Problem 96 - Fundamentals of Mathematics

Question

Find \(\frac{8}{9}\) of \(\frac{27}{2}\).

Step-by-Step Solution

Answer

The solution is 12.

Step 1: Understand the Problem

We need to find a fraction of another fraction. This means that we are going to multiply the two fractions together.

Step 2: Set Up the Multiplication

Write down the fractions that need to be multiplied: \(\frac{8}{9}\) and \(\frac{27}{2}\). We will calculate \(\frac{8}{9} \times \frac{27}{2}\).

Step 3: Multiply the Numerators

Multiply the numerators of the two fractions together: \(8 \times 27 = 216\).

Step 4: Multiply the Denominators

Multiply the denominators of the two fractions together: \(9 \times 2 = 18\).

Step 5: Form the New Fraction

Using the results from the previous steps, form a new fraction: \(\frac{216}{18}\).

Step 6: Simplify the Fraction

Find the greatest common divisor of 216 and 18, which is 18. Divide both the numerator and the denominator by 18: \(\frac{216 \div 18}{18 \div 18} = \frac{12}{1} = 12\).

Key Concepts

Simplifying Fractions

Simplifying fractions means making them as simple as possible. The idea is to have the smallest numbers in the numerator and the denominator without changing the value of the fraction. This makes them easier to work with in mathematical problems.

To simplify a fraction:

• Look for common factors in both the numerator and the denominator.
• Divide both the top number and the bottom number by these common factors.

Choosing common factors that can divide both numbers without a remainder is key. This will give you a new, simpler fraction. Remember, the simpler the fraction, the less work you have to do with large numbers! When you solve exercises like finding a fraction of a fraction - for instance, finding \( \frac{8}{9} \times \frac{27}{2} \), simplifying the result makes the math easier to handle.

Numerator and Denominator

A fraction is made of two parts: the numerator and the denominator.

• The numerator is the top number. It shows how many parts we have.
• The denominator is the bottom number. It tells us how many total parts make up a whole.

For example, in the fraction \(\frac{8}{9}\), 8 is the numerator, and 9 is the denominator. It tells us that we have 8 parts out of a total of 9. The numerator and the denominator help us understand exactly how much of something we have and how it compares to the whole. Swapping these two numbers changes the value of the fraction entirely, which is why keeping them in the right order is vital to correctly representing the ratio they provide.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. It is useful when simplifying fractions, as it helps reduce them to their simplest form.

To find the GCD, you could:

• List all divisors of each number and find the greatest one they share.
• Use the Euclidean algorithm, an efficient method to find the GCD.

For instance, in the result \(\frac{216}{18}\) from multiplying the fractions \(\frac{8}{9}\) and \(\frac{27}{2}\), the GCD of 216 and 18 is 18. Dividing both the numerator and the denominator by their GCD simplifies the fraction to \(\frac{12}{1}\) or just 12. Finding the GCD is crucial in ensuring fractions are presented in the simplest and most concise way possible.