Problem 1 - Prealgebra

Question

In Exercises 1-16, divide the numbers. \(\frac{39}{52}\)

Step-by-Step Solution

Answer

\(\frac{39}{52}\) simplifies to \(\frac{3}{4}\).

Step 1: Simplifying the Fraction

The first step in dividing\(\frac{39}{52}\)is to simplify it to its lowest terms by finding the greatest common divisor (GCD) of 39 and 52. The GCD of 39 and 52 is 13.

Step 2: Dividing Numerator and Denominator by GCD

Divide both the numerator and the denominator by their GCD, which is 13. Therefore,\(\frac{39}{52} = \frac{39 \div 13}{52 \div 13} = \frac{3}{4}\).

Key Concepts

Greatest Common Divisor

Fraction simplification often requires finding the Greatest Common Divisor (GCD). This number is the largest integer that divides both the numerator and the denominator without leaving a remainder. Determining the GCD is crucial because it helps reduce fractions to their simplest form. In our example with \(\frac{39}{52}\), the GCD is 13. Here's a simple way to find it:

• List the factors of 39: 1, 3, 13, 39
• List the factors of 52: 1, 2, 4, 13, 26, 52
• The greatest number common in both lists is 13.

With the GCD, you can accurately scale down both sides of the fraction and achieve a simplified version like \(\frac{3}{4}\). Remember, the GCD simplifies fractions efficiently, making calculations easier.

Numerator

In any fraction, the numerator is the number written above the fraction's division line. It represents how many parts of a whole are being considered. For instance, in \(\frac{39}{52}\), the numerator is 39. To simplify a fraction, knowing how to handle the numerator is essential.
The numerator plays a vital role when reducing the entire fraction. By dividing the numerator by the GCD, you effectively shrink down the fraction. For example:

• We calculated 39 divided by 13 to get 3.

This step ensures you maintain the correct ratio while simplifying the fraction. Keeping track of these numbers allows us to express complicated values more clearly as part of simpler fractions.

Denominator

The denominator, positioned below the line in a fraction, indicates how many equal parts the whole is divided into. In the fraction \(\frac{39}{52}\), the denominator is 52. Understanding the denominator's function is crucial while working to simplify a fraction.
To simplify, divide both the numerator and this denominator by their GCD. In our example:

• The denominator 52 divided by 13 results in 4!

Completing this division confirms the reduction process's accuracy, transforming the original fraction into a cleaner form of \(\frac{3}{4}\). This process is pivotal in simplifying fractions, making numbers manageable and easy to work with.