Problem 17 - Fundamentals of Mathematics

Question

Perform the following multiplications. Convert improper fractions to mixed numbers. $$2 \frac{2}{3} \cdot 2 \frac{1}{4}$$

Step-by-Step Solution

Answer

The product is 6.

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions.\For the mixed number \(2 \frac{2}{3}\), multiply the whole number \(2\) by the denominator \(3\) and add the numerator \(2\). This gives \((2 \times 3) + 2 = 6 + 2 = 8\). So, \(2 \frac{2}{3}\) becomes \(\frac{8}{3}\).\For the mixed number \(2 \frac{1}{4}\), multiply the whole number \(2\) by the denominator \(4\) and add the numerator \(1\). This gives \((2 \times 4) + 1 = 8 + 1 = 9\). So, \(2 \frac{1}{4}\) becomes \(\frac{9}{4}\).

Step 2: Multiply the Improper Fractions

Multiply the two improper fractions obtained: \(\frac{8}{3} \times \frac{9}{4}\). Multiply the numerators together and the denominators together.\\[\frac{8 \times 9}{3 \times 4} = \frac{72}{12}\]

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{72}{12}\). Divide the numerator and the denominator by their greatest common divisor, which is 12.\\[\frac{72 \div 12}{12 \div 12} = \frac{6}{1} = 6\]

Step 4: Convert Back to a Mixed Number (if needed)

Since \(6\) is a whole number, there is no need to convert back to a mixed number. The final product of the multiplication is \(6\).

Key Concepts

Improper Fractions

When we talk about improper fractions, we refer to fractions where the numerator is larger than the denominator. This means the fraction represents a value equal to or greater than 1.
For example, in the improper fraction \(\frac{8}{3}\), the numerator \(8\) is greater than the denominator \(3\). Improper fractions are especially handy when performing arithmetic operations like multiplication.
To convert a mixed number to an improper fraction, multiply the whole number by the fraction's denominator and then add the numerator. This procedure ensures the fraction accounts for both the whole and fractional parts of the number:

• Example: Convert \(2 \frac{2}{3}\) into an improper fraction:
  \((2 \times 3) + 2 = 8\)
  Result: \(\frac{8}{3}\)

Improper fractions make calculations easier, particularly for operations like multiplication or division.

Mixed Numbers

A mixed number is a combination of a whole number and a fraction, such as \(2 \frac{1}{4}\). Mixed numbers are useful in daily life, especially when dealing with measurements and quantities that aren't whole.
Mixed numbers can be converted into improper fractions to simplify computations, as you've seen in our example.
To do this, remember:

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator of the fraction.

For example, to convert \(2 \frac{1}{4}\):
\((2 \times 4) + 1 = 9\)
This means that \(2 \frac{1}{4}\) becomes \(\frac{9}{4}\) when expressed as an improper fraction. Converting back is just as simple: divide the numerator by the denominator to get the whole number, and the remainder becomes the new numerator.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two numbers without leaving a remainder. Finding the GCD is crucial for simplifying fractions. This step ensures the fraction is in its simplest form after calculations.
To determine the GCD of two numbers, you can use methods such as listing the divisors or utilizing the Euclidean algorithm. When simplifying the fraction \(\frac{72}{12}\), the GCD of 72 and 12 is 12 because it is the largest number that divides both exactly.

• List the divisors:
  Divisors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
  Divisors of 12: 1, 2, 3, 4, 6, 12

The common divisors are 1, 2, 3, 4, 6, and 12, with 12 being the largest. Using the GCD, we can simplify \(\frac{72}{12}\) by dividing both numerator and denominator by 12, resulting in \(\frac{6}{1}\) or 6.

Simplification of Fractions

The process of simplifying fractions involves reducing them to their simplest form. This means finding an equivalent fraction where the numerator and denominator have no common factors other than 1.
Simplifying fractions makes them easier to interpret and work with, especially in further calculations. After multiplying fractions and getting a product, simplifying ensures you're working with the smallest and most manageable form.

• To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD).
• For example, in our solution, we simplified \(\frac{72}{12}\) by dividing both by their GCD, which was 12, resulting in the simplified fraction \(\frac{6}{1}\).

This fraction translates to the whole number 6. Keeping fractions in a simplified form is crucial, making your numbers cleaner and arithmetic operations more straightforward.