Problem 70 - Prealgebra

Question

Reduce to lowest terms. $$\frac{220}{1,000}$$

Step-by-Step Solution

Answer

\( \frac{220}{1,000} \) simplifies to \( \frac{11}{50} \).

Step 1: Identify the Greatest Common Divisor (GCD)

Finding the GCD of the numerator (220) and the denominator (1000) will help simplify the fraction. To find the GCD, we can list the factors. Factors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220. Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000. The greatest common factor here is 20.

Step 2: Divide by the GCD

Now divide both the numerator and the denominator by the GCD we just found. Divide 220 and 1000 by 20.\( \frac{220}{20} = 11 \)\( \frac{1000}{20} = 50 \)

Step 3: Write the Simplified Fraction

After dividing both the numerator and the denominator by the GCD, we obtain the simplified fraction:\( \frac{11}{50} \)

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a key component in simplifying fractions. It is the largest number that can exactly divide both the numerator and the denominator without leaving a remainder. Finding the GCD involves examining the factors of each number. Factors are numbers that multiply together to give the original number.

For instance, consider the exercise problem of reducing \(\frac{220}{1000}\). To find the GCD:

• List factors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220
• List factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
• The GCD is the largest shared factor from both lists—in this case, 20.

This process helps us find how much we can reduce the fraction to its simplest form. By using the GCD, we ensure the reduction is done correctly and efficiently, leading to a more understandable expression.

Breaking Down Numerator and Denominator

In fractions, the numerator and denominator are the key players. A fraction is written as \(\frac{a}{b}\), where 'a' is the numerator and 'b' is the denominator. The numerator represents the number of parts considered, while the denominator shows the total number of equal parts in the whole.

Understanding these concepts is essential when simplifying fractions. For example, in \(\frac{220}{1000}\):

• The numerator is 220. This is the portion of the entire whole we are studying.
• The denominator is 1000. It reflects the complete set of parts that make up the whole.

When simplifying, both the numerator and the denominator are divided by their GCD. By dividing through the GCD, the fraction maintains its value but is more straightforward to interpret. This makes it easier to compare, compute, and fully understand the relationships between different fractions.

Mastering Fraction Reduction

Fraction reduction is a valuable skill that makes complex fractions simpler and more useful in calculations. By reducing fractions to their lowest terms, you present the fraction in its simplest form, which aids in understanding and comparison.

In the problem \(\frac{220}{1000}\), the steps to reduce it are:

• Identify the GCD, which we found to be 20.
• Divide both the numerator and denominator by 20: \(\frac{220}{20} = 11\) and \(\frac{1000}{20} = 50\).
• This results in the simplified fraction: \(\frac{11}{50}\).

Reduction through division by the GCD ensures that no further simplification is possible. The simplified form \(\frac{11}{50}\) is easier to comprehend and use, highlighting the fraction's true meaning without unnecessary complication. This process finds the most basic version of the fraction, enhancing clarity in both mathematical and everyday contexts.