Problem 5 - Prealgebra

Question

Complete the following tables by converting each fraction to a decimal. Fraction \(\frac{1}{4} \quad \frac{2}{4} \quad \frac{3}{4} \quad \frac{4}{4}\) Decimal

Step-by-Step Solution

Answer

The decimals are 0.25, 0.5, 0.75, and 1.0 respectively.

Step 1: Convert First Fraction to Decimal

To convert the fraction \( \frac{1}{4} \) to a decimal, divide the numerator by the denominator. Thus, \( 1 \div 4 = 0.25 \).

Step 2: Convert Second Fraction to Decimal

Convert \( \frac{2}{4} \) to a decimal. Simplify the fraction first: \( \frac{2}{4} = \frac{1}{2} \). Now divide the numerator by the denominator: \( 1 \div 2 = 0.5 \).

Step 3: Convert Third Fraction to Decimal

Convert \( \frac{3}{4} \) to a decimal by dividing 3 by 4. Thus, \( 3 \div 4 = 0.75 \).

Step 4: Convert Fourth Fraction to Decimal

Convert \( \frac{4}{4} \) to a decimal. Simplify the fraction \( \frac{4}{4} = 1 \). Hence, the decimal is \( 1.0 \).

Key Concepts

Simplifying Fractions

Simplifying fractions makes calculations easier and often sheds light on the relationships between fractions and their decimal counterparts. To simplify a fraction, you find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number.
For example, the fraction \( \frac{2}{4} \) can be simplified. We notice that both 2 (the numerator) and 4 (the denominator) can be divided evenly by 2, which is their GCD. So, \( \frac{2}{4} \) simplifies to \( \frac{1}{2} \).

• Simplifying \( \frac{2}{4} \) to \( \frac{1}{2} \) makes it easier to convert to a decimal.
• Simplified fractions are always more straightforward to work with in further calculations.
• Simplification helps illustrate equivalent fractions and makes their decimal conversion more evident.

Knowing how and when to simplify fractions can save time and make understanding their decimal form more intuitive.

Division

Division is the key operation for converting fractions into decimals. The process is simple: divide the numerator (the top part of the fraction) by the denominator (the bottom part). This gives the decimal equivalent of the fraction.
Let's take a few examples from the exercise:

• For \( \frac{1}{4} \), divide 1 by 4. You get 0.25 as the decimal representation.
• For \( \frac{3}{4} \), dividing 3 by 4 results in 0.75.
• When the numerator and denominator are the same, like in \( \frac{4}{4} \), dividing them gives you 1.0, since any number divided by itself is 1.

Understanding division is crucial, as it directly translates a fraction to a decimal form. Practice can greatly help in mastering this skill.

Decimal Representation

Decimal representation expresses fractions as numbers that have positions to the right of a decimal point. This format is particularly useful because it is how most measurements, prices, and quantities are represented in everyday life. Decimals are read by acknowledging each digit's position as a power of ten.

• Consider \( \frac{1}{4} \) which is converted to 0.25. Here, 2 represents \( \frac{2}{10} \) (or 2 tenths) and 5 represents \( \frac{5}{100} \) (or 5 hundredths).
• The decimal of 0.5 from \( \frac{1}{2} \) is read as 5 tenths, illustrating its immediate equivalence to 50 hundredths.
• Whole numbers, like the 1 from \( \frac{4}{4} \), can also be expressed as 1.0, though the decimal part here is zero.

Decimals provide a simple and consistent way of expressing non-whole numbers, especially when measurement precision is necessary. Understanding their link to fractions helps solidify basic numerical literacy.