Problem 22 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$-3 \frac{3}{4}$$

Step-by-Step Solution

Answer

The decimal representation is -3.75.

Step 1: Break Down the Mixed Number

First, let's take the mixed number \(-3 \frac{3}{4}\) and write it as the sum of its whole number and fractional parts: \(-3 + \frac{3}{4}\).

Step 2: Convert the Fraction to a Decimal

The fraction part \(\frac{3}{4}\) can be converted to a decimal by dividing the numerator by the denominator: \(3 \div 4 = 0.75\).

Step 3: Combine to Form a Decimal

Now combine the decimal and whole number: \(-3\) (whole number part) + \(0.75\) (fractional decimal part) equals \(-3.75\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It's like having a complete item and a piece of another. In mathematics, mixed numbers are widely used and are simply another way to express quantities. For example, if you have

• 3 full pizzas
• and 3/4 of another pizza,

you could say you have \(-3 \frac{3}{4}\) pizzas.To work with mixed numbers easily, you need to be able to break them into their whole and fractional parts. In our example,

• the whole number is -3,
• and the fractional part is \(\frac{3}{4}\).

This separation allows us to handle each part individually. This is especially useful when converting mixed numbers to decimal form, like in the original exercise.

Repeating Decimals

Repeating decimals are decimals that go on forever in a repeating pattern. They are commonly found when dividing numbers that do not divide evenly. To indicate that a decimal repeats, we use a bar over the repeating section. For instance,

• if 1 divided by 3 results in a decimal that continues as 0.3333...,

we can write it as 0.\(\overline{3}\).Understanding repeating decimals is vital when converting fractions to decimals, as some fractions result in repeating patterns when divided. However, not all fractions create repeating decimals. In the exercise, the fraction \(\frac{3}{4}\) converts to 0.75, a terminating decimal with no repeating part.

Numerator and Denominator Division

When converting fractions to decimals, dividing the numerator by the denominator is key. The numerator is the number above the fraction bar, representing how many parts or items you have. The denominator is below the bar and shows the total number of equal parts into which something is divided.For example, in the fraction \(\frac{3}{4}\):

• the numerator is 3,
• and the denominator is 4.

To convert it to a decimal, you perform the division \[3 \div 4 = 0.75\].This operation is straightforward when the division is exact, as is the case here. But if the division results in a repeating decimal, like

• \(\frac{1}{3}\) resulting in 0.\(\overline{3}\),

understanding numerator and denominator division helps you expect and manage these outcomes effectively.