Problem 94 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{1}{4}\right)^{2} \cdot \frac{16}{15}$$

Step-by-Step Solution

Answer

The product is \(\frac{1}{15}\).

Step 1: Square the Fraction

First, we need to square the fraction \(\left(\frac{1}{4}\right)^2\). To do this, we raise both the numerator and the denominator to the power of 2. Thus, \(\left(\frac{1}{4}\right)^2 = \frac{1^2}{4^2} = \frac{1}{16}\).

Step 2: Multiply the Fractions

Now, multiply the squared fraction \(\frac{1}{16}\) by \(\frac{16}{15}\). When multiplying fractions, we multiply the numerators together and the denominators together: \(\frac{1}{16} \times \frac{16}{15} = \frac{1 \times 16}{16 \times 15} = \frac{16}{240}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{16}{240}\) by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 16 and 240 is 16. Divide both the numerator and the denominator by their GCD: \(\frac{16 \div 16}{240 \div 16} = \frac{1}{15}\).

Key Concepts

Squaring Fractions

Squaring a fraction is a simple process that involves multiplying the fraction by itself. To do this, you take both the numerator (the top number) and the denominator (the bottom number) and square each of them individually. For example, if you have the fraction \(\frac{1}{4}\), squaring it means calculating \(\left(\frac{1}{4}\right)^2\). This is equivalent to multiplying the numerator by itself, \(1 \times 1\), and doing the same with the denominator, \(4 \times 4\). Thus, \(\left(\frac{1}{4}\right)^2 = \frac{1^2}{4^2} = \frac{1}{16}\).

• Square the numerator: Just multiply the numerator by itself.
• Square the denominator: Do the same for the denominator.
• The squared fraction will have the squared numerator over the squared denominator.

After squaring the fraction, the product is still another fraction. This operation does not change the general structure of the fraction but it might lead to a simpler number depending on the values involved.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator are as small as possible. You simplify a fraction by dividing both its numerator and denominator by their greatest common divisor (GCD). For instance, when you reach \(\frac{16}{240}\), even though you've correctly multiplied and squared as needed, the fraction might not be reduced to its simplest form.
To simplify, you seek the greatest number that exactly divides both numbers. In this case, the GCD of 16 and 240 is 16. Dividing both the numerator and denominator by this number reduces the fraction to \(\frac{1}{15}\).

• Find the GCD of the numerator and denominator.
• Divide both by this number.
• The simplified fraction is much easier to understand and is its simplest form.

Simplifying not only makes fractions easier to work with in future calculations, but also clarifies comparisons between different fractions.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the highest number that can exactly divide two or more numbers without leaving a remainder. Finding the GCD is often key to simplifying fractions. When you want to simplify a fraction like \(\frac{16}{240}\), knowing the GCD can greatly assist in reducing this fraction to its simplest form.
To find the GCD, you might consider a few methods:

• **Listing Factors:** Write down all factors of each number and find the largest common one.
• **Prime Factorization:** Break down each number into its prime factors and choose the largest common set of these.
• **Euclidean Algorithm:** A more advanced method that repeatedly applies division to find the GCD.

For \(16\) and \(240\), listing factors gives us 1, 2, 4, 8, 16 for 16 and 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, etc., for 240, revealing that 16 is the largest number they have in common.
Understanding the GCD is important, not just for simplifying fractions but also in other areas of math like solving equations and factoring expressions.