Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{67}{10}=\) ______

Short answer

\( 6 \frac{7}{10} \) is the mixed number.

Step-by-step solution

Divide the Numerator by the Denominator

To convert the improper fraction \( \frac{67}{10} \) to a mixed number, first divide 67 by 10. Performing the division, \( 67 \div 10 \), gives a quotient of 6 and a remainder of 7.

Form the Mixed Number

The quotient (6) becomes the whole number part of the mixed number. The remainder (7) becomes the numerator of the fractional part, while the original denominator (10) stays the same, forming the mixed number \( 6 \frac{7}{10} \).

Simplify the Fractional Part (if possible)

Check if the fraction \( \frac{7}{10} \) can be simplified. The greatest common divisor of 7 and 10 is 1, so the fraction is already in its simplest form.

Key concepts

Mixed Numbers

A mixed number is a way of expressing improper fractions as a combination of a whole number and a proper fraction. It's an intuitive way to understand values greater than one. When you see an improper fraction like \(\frac{67}{10}\), this means 67 parts out of 10, which immediately tells us that there's more than one whole group of 10 in 67.

To convert an improper fraction to a mixed number, you follow a straightforward process:

• Division: Divide the numerator by the denominator.
• Whole number part: The quotient becomes the whole number.
• Proper fraction part: The remainder becomes the numerator of the proper fraction, with the original denominator remaining unchanged.

Using our example, when we divide 67 by 10, we get a quotient of 6 and a remainder of 7. Thus, \(\frac{67}{10}\) can be written as the mixed number \(6 \frac{7}{10}\). This way of representation is useful because it provides a quick snapshot of the number's size.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is a fundamental concept in fraction simplification. It refers to the largest number that can divide two given numbers without leaving a remainder.

When you're simplifying fractions, finding the GCD of the numerator and the denominator is essential because it helps you reduce the fraction to its simplest form. For example, if you have a fraction \(\frac{8}{12}\), the GCD of 8 and 12 is 4. Simplifying the fraction by dividing both the numerator and the denominator by their GCD gives us \(\frac{2}{3}\).

In our example \(\frac{7}{10}\), the numbers 7 and 10 have a GCD of 1, meaning they do not share any common factors other than 1. That informs us that \(\frac{7}{10}\) is already simplified, as its parts cannot be reduced further.

Fraction Simplification

Simplifying fractions is an important step that often follows the initial writing of a fraction. It makes fractions easier to understand, compare, and use in calculations. To simplify a fraction, you must divide both the numerator and the denominator by their greatest common divisor (GCD).

Here's a straightforward approach to simplify fractions:

• Determine the GCD: Find the greatest common divisor of the numerator and the denominator.
• Divide: Divide both the numerator and the denominator by this GCD.

For instance, the fraction \(\frac{15}{25}\) has a GCD of 5. Dividing both the numerator and the denominator by 5 simplifies the fraction to \(\frac{3}{5}\).

In cases where the GCD is 1, such as \(\frac{7}{10}\) from our earlier example, the fraction is considered already in its simplest form, because no further division can reduce the fraction to a simpler expression. Understanding and applying simplification smoothly is key to working efficiently with fractions in math.