Question

Change the following mixed numbers to improper fractions. \(12 \frac{3}{4}=\) ______

Short answer

The improper fraction is \(\frac{51}{4}\).

Step-by-step solution

Identify Components

A mixed number like \(12 \frac{3}{4}\) consists of two parts: the whole number, which is 12, and the fractional part, which is \(\frac{3}{4}\). We need to convert both parts into an improper fraction.

Convert Whole Number

To convert the whole number 12 into a fraction with the same denominator as \(\frac{3}{4}\), multiply 12 by the denominator 4: \[12 \times 4 = 48\] This gives us \(\frac{48}{4}\) as the fractional equivalent of 12.

Add Fractions

Now, add the improper fraction form of the whole number to the fractional part of the mixed number:\[\frac{48}{4} + \frac{3}{4} = \frac{51}{4}\]

Combine into Improper Fraction

The sum \(\frac{51}{4}\) is the improper fraction representation of the mixed number \(12 \frac{3}{4}\).

Key concepts

Mixed Numbers

Mixed numbers are a way to express quantities greater than a whole number using fractions. They consist of two components.

• A whole number part
• A fractional part

When you see a mixed number, such as \(12 \frac{3}{4}\), it integrates both a whole and a fraction.

The whole number in this case is 12, and the fractional part is \(\frac{3}{4}\).To fully appreciate why we use mixed numbers, think about scenarios involving real-world measurements.For instance, if you are baking and need 12 and three-quarters cups of flour, expressing it as \(12 \frac{3}{4}\) makes the measuring process clear and straightforward. By converting it into a mixed number, we provide an intuitive way to understand and visualize measurements that aren't full integers.

Improper Fractions

Improper fractions are an essential concept in understanding the number representation involving fractions.In improper fractions, the numerator (top number) is larger than or equal to the denominator (bottom number).

For example, the fraction \(\frac{51}{4}\) is an improper fraction because 51 is greater than 4.This kind of fraction is particularly useful for calculations and simplifications.They also present numbers in a form that is purely fractional, without separating out whole numbers from fractional values.

Rather than seeing 12 whole parts with a portion left over, like we did in the mixed number \(12 \frac{3}{4}\), the entire number is expressed as a single fraction, \(\frac{51}{4}\).Thus, improper fractions are often preferred when performing arithmetic operations, as they eliminate the need to separately manage whole numbers and fractional parts.

Mathematical Conversion

Mathematical conversion, in this context, refers to changing one form of a number to another; specifically, from a mixed number to an improper fraction.This process involves several steps that ensure accuracy and retention of the value represented by the original number.

To convert a mixed number such as \(12 \frac{3}{4}\) to an improper fraction:

• First, identify the whole number and the fraction part. Here, it’s 12 and \(\frac{3}{4}\).
• Next, multiply the whole number by the denominator of the fractional part. This means calculating \(12 \times 4 = 48\).
• Then, add this result to the numerator of the fractional part, \(48 + 3 = 51\).
• Finally, place this sum over the original denominator to complete the conversion, resulting in \(\frac{51}{4}\).

Conversion between these forms is crucial in various mathematical and real-life applications, as it helps provide a unified way to work with different forms of numbers in equations and problems.