Problem 57 - Prealgebra

Question

Find each product. Write in simplest form. $$\frac{3}{5} \cdot \frac{1}{3}$$

Step-by-Step Solution

Answer

The product is \( \frac{1}{5} \) in simplest form.

Step 1: Understand the Problem

We need to multiply two fractions, \( \frac{3}{5} \) and \( \frac{1}{3} \), and then simplify the resulting fraction to its simplest form.

Step 2: Multiply the Fractions

Multiply the numerators: \( 3 \times 1 = 3 \). Multiply the denominators: \( 5 \times 3 = 15 \). The resulting fraction is \( \frac{3}{15} \).

Step 3: Simplify the Fraction

To simplify \( \frac{3}{15} \), find the greatest common divisor (GCD) of 3 and 15, which is 3. Divide both the numerator and denominator by 3: \( \frac{3 \div 3}{15 \div 3} = \frac{1}{5} \).

Step 4: Verify the Simplification

Confirm that \( \frac{1}{5} \) is in its simplest form. Since 1 and 5 have no common divisors other than 1, \( \frac{1}{5} \) is indeed simplified.

Key Concepts

Simplifying Fractions

Fractions can sometimes look a bit complex, especially when they have big numbers. That's why simplifying them is a handy trick! When we simplify a fraction, we're looking to make it as straightforward as possible.

Take the fraction \( \frac{3}{15} \) from our example. We want to reduce it to a version that has smaller numbers, but that still means the same amount. Simplifying a fraction makes it easier to think about and easier to work with in math problems.

To do this, you need to look for a number called the Greatest Common Divisor, or GCD for short. The GCD is the biggest number that can be divided into both the top and bottom of the fraction without leaving a fraction behind. Once you find this number, simply divide the numerator (the top part) and the denominator (the bottom part) by this GCD. Apply these to your fractions and see them shrink into an easier form to handle!

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a key player in simplifying fractions. Think of it as the secret ingredient that makes fractions simpler. Here’s how it works:

• First, list out the factors for both numbers in the fraction. Factors are numbers that can divide another number cleanly, leaving no remainder.
• For example, in the fraction \(\frac{3}{15}\), the factors of 3 are 1, 3. The factors of 15 are 1, 3, 5, 15.
• Next, find the biggest number common in both lists. Here, it's 3.
• That number is your GCD, and it’s what you’ll use to simplify your fraction.

Once you've found this GCD, use it to divide both the numerator and the denominator. Simplifying \(\frac{3}{15}\) with a GCD of 3 gives us \(\frac{1}{5}\), which is much cleaner and easier to handle. It's like magic for numbers!

Numerators and Denominators

Whenever you're working with fractions, you'll often hear about numerators and denominators. These are the parts that make up every fraction and understanding them is essential.The **numerator** is the top number. It represents how many parts of a whole you have. For example, in \( \frac{3}{5} \), the 3 tells you there are 3 parts of the whole.

The **denominator** is the bottom number. It tells you into how many parts the whole is divided. So in \( \frac{3}{5} \), the 5 means the whole is split into 5 equal parts.

Why are these important? Because when multiplying fractions, like in our exercise, you multiply the two numerators together, and the two denominators together, to get a new fraction. But remember, after multiplying, you might need to simplify the result. That’s where recognizing and working with numerators and denominators becomes incredibly useful. Knowing these parts makes it much easier to follow the steps for multiplication and simplification.