Problem 30 - Prealgebra

Question

Write each decimal as a mixed number. $$8.4$$

Step-by-Step Solution

Answer

The mixed number is \(8\frac{2}{5}\).

Step 1: Identify Whole and Decimal Parts

First, we need to identify the whole number part and the decimal part of the given number \(8.4\). In this case, the whole number part is \(8\) and the decimal part is \(0.4\).

Step 2: Convert Decimal to Fraction

Convert the decimal \(0.4\) to a fraction. The decimal \(0.4\) is equivalent to \(\frac{4}{10}\) because it is read as 'four-tenths'. This can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is \(2\). Thus, \(\frac{4}{10} = \frac{2}{5}\).

Step 3: Formulate the Mixed Number

Combine the whole number part \(8\) with the simplified fraction \(\frac{2}{5}\) to form the mixed number. Thus, \(8.4\) as a mixed number is \(8\frac{2}{5}\).

Key Concepts

Converting Decimals to Fractions

To understand how to convert decimals to fractions, it's crucial to recognize what a decimal represents. For instance, the decimal number "0.4" indicates four-tenths. Here, the digit "4" is in the tenths place. When converting, write this as a fraction with the decimal number over the corresponding power of 10.
For "0.4", it becomes \(\frac{4}{10}\) because it is equivalent to four parts out of ten. Once you have the fraction, the next step involves simplifying it to its simplest form, which we'll explore in the next section.
This process works with any decimal, just by counting the decimal places and using the right power of ten, you can convert any decimal into a fraction.

Simplifying Fractions

To simplify fractions, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Simplifying fractions makes them easier to understand and work with.
In our example, we started with the fraction \(\frac{4}{10}\). Now look for the largest number that can divide both 4 and 10 without leaving a remainder. Here, that number is 2.
Divide both the numerator and the denominator by the GCD:

• Numerator: \(4 \div 2 = 2\)
• Denominator: \(10 \div 2 = 5\)

So, \(\frac{4}{10}\) simplifies to \(\frac{2}{5}\). Achieving the simplest form of a fraction is key in understanding the relationship between the parts being represented.

Mixed Numbers

Mixed numbers are used when you have a whole number combined with a fraction. They provide a way to express numbers that are more than whole, yet less than another whole number, in an intuitive manner.
Utilizing the number 8.4, we first separated the whole number 8 from the decimal part 0.4. The 0.4 was converted to the fraction \(\frac{2}{5}\) and simplified as explained before.
To write this as a mixed number, you simply combine the whole and the fraction: your whole number "8" and the simplified fraction "\(\frac{2}{5}\)" results in the mixed number \(8\frac{2}{5}\).
Mixed numbers efficiently communicate quantifiable values that include both entire units and partial units.