Problem 65 - Prealgebra

Question

Convert the given decimal to a fraction. Reduce your answer to lowest terms. 0.98

Step-by-Step Solution

Answer

The decimal 0.98 in fraction form is \(\frac{49}{50}\).

Step 1: Identify the Decimal

The given decimal is 0.98. This is a decimal number with two digits following the decimal point.

Step 2: Convert Decimal to Fraction

To convert the decimal 0.98 to a fraction, recognize that 0.98 is equivalent to 98 hundredths, or \[ \frac{98}{100}. \]

Step 3: Simplify the Fraction

To simplify the fraction \(\frac{98}{100}\), find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 98 and 100 is 2. Divide both the numerator and the denominator by their GCD:\[ \frac{98 \div 2}{100 \div 2} = \frac{49}{50}. \]

Step 4: Verify the Reduced Fraction

Ensure that the fraction \(\frac{49}{50}\) cannot be simplified further. Since 49 and 50 have no common divisors other than 1, the fraction is in its lowest terms.

Key Concepts

Decimal to Fraction Conversion

Converting a decimal to a fraction might seem challenging at first, but it boils down to understanding place value. When you have a decimal like 0.98, it can be read as "98 hundredths." This means the decimal point is followed by two digits, signifying hundredths.
Consequently, the decimal 0.98 is equivalent to the fraction \( \frac{98}{100} \) because 98 sits in the hundredths place. For every digit after the decimal, you multiply by 10; thus, two decimal places equal 100 (10 times 10).

• Identify the digits after the decimal.
• Use a denominator that corresponds to the place value (100 for hundredths).
• Write the fraction from the decimal form.

Once you've understood this fundamental principle, turning decimals into fractions becomes straightforward.

Simplifying Fractions

Once we have our fraction \( \frac{98}{100} \), the next step is simplification. Simplifying fractions helps to present them in their simplest form, making it easier to understand and compare.
The key to simplification is to reduce both the numerator (top number) and the denominator (bottom number) by their greatest common factor (GCF), also known as the greatest common divisor (GCD).

• The fraction in its simplest form still represents the same value as its original.
• It often makes calculations and comparisons easier.

To simplify \( \frac{98}{100} \), we will need to determine the GCF.

Understanding Greatest Common Divisor (GCD)

Finding the greatest common divisor is a crucial step in reducing fractions. The GCD of two numbers is the largest integer that evenly divides both numbers without leaving a remainder. To find it, you can list the factors of each number and identify the largest one they share.
For the numbers 98 and 100, the GCD is 2, because 2 is the highest number that can divide both without leaving a remainder.

• List all factors of each number.
• Identify the common factors.
• The greatest of these is the GCD.

In the case of 98 and 100, after we find that their GCD is 2, we will use it to reduce the fraction.

Reducing Fractions to Lowest Terms

Reducing a fraction to its lowest terms means simplifying it as much as possible using the GCD. This ensures that the fraction is in its simplest form, not alterable by any common divisor other than 1.
To reduce \( \frac{98}{100} \) using the GCD of 2, divide both the numerator and the denominator by 2:

• Divide the numerator: \( 98 \div 2 = 49 \).
• Divide the denominator: \( 100 \div 2 = 50 \).

This results in the reduced fraction \( \frac{49}{50} \). At this point, no further simplification is possible because 49 and 50 do not share any common divisors other than 1. Therefore, \( \frac{49}{50} \) is the fraction in its lowest terms, neatly representing the original decimal.