Problem 3 - Fundamentals of Mathematics

Question

\(\frac{11}{16}\) of \(\frac{8}{33}\) is what number?

Step-by-Step Solution

Answer

\(\frac{11}{16} \times \frac{8}{33} = \frac{1}{6}\).

Step 1: Understand the Problem

We have a problem that involves finding a fraction of another fraction. The problem asks us to find \( \frac{11}{16} \) of \( \frac{8}{33} \).

Step 2: Set Up the Multiplication

To find what \( \frac{11}{16} \) of \( \frac{8}{33} \) is, we need to multiply the two fractions together. This means setting up the multiplication: \[ \frac{11}{16} \times \frac{8}{33} \].

Step 3: Multiply the Numerators

We multiply the numerators of the fractions together. \( 11 \times 8 = 88 \).

Step 4: Multiply the Denominators

Now, we multiply the denominators of the fractions. \( 16 \times 33 = 528 \).

Step 5: Form the New Fraction

Combine the results from Steps 3 and 4 to form a new fraction: \( \frac{88}{528} \).

Step 6: Simplify the Fraction

To simplify \( \frac{88}{528} \), we find the greatest common divisor (GCD) of 88 and 528, which is 8. Divide both the numerator and the denominator by their GCD: \( \frac{88 \div 8}{528 \div 8} = \frac{11}{66} \).

Step 7: Verify the Simplified Form

Check if \( \frac{11}{66} \) can be simplified further. After verification, it is found that 11 is the GCD of 11 and 66. Simplify again to get: \( \frac{1}{6} \).

Key Concepts

Numerators and Denominators

Whenever you encounter a fraction, it consists of two essential parts: the numerator and the denominator. Both terms play a crucial role in the meaning and manipulation of a fraction. The numerator is the number above the fraction line. It tells us how many parts of a whole or group we have. In our exercise, for example, the numerators were 11 and 8.
The denominator is the number underneath the fraction line. It signifies the total number of equal parts into which the whole is divided. In our problem, the denominators were 16 and 33.

• The numerator indicates the number of pieces taken.
• The denominator shows how many pieces make up a whole.

Understanding numerators and denominators helps you visualize precisely what a fraction represents. This conceptual clarity is the first step in both multiplying fractions and simplifying them thereafter.

Simplifying Fractions

Simplifying fractions is all about making a fraction as simple as possible without changing its value. This is done by finding a common factor of both the numerator and the denominator that is greater than 1, and reducing them while keeping the fraction equivalent. In our example, after finding the product fraction \( \frac{88}{528} \), we need to simplify it.

• First, identify any common factors between the numerator and the denominator.
• Divide both by that factor to reduce the fraction.

For instance, the initial fraction \( \frac{88}{528} \) was simplified by dividing both the numerator and the denominator by their common factor, 8, to get \( \frac{11}{66} \). Then, by determining that 11 is also a common factor, you can achieve complete simplification, which is \( \frac{1}{6} \).
Simplifying not only makes fractions easier to understand but can also be essential in making subsequent mathematical operations much more straightforward.

Greatest Common Divisor

The greatest common divisor (GCD) is key when you're simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the GCD allows us to reduce fractions to their simplest form efficiently. Here's how it assists fraction simplification:

• Identify all divisors of the numerator and the denominator.
• Determine the largest number common to both sets of divisors.

In our solved exercise, the GCD was crucial in simplifying \( \frac{88}{528} \). Initially, we identified 8 as the GCD, simplifying to \( \frac{11}{66} \). Yet, further inspection showed another GCD of 11, leading to the simplest form, \( \frac{1}{6} \).
Using the GCD will save you time and effort, ensuring your fraction work is both accurate and as simple as possible.