Problem 63 - Prealgebra

Question

Find quotient. Write in simplest form. \(\frac{2}{5} \cdot \frac{3}{4}\)

Step-by-Step Solution

Answer

The simplest form of the product is \(\frac{3}{10}\).

Step 1: Multiply the numerators

To multiply fractions, we start by multiplying the numerators of the two fractions. Here, multiply 2 by 3. \(2 imes 3 = 6\)

Step 2: Multiply the denominators

Next, multiply the denominators of the two fractions. Multiply 5 by 4. \(5 imes 4 = 20\)

Step 3: Combine the products

After calculating the products of the numerators and denominators, we write the fraction with these numbers: \(\frac{6}{20}\)

Step 4: Simplify the fraction

To simplify the fraction, find the greatest common divisor of 6 and 20, which is 2. Divide both the numerator and the denominator by 2. \(\frac{6 \div 2}{20 \div 2} = \frac{3}{10}\)

Key Concepts

Simplifying Fractions

Simplifying fractions is all about making things easier. Imagine having a huge candy bar and breaking it down into smaller, equal pieces to share with friends. That's similar to simplifying. We take a fraction and try to make it as simple as possible without changing its value.

For example, take the fraction \( \frac{6}{20} \). What we do is find if there's a number that both the top (numerator) and the bottom (denominator) can be evenly divided by. Doing this, we'll have a simpler fraction that's equivalent to the original. This simpler form is usually much easier to understand at a glance, just like smaller portions of a candy bar.

• A fraction is fully simplified when the only common factor between the numerator and denominator is 1.
• Always check if a fraction can be made simpler by checking the divisibility of numbers.

Greatest Common Divisor

To simplify fractions, we need to talk about the greatest common divisor (GCD). The GCD is the largest number that evenly divides both the numerator and denominator of a fraction. It's like finding the biggest piece that can fit perfectly into each part without leftovers.

Let's revisit \( \frac{6}{20} \). Here, the GCD of 6 and 20 is 2. This means both the numerator (6) and the denominator (20) can be divided by 2 without any leftovers.

• When simplifying a fraction, start by identifying the GCD of the numerator and denominator.
• Divide both by the GCD to achieve the simplest form of the fraction.
• This step ensures that the fraction is reduced to its simplest and most easy-to-read form.

Numerators and Denominators

Understanding numerators and denominators is key to gripping the whole concept of fractions. Let's break it down:

The numerator is the top number in a fraction, showing how many parts of a whole we have. Meanwhile, the denominator is the bottom number, indicating how many equal parts the whole is divided into. In the fraction \( \frac{2}{5} \), 2 is the numerator and 5 is the denominator.

• A higher numerator compared to the denominator represents more parts of the whole, and vice versa.
• In multiplication, numerators and denominators are multiplied separately to create a new fraction.
• Maintaining the relationship between numerators and denominators is vital for accuracy in fraction operations.

When multiplying fractions, both the numerators and denominators are multiplied separately, forming a new fraction. Always remember, these two parts together tell you precisely what portion of the whole you have.