Problem 75 - Prealgebra

Question

Reduce to lowest terms. $$\frac{38}{100}$$

Step-by-Step Solution

Answer

The reduced form of \(\frac{38}{100}\) is \(\frac{19}{50}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction \(\frac{38}{100}\) to its lowest terms, first identify the greatest common divisor (GCD) of the numerator (38) and the denominator (100). The greatest common divisor is the largest number that divides both numbers without leaving a remainder. We find that both 38 and 100 are divisible by 2.

Step 2: Divide the Numerator and Denominator by the GCD

Now that we know the GCD is 2, divide the numerator and the denominator by the GCD to simplify the fraction: \(\frac{38 \div 2}{100 \div 2} = \frac{19}{50}\).

Step 3: Verify the Resulting Fraction

After simplifying, you get \(\frac{19}{50}\). Verify that 19 and 50 have no common divisors other than 1. Since 19 is a prime number and does not divide 50, \(\frac{19}{50}\) is in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

When we talk about the Greatest Common Divisor, or GCD, we mean the largest number that can divide two numbers evenly. By evenly, we mean without leaving any leftovers or remainders.
To figure out the GCD of any two numbers, we need to find the factors of each number. Factors are numbers that multiply together to give the original number.
For example, if we want to find the GCD of 38 and 100, we start by finding the factors:

• Factors of 38: 1, 2, 19, 38.
• Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.

The largest factor that appears in both lists is 2. This is how we know that the GCD of 38 and 100 is 2. With this information, we can simplify our fractions much more easily!

Prime Numbers

Prime numbers have a central role in simplifying fractions. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.
Think of prime numbers as the building blocks of all numbers. Common examples include 2, 3, 5, 7, 11, 13, 17, and 19.
In our exercise with the fraction \(\frac{38}{100}\), we noted that 19 is a prime number.

• Because 19 is prime, it cannot be divided by any number other than 1 and 19.

Knowing whether a number is prime is helpful when checking if a fraction is in its simplest form. Why? Because if the numerator is prime and does not divide the denominator, you are likely done with simplifying!

Simplifying Fractions

Simplifying fractions is about reducing them to the lowest form possible. This process helps us to express the fraction in its simplest and most understandable form.
To simplify a fraction, such as \(\frac{38}{100}\), the first step is finding the GCD, like we did with 2. Then, we divide both the numerator and the denominator by this number. So:

• 38 divided by 2 equals 19.
• 100 divided by 2 equals 50.

This gives us the new fraction \(\frac{19}{50}\). The last step is verifying that no other common divisors exist between the new numerator and the denominator. Since 19 is a prime number and does not divide into 50, this means \(\frac{19}{50}\) is the simplest form. By following these steps, you ensure that your fractions are always easy to understand and correctly reduced!