Problem 55 - Prealgebra

Question

Perform the indicated operations. $$\frac{2}{3} \cdot \frac{3}{4} \div \frac{5}{8}$$

Step-by-Step Solution

Answer

The result of the operation is \( \frac{4}{5} \).

Step 1: Perform the Multiplication

The first operation we need to perform is the multiplication. Multiply the fractions \( \frac{2}{3} \) and \( \frac{3}{4} \). To do this, multiply the numerators and multiply the denominators: \( \frac{2 \times 3}{3 \times 4} = \frac{6}{12} \). Simplify \( \frac{6}{12} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 6: \( \frac{6 \div 6}{12 \div 6} = \frac{1}{2} \).

Step 2: Interpret the Division

Now, take the result from Step 1, \( \frac{1}{2} \), and divide by \( \frac{5}{8} \). Division of fractions is equivalent to multiplying by the reciprocal. The reciprocal of \( \frac{5}{8} \) is \( \frac{8}{5} \). So, the operation becomes \( \frac{1}{2} \times \frac{8}{5} \).

Step 3: Perform the Multiplication with Reciprocal

Multiply the result from Step 1, \( \frac{1}{2} \), by the reciprocal of the divisor, \( \frac{8}{5} \). Multiply the numerators, \( 1 \times 8 = 8 \), and the denominators, \( 2 \times 5 = 10 \), giving us \( \frac{8}{10} \). Simplify \( \frac{8}{10} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \( \frac{8 \div 2}{10 \div 2} = \frac{4}{5} \).

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward operation. Here's how it works:

• Multiply the numerators (the numbers on top of the fraction).
• Multiply the denominators (the numbers on the bottom of the fraction).

So, if you are given fractions \( \frac{2}{3} \) and \( \frac{3}{4} \), you multiply the numerators \(2 \times 3\) to get \(6\). Then multiply the denominators \(3 \times 4\) to get \(12\).
After multiplying, you form a new fraction, \( \frac{6}{12} \). The process is complete for multiplication, but there's usually one more important step to finish things off. That's simplifying the fraction, which we'll talk about soon. But remember, the key to multiplying fractions is multiply straight across — numerator with numerator, and denominator with denominator.

Dividing Fractions

Dividing fractions can seem tricky, but it's simple once you know the key idea: flip and multiply. Here's what you do:

• Flip the second fraction (also known as finding the reciprocal)
• Change the division sign to multiplication
• Multiply the fractions

For example, if you have \( \frac{1}{2} \) divided by \( \frac{5}{8} \), you take the reciprocal of \( \frac{5}{8} \) to get \( \frac{8}{5} \). Then, instead of dividing, you multiply \( \frac{1}{2} \times \frac{8}{5} \). From here, it's just like multiplying fractions: \( 1 \times 8 = 8 \) for the numerator, and \( 2 \times 5 = 10 \) for the denominator, resulting in the fraction \( \frac{8}{10} \).
It's a clever conversion that simplifies the process substantially. Always remember: flip the fraction you're dividing by and multiply, and you're off to the correct path.

Simplifying Fractions

Simplifying fractions is all about finding the smallest, most reduced form. To do this, you look for the greatest common divisor (GCD) of the numerator and the denominator. Here's a simple guide:

• Find the GCD of the two numbers.
• Divide both the numerator and the denominator by this common divisor.

Let's break this down with an example: for the fraction \( \frac{6}{12} \), both 6 and 12 are divisible by 6. So you divide the numerator and the denominator by their GCD, 6, to get \( \frac{1}{2} \).
Another example with \( \frac{8}{10} \), where the GCD is 2. By dividing both the numerator and denominator by 2, you simplify \( \frac{8}{10} \) to \( \frac{4}{5} \).
By reducing fractions, you make them cleaner and easier to work with in equations. Simplifying is not just a final touch; it's an essential part of working with fractions in mathematics.