Problem 17 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{5}{10}$$

Step-by-Step Solution

Answer

The fraction \( \frac{5}{10} \) reduces to \( \frac{1}{2} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{5}{10} \) to its lowest terms, start by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For the numbers 5 and 10, the divisors of 5 are 1 and 5, while the divisors of 10 are 1, 2, 5, and 10. Therefore, the GCD of 5 and 10 is 5.

Step 2: Divide Both Numerator and Denominator by the GCD

Now that we have found the GCD, divide both the numerator and the denominator of the fraction by this number. For \( \frac{5}{10} \), divide both 5 (numerator) and 10 (denominator) by their GCD, which is 5: \( \frac{5}{5} = 1 \) and \( \frac{10}{5} = 2 \).

Step 3: Write the Reduced Fraction

After dividing, you get the new numerator and denominator. Write them as a fraction. So, \( \frac{5}{10} \) reduced is \( \frac{1}{2} \).

Key Concepts

Greatest Common Divisor

The concept of the greatest common divisor (GCD) is a foundational element in simplifying fractions. It might sound complicated, but it's simply the largest number that evenly divides two or more numbers without any leftover.

• For example, if you have the numbers 5 and 10, you look at the divisors of each. The divisors of 5 are 1 and 5. For 10, they are 1, 2, 5, and 10.
• The greatest number in both lists is the GCD. Here, that's 5.

Finding the GCD is crucial as it helps simplify the fraction by reducing the numbers. Always remember, using the GCD helps you to find the simplest form of a fraction quickly and efficiently.When you simplify a fraction, like turning \( \frac{5}{10} \) into \( \frac{1}{2} \), the GCD has been used to transform it into a more manageable size, making it easier to understand and use.

Reducing Fractions

Reducing fractions means simplifying them to their lowest terms, where the numerator and denominator are as small as possible yet still have the same value as the original fraction.

• Take the fraction \( \frac{5}{10} \). Here, we identified its GCD, which was 5.
• By dividing the numerator and the denominator by 5, we get: \( \frac{5}{5} = 1 \) and \( \frac{10}{5} = 2 \).
• The simplified version of this fraction is \( \frac{1}{2} \).

Reducing fractions is particularly useful for keeping math as simple and comprehensible as possible. When preparing fractions for further calculations, having them in their simplest form makes your operations smoother and more intuitive.

Numerator and Denominator Division

The numerator and the denominator are two key components of any fraction. By understanding how to manipulate these parts, you can easily reduce any fraction.
The numerator is the top number, while the denominator is the bottom number of a fraction:

• In \( \frac{5}{10} \), 5 is the numerator, and 10 is the denominator.
• To simplify this fraction, use the GCD you found earlier which is 5.
• By dividing both 5 and 10 by the GCD (5), you transform the fraction into \( \frac{1}{2} \).

This division step is the crux of reducing fractions. It's like shrinking the size of the fraction components so they are at their most "simplified" form, while still representing the same value as the original fraction. This makes your math more straightforward!