Problem 38 - Prealgebra

Question

Simplify each fraction. $$\frac{78}{100}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{78}{100}\) is \(\frac{39}{50}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify a fraction, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). For the fraction \(\frac{78}{100}\), we first find the GCD of 78 and 100. The factors of 78 are 1, 2, 3, 6, 13, 26, 39, and 78. The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The common factors are 1 and 2. Therefore, the GCD is 2.

Step 2: Divide the Numerator and Denominator by the GCD

Now that we have the GCD, which is 2, we divide both the numerator and the denominator of the fraction \(\frac{78}{100}\) by 2. This gives us: \(\frac{78 \div 2}{100 \div 2} = \frac{39}{50}\).

Step 3: Simplify and Verify

After simplification, we have \(\frac{39}{50}\). To verify, we check if \(\frac{39}{50}\) can be simplified further. The common factors of 39 and 50 are only 1, confirming that \(\frac{39}{50}\) is in its simplest form.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. Imagine you have a pair of matching gloves, several pairs in fact, with different colors. To organize them neatly, you want to find the largest pair that is common among them. Similarly, the GCD is like finding the largest "pair" or number that divides two numbers (in our case, the numerator and denominator of the fraction). To find the GCD, follow these steps:

• List all factors of each number.
• Identify the common factors from both lists.
• Select the largest factor that both numbers share. That's your GCD!

For 78 and 100, the GCD is 2. This number helps us reduce the fraction to its simplest form by evenly dividing both the numerator and the denominator.

Numerator

The numerator is the number placed at the top of the fraction. It's like the part of a whole pizza that has already been eaten or the amount of a task that has been completed. It's essential to understand that the numerator represents what you're focusing on — in this exercise, that's the number 78.When simplifying a fraction, the numerator plays a crucial role. You divide this number by the GCD, as we've done with 78 using the GCD of 2. So, \( 78 \div 2 = 39 \). By dividing the numerator, you are essentially scaling down the fraction while maintaining its value.

Denominator

The denominator is the number at the bottom of the fraction. It tells you what the "whole" is — kind of like the entire pizza before you take any slices out. In any fraction, the denominator never can be zero, because you can't divide by zero. Here, our denominator is 100.In the simplification process, similar to the numerator, the denominator also gets divided by the GCD. For our example:\( 100 \div 2 = 50 \). This action adjusts the fraction, keeping its value consistent, while making it easier to understand and compare to other fractions.

Simplification Process

Simplifying fractions is about making them as simple as possible without changing their value. It's like shrinking a bulky sweater to fit better, only in this case, you're adjusting numbers!Here's how to tackle it:

• Find the GCD, as we did with 78 and 100, discovering it to be 2.
• Divide both the numerator and the denominator by this GCD.
• After division, you get a new fraction that is equivalent to the original: \( \frac{39}{50} \).
• Check if the new fraction can be further simplified. If the only common factor is 1, your work is done!

Following these steps ensures you've simplified the fraction, which helps maintain clarity and precision in mathematical expressions.