Problem 93 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{3}{5}\right)^{2} \cdot \frac{20}{3}$$

Step-by-Step Solution

Answer

The product is \( \frac{12}{5} \).

Step 1: Square the Fraction

First, we need to square the fraction \( \left(\frac{3}{5}\right)^2 \). This means multiplying the fraction by itself: \( \frac{3}{5} \times \frac{3}{5} \). The result is \( \frac{3 \times 3}{5 \times 5} = \frac{9}{25} \).

Step 2: Multiply the Fractions

Now, multiply the squared result \( \frac{9}{25} \) by \( \frac{20}{3} \). The multiplication of fractions involves multiplying the numerators and the denominators separately: \( \frac{9 \times 20}{25 \times 3} = \frac{180}{75} \).

Step 3: Reduce the Fraction

To reduce the fraction \( \frac{180}{75} \), we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 180 and 75 is 15. Now, divide both the numerator and the denominator by 15: \( \frac{180 \div 15}{75 \div 15} = \frac{12}{5} \).

Key Concepts

Squaring Fractions

When it comes to squaring fractions, the process is similar to squaring whole numbers, but it involves multiplying the fraction by itself. Let's break it down: given a fraction \( \frac{3}{5} \), squaring it means you multiply \( \frac{3}{5} \) by \( \frac{3}{5} \) again. Here's how it goes:

• Multiply the numerators together. In this case, \( 3 \times 3 = 9 \).
• Multiply the denominators together. Here, \( 5 \times 5 = 25 \).

So, \( \left(\frac{3}{5}\right)^2 = \frac{9}{25} \). This is the new fraction after squaring. Keep in mind that squaring changes both the numerator and denominator, but the fraction's core concept remains intact: you've simply enhanced it by multiplication. Squaring fractions is a straightforward way to handle mathematical operations involving fractional exponents.

Multiplication of Fractions

Multiplying fractions is a breeze once you understand the process. It involves straightforward multiplication of the numerators and denominators across two fractions. Consider the expression \( \frac{9}{25} \times \frac{20}{3} \). Let's break this down:

• Multiply the top numbers (numerators): \( 9 \times 20 = 180 \).
• Multiply the bottom numbers (denominators): \( 25 \times 3 = 75 \).

Thus, the result of multiplying these fractions is \( \frac{180}{75} \). This process shows how clean and direct fraction multiplication can be, without any additional steps or conversions. It's simply a matter of handling basic multiplication and then simplifying the result if necessary.

Reducing Fractions

Once multiplication is complete, you might end up with a fraction that can be simplified or reduced. This makes the fraction easier to work with and often tidier in appearance. Reducing involves finding the greatest common divisor (GCD) of the numerator and denominator. Using our fraction \( \frac{180}{75} \), the steps to reduce are:

• Identify the GCD of 180 and 75, which is 15.
• Divide both the numerator and the denominator by their GCD: \( \frac{180 \div 15}{75 \div 15} \).

Performing this division, you arrive at \( \frac{12}{5} \). That's your reduced fraction, simpler and more elegant! Reducing fractions not only simplifies calculations but also allows for an easier comparison between fractions. It's an essential skill that ensures all resultant fractions are in their simplest and most recognizable form.