Question

Change the following mixed numbers to improper fractions. \(7 \frac{3}{8}=\) ______

Short answer

\(7 \frac{3}{8}\) as an improper fraction is \(\frac{59}{8}\).

Step-by-step solution

Understand the Mixed Number

A mixed number consists of a whole number and a fraction. In the given problem, the mixed number is \(7 \frac{3}{8}\), where 7 is the whole number and \(\frac{3}{8}\) is the fractional part.

Convert the Whole Number to a Fraction

To change the whole number into a fraction, multiply the whole number by the denominator of the fraction. Here, multiply 7 by 8. So, \(7 \times 8 = 56\). Now, write it as \(\frac{56}{8}\).

Add the Fractions

Add the fraction from Step 2 to the original fractional part of the mixed number. So you have \(\frac{56}{8} + \frac{3}{8}\). Since the denominators are the same, add the numerators: \(56 + 3 = 59\).

Write as Improper Fraction

Once the fractions are combined, write it as a single fraction: \(\frac{59}{8}\). This is the improper fraction form of \(7 \frac{3}{8}\).

Key concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. Imagine you have 7 whole pizzas and a slice that equals three-eighths of another pizza. This is a mixed number written as \(7 \frac{3}{8}\). Here:

• Whole Number: The "7" represents the whole pizzas.
• Fractional Part: The "\(\frac{3}{8}\)" indicates the additional pizza slice.

Mixed numbers make it easier to express quantities greater than one in a simple format by separating out the complete wholes from any parts.

Improper Fractions

An improper fraction is one where the numerator is larger than or equal to the denominator. It may seem unusual because typically, a fraction represents a part less than one. However, with improper fractions, it indicates a value equal to or greater than one. Converting a mixed number like \(7 \frac{3}{8}\) into an improper fraction involves transforming the whole number portion into a fraction and then combining it with the existing fractional part. This gives us a simple way to handle operations like addition, subtraction, multiplication, or division.

Denominator

The denominator in a fraction is the bottom number. It tells us into how many equal parts the whole is divided. In the example \(7 \frac{3}{8}\), the denominator is 8.
Here are some important roles of the denominator:

• Defines the unit size: In \(\frac{3}{8}\), each part is 1/8 of the whole.
• It must remain consistent when adding fractions, as seen in the original conversion.

The denominator is crucial because it defines the frame of reference for the fraction and helps ensure all fractions are measured rightly against each other.

Numerator

The numerator is the top number in a fraction and represents how many parts we have out of the total defined by the denominator. In \(\frac{3}{8}\), 3 is the numerator.
Here’s what the numerator does:

• Counts the parts: It tells us that we have 3 parts out of 8.
• Changes with operations: When fractions are added or subtracted, the numerator adjusts accordingly, like in the conversion to an improper fraction.

Understanding numerators helps you understand how much of the whole you have, whether it's a part of one or many wholes.