Problem 89 - Elementary and Intermediate Algebra

Question

Explain how to find the decimal equivalent of a fraction.

Step-by-Step Solution

Answer

To convert a fraction to a decimal, divide the numerator by the denominator.

Step 1: Understanding Fractions to Decimals

To convert a fraction to a decimal, you need to understand that a fraction is a division problem: the numerator (top number) is divided by the denominator (bottom number). For example, the fraction \( \frac{3}{4} \) is equivalent to the division \( 3 \div 4 \).

Step 2: Performing the Division

Use long division to divide the numerator by the denominator. If you take the fraction \( \frac{3}{4} \), divide \( 3 \) by \( 4 \). Here, \( 3 \div 4 = 0.75 \). This is done by seeing how many times \( 4 \) can fit into \( 3 \), which is \( 0 \), then considering how many times \( 4 \) can go into \( 30 \) (since we add a zero), which is \( 7 \) times with a remainder of \( 2 \). Thus, \( 4 \) goes into \( 20 \) five times exactly.

Step 3: Refining the Result

If you have a repeating decimal, you will notice the division has a pattern. For non-terminating decimals, continue division until the desired precision, and for repeating numbers, note the repeating sequence.

Step 4: Finalizing the Answer

The result obtained from the division gives you the decimal equivalent of your fraction. If dividing \( 3 \) by \( 4 \) gives \( 0.75 \), this is the decimal equivalent of \( \frac{3}{4} \).

Key Concepts

Long Division

Long division is a method used to transform fractions into decimals. It involves dividing the numerator by the denominator meticulously. Imagine you have the fraction \( \frac{3}{4} \), and you want to convert it to a decimal. Here’s how long division helps:

• Start by seeing how many times the denominator (4) can fit into the first digit of the numerator (3). It fits 0 times, so you write 0 and a decimal point.
• Next, consider the entire numerator by bringing down a zero, making it 30. Now, 4 fits into 30, seven times, precisely giving a quotient of 7.
• The remainder so far is 2, and when you add another 0, making it 20, 4 can fit into it 5 times without any remainder. Thus, hence the decimal is 0.75.

Long division helps keep calculations orderly, making it clearer to understand how fractions can be converted into decimal numbers.

Numerator and Denominator

In the world of fractions, it's vital to understand the roles of the numerator and the denominator. When you have a fraction like \( \frac{3}{4} \):

• The numerator is 3. It signifies how many parts of a whole are being considered.
• The denominator is 4. It indicates into how many parts the whole is divided.

Recognizing that a fraction represents a division—specifically, the numerator divided by the denominator—is crucial. In the context of converting to decimals, this implies dividing the top number by the bottom number, using techniques like long division to achieve an accurate decimal result.

Repeating Decimal

When converting fractions to decimals, sometimes you'll encounter a repeating decimal. These are decimals in which one or more digits keep repeating infinitely. For instance, \( \frac{1}{3} \) converts to a decimal of 0.333..., where the digit 3 repeats continuously.There are some key points about repeating decimals:

• Identifying Patterns: During division, if you see the remainder start to repeat at any step, the sequence of numbers in the quotient from this point will keep repeating.
• Notation: To represent a repeating decimal concisely, a line or a dot is placed over the digit(s) that repeat, such as \( 0.\overline{3} \) for a repeating 3.

Repeating decimals are a fascinating area of study because they depict the infinite nature of some rational numbers when expressed in decimal form. It’s important to be comfortable spotting and working with them during conversions.