Problem 46 - Elementary and Intermediate Algebra

Question

Simplify. $$ \frac{170}{34} $$

Step-by-Step Solution

Answer

5

Step 1: Identify the numbers

The fractions asks to simplify \( \frac{170}{34} \). The numerator is 170 and the denominator is 34.

Step 2: Find the greatest common divisor

To simplify the fraction, determine the greatest common divisor (GCD) of 170 and 34. Both numbers are divisible by 34 itself.

Step 3: Divide numerator and denominator by the GCD

Divide both the numerator and the denominator by their GCD of 34. \( \frac{170 \div 34}{34 \div 34} = \frac{5}{1} \)

Step 4: Simplify the fraction

The result \( \frac{5}{1} \) can be simplified to 5.

Key Concepts

Greatest Common Divisor (GCD)

Simplifying fractions often involves finding the greatest common divisor (GCD). The GCD of two numbers is the largest number that evenly divides both of them. For example, given the fraction \(\frac{170}{34}\), we need to determine the GCD of 170 and 34. Here's how you can find it:

• List the factors of each number.
• 170: 1, 2, 5, 10, 17, 34, 85, 170
• 34: 1, 2, 17, 34

The highest number that appears in both lists is 34, so the GCD is 34. Understanding the GCD helps in reducing fractions to their simplest form. In this case, dividing both the numerator and the denominator by 34 gives us \(\frac{5}{1}\), which simplifies to 5. This makes arithmetic operations easier and results more understandable.

Numerator and Denominator

In a fraction, the top number is called the numerator, and the bottom number is the denominator. The numerator represents how many parts we have, while the denominator represents how many parts make up a whole. In the fraction \(\frac{170}{34}\), 170 is the numerator and 34 is the denominator. Knowing how to identify and manipulate these components is crucial in simplifying fractions.

• The numerator (170) tells us how many parts we're dealing with.
• The denominator (34) tells us the size of each part relative to a whole.

To simplify the fraction, we divide both the numerator and the denominator by their GCD, in this case, 34. This gives us \(\frac{170 \times 1}{34 \times 1} = \frac{5}{1}\), which can be further simplified to just 5. Understanding numerators and denominators helps make the concept of fractions clearer and their simplification more intuitive.

Basic Algebra

Simplifying fractions is a fundamental concept in basic algebra. Algebra often involves manipulating expressions to make them simpler or to solve equations. Here’s how basic algebra principles apply to our task:
1. **Identify the components:** Recognize the numerator and the denominator. In our case, 170 and 34.
2. **Apply the GCD:** Use the greatest common divisor to reduce the fraction. Here, the GCD of 170 and 34 is 34.
3. **Perform the division:** Divide both numerator and denominator by the GCD. \(\frac{170 \text{ ÷ } 34}{34 \text{ ÷ } 34} = \frac{5}{1}\)
4. **Simplify further if needed:** The result \(\frac{5}{1}\) is actually just 5.
Basic algebra makes dealing with fractions, especially large ones, far less daunting. By practicing these methods, students become more comfortable with a wide range of math problems.