Problem 102 - Algebra 1: Concepts and Skills

Question

Write the fraction as a terminating or repeating decimal. $$ \frac{3}{4} $$

Step-by-Step Solution

Answer

The fraction \( \frac{3}{4} \) converts to 0.75 in decimal, and it is a terminating decimal.

Step 1: Identify the Fraction to Convert

The fraction to convert into a decimal form is \( \frac{3}{4} \).

Step 2: Convert the Fraction into a Decimal

To convert a fraction to a decimal, divide the numerator by the denominator. Therefore, calculate 3 divided by 4.

Step 3: Determine the Decimal Type

After performing the division, one obtains a number with finite decimals, hence it is a terminating decimal.

Key Concepts

Terminating Decimals

A terminating decimal is a decimal number that has a finite number of digits after the decimal point. It results from a division that comes out even; meaning, after a certain point, you don't continue to get more non-zero digits. This happens when the denominator of the fraction, after removing all factors of 2 and 5, is 1. In our example, the fraction \( \frac{3}{4} \) is converted to a decimal resulting in 0.75. Since there are a finite number of digits (7 and 5) after the decimal point, this is a terminating decimal.

How can you tell if another fraction will be a terminating decimal? Well, check the denominator: if it can be written as a product of only 2s and 5s, then the fraction will produce a terminating decimal when expressed as a decimal. For instance, \( \frac{5}{8} \) will be terminating because 8 is a power of 2.

Repeating Decimals

In contrast to terminating decimals, repeating decimals are decimals that have one or several digits that repeat infinitely. Take the fraction \( \frac{1}{3} \), for instance, which equals 0.333... In this case, the digit 3 repeats indefinitely, which is why the decimal is termed 'repeating'.

To represent such decimals in a more concise way, a bar is often placed over the repeating digit or group of digits. The fraction \( \frac{1}{6} \), which equals 0.1666..., would be written as 0.1\( \bar{6} \). This indicates that the 6 repeats without end. Repeating decimals occur when the denominator contains other primes than 2 or 5, as these cannot be converted into a finite decimal.

Decimal Conversion

The process of decimal conversion involves rewriting a fraction as a decimal. To do this, we simply divide the numerator (the top number) by the denominator (the bottom number). This can be done through long division, or by using a calculator. For example, the fraction \( \frac{3}{4} \) is converted to decimal by dividing 3 by 4, yielding 0.75.

For conversion, remember that whole numbers or mixed fractions must first be separated into their whole and fractional parts. The fractional part can then be converted to a decimal and added back to the whole number. For instance, the mixed fraction 1\( \frac{1}{2} \) becomes 1.5 when converted to a decimal.

Numerators and Denominators

Every fraction consists of a numerator and a denominator. The numerator, located above the fraction line, indicates how many parts we have, while the denominator, below the line, defines the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{3}{4} \), 3 is the numerator showing that we have three parts, and 4 is the denominator indicating that the whole is divided into four equal parts. When converting this fraction to a decimal, we are dividing 3 (the number of parts) by 4 (the total number of equal parts). Understanding the roles of numerators and denominators is crucial in visualizing fractions and their decimal equivalents.