Problem 48 - Fundamentals of Mathematics

Question

For problems 48-60, reduce, if possible, each fraction. $$\frac{10}{25}$$

Step-by-Step Solution

Answer

The reduced form of \( \frac{10}{25} \) is \( \frac{2}{5} \).

Step 1: Identify the Greatest Common Divisor (GCD)

The first step in simplifying the fraction \( \frac{10}{25} \) is to find the greatest common divisor (GCD) of the numerator (10) and the denominator (25). The GCD is the largest number that evenly divides both numbers. To find the GCD, list the factors of 10: 1, 2, 5, 10, and the factors of 25: 1, 5, 25. The largest common factor between them is 5.

Step 2: Divide Both Numerator and Denominator by the GCD

Once the GCD is identified as 5, divide both the numerator and the denominator of \( \frac{10}{25} \) by 5. This results in: \( \frac{10 \div 5}{25 \div 5} = \frac{2}{5} \). This is the reduced form of the original fraction.

Key Concepts

Greatest Common Divisor

When simplifying fractions, one crucial step is identifying the greatest common divisor (GCD). The GCD is the largest whole number that can divide both the numerator and the denominator without leaving a remainder. To find this number, list the factors of each number:

• For the numerator 10, the factors are 1, 2, 5, and 10.
• For the denominator 25, the factors are 1, 5, and 25.

Next, determine which factors they have in common. In this case, both 10 and 25 share the factor 5. Since 5 is the largest of these shared factors, it is the greatest common divisor of 10 and 25. Using the GCD to reduce a fraction helps you arrive at its simplest form quickly.

Numerator

The numerator is the top number in a fraction and represents how many parts are being considered. For instance, in the fraction \( \frac{10}{25} \), the numerator is 10. This means we are looking at 10 parts of whatever whole is being described. When simplifying fractions, the numerator is divided by the greatest common divisor (GCD) to make the fraction easier to understand. For example, if you divide 10 by the GCD, which is 5, you simplify the numerator to 2. This division results in a simpler fraction, \( \frac{2}{5} \), which is easier to work with and interpret. It's essential to understand the role of the numerator in expressing a fraction as it helps comprehend the quantity we are dealing with.

Denominator

In a fraction, the denominator is the bottom number. This number indicates the total number of equal parts in the whole. For example, in the initial fraction \( \frac{10}{25} \), the denominator is 25. This tells us we have 10 parts out of a total of 25.When simplifying, you also divide the denominator by the greatest common divisor (GCD). In our example, dividing the denominator 25 by its GCD with 10, which is 5, simplifies it to 5.This results in the simplified fraction \( \frac{2}{5} \). A simpler denominator makes it easier to understand how many parts make up the whole. Remember, reducing a fraction does not change its value; it simply presents it in a more comprehensible form.