Problem 42 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{3 y}{8} \div \frac{3}{2}\)

Step-by-Step Solution

Answer

The simplified result is \( \frac{y}{4} \).

Step 1: Understand Division of Fractions

When dividing fractions, such as \( \frac{3y}{8} \div \frac{3}{2} \), remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, our expression becomes \( \frac{3y}{8} \times \frac{2}{3} \).

Step 2: Multiply Fractions

Now, multiply the numerators and the denominators. For the numerators, \(3y \times 2 = 6y\). For the denominators, \(8 \times 3 = 24\). This gives us the fraction \(\frac{6y}{24}\).

Step 3: Simplify the Fraction

To simplify \(\frac{6y}{24}\), find the greatest common divisor (GCD) of 6 and 24, which is 6 in this case. Divide both your numerator and denominator by 6. This results in \(\frac{y}{4}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a key skill in algebra that makes working with numbers and expressions much easier. When you have a fraction, simplifying it involves reducing it to its smallest or simplest form. This means making both the numerator and the denominator as small as possible while maintaining the equivalency of the fraction. For example, if you end up with \(\frac{6y}{24}\) after a calculation, you simplify it by finding a number that divides both, which in this case is 6.
To simplify:

• Identify the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide the numerator and the denominator by the GCD to reduce the fraction to its simplest form.

So, dividing both 6y and 24 by 6, you get \(\frac{y}{4}\), which is much easier to work with.

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in simplifying fractions. It's the largest number that can evenly divide both the numerator and the denominator of a fraction. In the case of our example with the fraction \(\frac{6y}{24}\), the GCD is 6.
To find the GCD, you can:

• List all the factors of both numbers.
• Find the highest factor that appears in both lists.

Let's look at another way. You could use the Euclidean algorithm, which is a systematic way to find the GCD of two numbers by repeatedly subtracting the smaller number from the larger one until you reach zero. However, for small numbers, simply listing factors is often quicker and easier.

Reciprocal in Fractions

The concept of a reciprocal is crucial for dividing fractions. A reciprocal is what you get when you flip a fraction upside down. So, the reciprocal of a fraction like \(\frac{3}{2}\) is \(\frac{2}{3}\).
When you're dividing fractions, rather than doing division directly, you multiply by the reciprocal of the second fraction. This turns division into multiplication, which is a more straightforward operation. Here's how it works:

• Replace the division sign with a multiplication sign.
• Flip the second fraction to get its reciprocal.
• Proceed with multiplying the numerators and the denominators as you would normally do in multiplication.

This process simplifies the problem and makes division of fractions manageable through multiplication.