Problem 12 - Elementary Algebra

Question

For exercises 1-12, simplify. $$ \frac{38}{57} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{2}{3}\).

Step 1 - Identify the greatest common divisor (GCD)

To simplify the fraction \(\frac{38}{57}\), find the greatest common divisor (GCD) of the numerator (38) and the denominator (57). This is the largest number that divides both 38 and 57 without leaving a remainder. The GCD of 38 and 57 is 19.

Step 2 - Divide numerator and denominator by GCD

Divide both the numerator and the denominator by the GCD (19). \(\frac{38 \/ 19}{57 \/ 19} = \frac{2}{3}\).

Step 3 - Write the simplified fraction

After dividing both the numerator and the denominator by 19, the fraction simplifies to \(\frac{2}{3}\).

Key Concepts

Greatest Common Divisor

To simplify a fraction, we first need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder.

For example, to simplify \(\frac{38}{57}\), we need the GCD of 38 and 57.

We can find the GCD using the following steps:

• List the factors of each number. The factors of 38 are 1, 2, 19, and 38. The factors of 57 are 1, 3, 19, and 57.
• Identify the largest common factor. In this case, it's 19.

So, the GCD of 38 and 57 is 19. This is important because we will use it to determine how to simplify the fraction.

Numerator and Denominator

A fraction consists of two parts: the numerator and the denominator.

The numerator is the top number, which represents how many parts we have. The denominator is the bottom number, which represents the total number of equal parts the whole is divided into.

In our example, \(\frac{38}{57}\), 38 is the numerator and 57 is the denominator.

Understanding these parts is essential for simplifying fractions because we need to manipulate both the numerator and the denominator similarly to maintain the value of the fraction.

Fraction Simplification

Simplifying a fraction means making the fraction as simple as possible by ensuring the numerator and denominator are as small as possible while retaining the same value.

We can achieve this by dividing both the numerator and the denominator by their GCD.

Let's apply this to our fraction \(\frac{38}{57}\):

• We already know the GCD of 38 and 57 is 19.
• Divide the numerator by 19: \(\frac{38}{19} = 2\).
• Divide the denominator by 19: \(\frac{57}{19} = 3\).

When we perform these divisions, we get the simplified fraction: \(\frac{2}{3}\).

Now, \(\frac{2}{3}\) is in its simplest form, and it shows that 38 parts out of 57 are equivalent to 2 parts out of 3.