Problem 51 - Prealgebra

Question

Reduce each fraction to lowest terms. a. \(\frac{2}{90}\) b. \(\frac{3}{90}\) c. \(\frac{5}{90}\) d. \(\frac{6}{90}\) e. \(\frac{9}{90}\)

Step-by-Step Solution

Answer

a. \(\frac{1}{45}\); b. \(\frac{1}{30}\); c. \(\frac{1}{18}\); d. \(\frac{1}{15}\); e. \(\frac{1}{10}\).

Step 1: Analyze Fraction a

Identify the greatest common divisor (GCD) of 2 and 90. Both numbers share a GCD of 2. Divide both the numerator and the denominator by 2 to reduce the fraction.\[\frac{2}{90} = \frac{2 \div 2}{90 \div 2} = \frac{1}{45}\]

Step 2: Simplify Fraction b

Determine the GCD of 3 and 90. The GCD is 3. Reduce the fraction by dividing both terms by 3.\[\frac{3}{90} = \frac{3 \div 3}{90 \div 3} = \frac{1}{30}\]

Step 3: Simplify Fraction c

Find the GCD of 5 and 90, which is 5. Divide both the numerator and the denominator by 5 to simplify the fraction.\[\frac{5}{90} = \frac{5 \div 5}{90 \div 5} = \frac{1}{18}\]

Step 4: Analyze Fraction d

Calculate the GCD of 6 and 90, which is 6. Use this to reduce the fraction.\[\frac{6}{90} = \frac{6 \div 6}{90 \div 6} = \frac{1}{15}\]

Step 5: Simplify Fraction e

Find the GCD of 9 and 90, which is 9. Simplify the fraction by dividing both the numerator and the denominator by 9.\[\frac{9}{90} = \frac{9 \div 9}{90 \div 9} = \frac{1}{10}\]

Key Concepts

Greatest Common Divisor

When reducing fractions, the greatest common divisor (GCD) plays a crucial role. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. It allows us to determine the most simplified form of a fraction. Finding the GCD can initially seem tricky, but it's pretty simple with a few steps:

• List the factors of each number.
• Identify the common factors shared by both numbers.
• The largest of these shared factors is the GCD.

For example, to find the GCD of 6 and 90:

• Factors of 6: 1, 2, 3, 6
• Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
• Common factors: 1, 2, 3, 6
• GCD: 6

Using the GCD helps in reducing fractions quickly and effectively.

Simplifying Fractions

Simplifying fractions involves dividing both the numerator and the denominator by their GCD to turn them into the simplest form. This process makes fractions easier to understand and work with. Here’s how you simplify a fraction step by step:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Write down the resulting fraction, which should be in its simplest form.

For example, let's simplify the fraction \(\frac{9}{90}\):

• GCD of 9 and 90 is 9.
• Divide both 9 and 90 by 9: \(\frac{9 \div 9}{90 \div 9} = \frac{1}{10}\).

There you have it! A simplified fraction is easier to read and use in calculations.

Prealgebra Concepts

Understanding prealgebra concepts like reducing fractions is fundamental to building a strong mathematical foundation. Prealgebra introduces students to the idea of breaking down numbers, which is essential for more advanced topics in math.
Here are a few basic yet important ideas from prealgebra:

• Understanding Numbers: Recognizing the role of divisors and multiples can aid in various mathematical operations.
• Factors: Knowing how to find factors of numbers is a skill that's very helpful for identifying the GCD.
• Fractions: Learning how to manipulate and simplify fractions lays down the groundwork for algebraic operations.

These foundational skills taught in prealgebra make it easier to tackle more complex mathematical problems later on. As you become comfortable with these concepts, you'll be well-prepared for higher-level courses.