Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{10}{8}=\) ______

Short answer

\(1 \frac{1}{4}\)

Step-by-step solution

Write as a Division Problem

The improper fraction \(\frac{10}{8}\) can be interpreted as "10 divided by 8". This means how many whole times 8 can fit into 10, and what would be the remainder.

Perform the Division

Divide 10 by 8. The whole number result is 1 because 8 fits into 10 once. Calculate the remainder by subtracting \(8 \times 1 = 8\) from 10, which results in a remainder of 2.

Formulate the Mixed Number

Now that we have a quotient of 1 and a remainder of 2, write the quotient as the whole number, and use the remainder as the new numerator over the original denominator. This gives us \(1\frac{2}{8}\).

Simplify the Fraction

Simplify the fraction \(\frac{2}{8}\). Find the greatest common divisor (GCD) of 2 and 8, which is 2. Divide both numerator and denominator by the GCD: \(\frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4}\).

Write the Final Answer

Combine the whole number with the simplified fraction to give the final answer: \(1\frac{1}{4}\).

Key concepts

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means the fraction is "top-heavy" since the part is greater than or equal to the whole. For example, in the fraction \( \frac{10}{8} \), 10 (the numerator) is larger than 8 (the denominator). Improper fractions are quite common in mathematics, especially when dealing with measurements or dividing large quantities. They can also be expressed as mixed numbers or simplified further for different applications. To handle improper fractions effectively, it often helps to translate them into a more intuitive form, such as a mixed number, which can give additional insight into what the fraction represents.

Mixed Numbers

When you encounter an improper fraction, a good approach is to convert it into a mixed number. Mixed numbers consist of a whole number and a proper fraction. For instance, with the improper fraction \( \frac{10}{8} \), converting this into a mixed number involves dividing 10 by 8. You determine how many times the denominator (8) fits into the numerator (10). In this case, 8 fits once into 10, indicating a whole number of 1.

• The remainder after dividing 10 by 8 is 2, which becomes the new numerator in the fraction part of the mixed number.
• The denominator of the mixed number remains unchanged at 8.

Thus, \( \frac{10}{8} \) becomes \( 1\frac{2}{8} \). Mixed numbers offer a clearer picture of the quantity by separating whole units from fractional parts.

Division

Division plays a crucial role in converting improper fractions to mixed numbers. To divide means to ask how many times one number fits into another. For improper fractions, this determines the whole number part of the mixed number. With \( \frac{10}{8} \), we divided 10 by 8. The quotient or result is 1, which represents how many full groups of 8 are within 10. The remainder of 2 indicates what is left over after forming those full groups. Using division in this way helps make sense of fractions and convert them into formats that can be more easily simplified or interpreted. Practicing division with fractions can strengthen your overall math skills, as it is a foundational operation in many areas of math, particularly in dealing with measurements and real-world problems.

Simplification

Simplifying fractions makes them more straightforward and easier to understand. After converting an improper fraction to a mixed number, the next step is often simplification. This involves reducing the fractional part of the mixed number to its simplest form. For example, the mixed number part \( \frac{2}{8} \) can be simplified.

• Identify the greatest common divisor (GCD), which is the largest number that divides both the numerator and the denominator without leaving a remainder.
• For \( \frac{2}{8} \), the GCD is 2.
• Divide both the numerator and the denominator by this GCD: \( \frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4} \).

Thus, \( 1\frac{2}{8} \) simplifies to \( 1\frac{1}{4} \). Simplification helps reveal the most elementary form of a number, making it easier to compare, add, or subtract fractions.