Problem 49 - Algebra A Combined Function

Question

Write each fraction as a decimal. If the decimal is a repeating decimal, write using the bar notation and then round to the nearest hundredth. $$ \frac{3}{4} $$

Step-by-Step Solution

Answer

The fraction \( \frac{3}{4} \) as a decimal is 0.75.

Step 1: Fraction Division

Divide the numerator by the denominator to convert the fraction \( \frac{3}{4} \) into its decimal form: 3 divided by 4.

Step 2: Perform the Division

Calculate \( 3 \div 4 = 0.75 \). The result shows that the fraction \( \frac{3}{4} \) is equal to the decimal 0.75.

Step 3: Determine Decimal Type

Since \( 0.75 \) has just two decimal places and does not continue indefinitely, it is a terminating decimal, not a repeating decimal.

Step 4: Round if Needed

Since the decimal 0.75 has only two decimal places and there is no repeating part, rounding is unnecessary. 0.75 is already rounded to the nearest hundredth.

Key Concepts

Terminating Decimal

When you convert a fraction into a decimal, you'll encounter one of two types: terminating or repeating decimals. A terminating decimal is simply a decimal that has a finite number of digits. It does not go on forever. Take the fraction \( \frac{3}{4} \) as an example. After dividing 3 by 4, you get 0.75.

Here, the decimal stops after two digits. Because it has a clear endpoint, we call it a terminating decimal. Generally, if you can express the fraction's denominator as a product of 2's and 5's (like 4 is \( 2^2 \)), the decimal will terminate.

Understanding whether a decimal terminates or repeats helps you in various mathematical settings, such as rounding or performing further calculations efficiently.

Decimal Notation

Decimal notation is the representation of numbers in a base-10 number system, which uses powers of 10. This method is familiar because it aligns with our everyday counting system. We often use it to make fractions easier to understand and work with. For instance, while \( \frac{3}{4} \) might not be instantly recognizable, the decimal 0.75 is more intuitive for many people.

Converting fractions into decimal notation involves division. You divide the numerator by the denominator to get a number with a whole part and a fractional part, separated by a decimal point. In our example, dividing 3 by 4 provided 0.75, a neat representation using only decimal notation.

Decimal notation simplifies comparison and arithmetic operations, especially addition and subtraction, making it a vital concept in math.

Division of Fractions

The process of turning a fraction into a decimal is essentially a division task. It involves dividing the numerator by the denominator. Take \( \frac{3}{4} \). This translates to performing the division: 3 divided by 4.

Before diving into the division, ensure that both numbers are in their proper places. The numerator (3) is divided by the denominator (4). Understanding this division is fundamental for converting fractions into their decimal equivalents.

Here's a simplified approach:

• Start by determining how many times the denominator fits into the numerator.
• In our example, figure out how many times 4 goes into 30, in this case, which results in 0.75.
• Track the remainder until you can't divide any further without repeating figures.

By mastering this division technique, you will find converting any fraction to a decimal much easier.