as \(\frac{64}{15}\), the numerator is larger, meaning there are more parts than a single whole can contain.
In the division process, the numerator gets divided by the denominator, resulting in a whole number (if applicable) and a remainder. The numerator of your new fraction part is just this remainder, and the denominator stays the same.