Problem 4 - Prealgebra

Question

Convert the given fraction to a terminating decimal. \(\frac{21}{4}\)

Step-by-Step Solution

Answer

The fraction \( \frac{21}{4} \) converts to the decimal 5.25.

Step 1: Understand the Problem

To convert the fraction \( \frac{21}{4} \) into a decimal, we need to perform division: divide the numerator (21) by the denominator (4). A terminating decimal is a decimal that ends, or has a finite number of digits after the decimal point.

Step 2: Perform the Division

Divide 21 by 4. Start with 21 as the dividend and 4 as the divisor. The whole number division gives 5, since 4 goes into 21 five times (4 * 5 = 20) with a remainder of 1.

Step 3: Continue Division for Decimals

After subtracting 20 from 21, you have a remainder of 1. Bring down a 0 to make it 10. Divide 10 by 4, which gives 2, since 4 goes into 10 twice (4 * 2 = 8) with a remainder of 2. Bring down another 0 to make it 20.

Step 4: Complete the Division

With 20, divide by 4 to get 5, because 4 goes into 20 five times (4 * 5 = 20) with a remainder of 0. This indicates the division is complete and results in a terminating decimal.

Step 5: Compile Final Answer

The decimal form of \( \frac{21}{4} \) is 5.25, since division ends when the remainder becomes zero after the second decimal place.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction to a decimal involves a straightforward process, mainly focusing on performing division between the numerator and the denominator. When you have a fraction like \( \frac{21}{4} \), it means you need to divide the number 21 by 4. The outcome is what we call a decimal form. This conversion is useful in many situations where a decimal representation is more convenient or practical. A particular type of decimal resulting from this conversion is a terminating decimal, which means the decimal has a definite end, rather than continuing indefinitely. In our example, you can see that \( \frac{21}{4} \) turns into 5.25, a terminating decimal with two decimal places.

Division

Division is the mathematical operation required to convert the fraction \( \frac{21}{4} \) into a decimal. The division process involves repeatedly subtracting the divisor from the dividend until what remains is less than the divisor.
In our case:

• 21 is the dividend (the number being divided).
• 4 is the divisor (the number you are dividing by).

First, divide 21 by 4. The nearest whole number you can obtain is 5, because 4 goes into 21 five times (4 * 5 = 20), leaving a remainder of 1. To continue the division:

• Add a decimal point.
• Bring down a 0 next to the 1 to make 10.

Repeat the division with 10 as your new dividend, which results in 2 remainder 2. Finally, bring down another 0 to make 20, resulting in a final division where the remainder reaches 0, confirming the end of our procedure, yielding the decimal 5.25.

Numerator and Denominator

Numerator and denominator are essential terms you'll often hear in the context of fractions. The numerator is the top number of a fraction, while the denominator is at the bottom.

• In the fraction \( \frac{21}{4} \), 21 is the numerator.
• 4 is the denominator.

These terms help in the clear expression of a part-whole relationship. The numerator signifies the portion of the whole being considered, while the denominator indicates into how many parts the whole is divided.
When converting a fraction to a decimal, focus on the relationship between these two numbers. The numerator represents how many parts you have, and the denominator shows how many parts make up a whole. Thus, dividing the numerator by the denominator as described transforms the fractional relationship into a decimal.