Problem 70 - Fundamentals of Mathematics

Question

(Section 4.5) Find \(\frac{8}{9}\) of \(6 \frac{3}{4}\).

Step-by-Step Solution

Answer

The result is 6.

Step 1: Convert the Mixed Number to an Improper Fraction

The mixed number given is \(6 \frac{3}{4}\). First, convert it into an improper fraction. Multiply the whole number \(6\) by the denominator \(4\), then add the numerator \(3\). Thus, \(6 \times 4 + 3 = 27\). The improper fraction is \(\frac{27}{4}\).

Step 2: Multiply the Fractions

We need to find \(\frac{8}{9}\) of \(\frac{27}{4}\). This is done by multiplying the two fractions. \(\frac{8}{9} \times \frac{27}{4}\).

Step 3: Simplify Before Multiplying

Before multiplying, check if any numbers can be simplified. The 8 and 4 can be simplified: \(8 \div 4 = 2\) and \(4 \div 4 = 1\). Similarly, divide 27 and 9 by 9: \(27 \div 9 = 3\) and \(9 \div 9 = 1\). The problem now is \(\frac{2}{1} \times \frac{3}{1}\).

Step 4: Perform the Multiplication

With simplifications done, multiply the numerators and the denominators: \(2 \times 3 = 6\) and \(1 \times 1 = 1\). The result is \(\frac{6}{1}\) or simply \(6\).

Step 5: Confirm the Simplified Result

Since \(\frac{6}{1}\) simplifies to 6, the final answer is correct.

Key Concepts

mixed numbers

A mixed number is a combination of a whole number and a fraction, such as \(6 \frac{3}{4}\). It's a way of expressing a number that is greater than a whole number but not an integer. The whole number 6 in our mixed number represents complete units, while the fraction \(\frac{3}{4}\) represents an additional fractional part.

• Whole Numbers: The number before the fraction.
• Fractions: The part of the mixed number that is less than one.

To work with mixed numbers, like performing multiplication involving them, it’s often easier to convert them into improper fractions first. This makes mathematical operations more straightforward.

improper fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. For example, the mixed number \(6 \frac{3}{4}\) converts to the improper fraction \(\frac{27}{4}\). This happens when you multiply the whole number (in this case, 6) by the fraction's denominator (4), and then add the fraction's numerator (3) to get 27.

The conversion steps look like this:

• Multiply the whole number and the denominator: \(6 \times 4 = 24\)
• Add the numerator: \(24 + 3 = 27\)
• This results in \(\frac{27}{4}\)

Improper fractions are useful because they allow you to perform operations like multiplication directly without needing separate steps for the whole and fractional parts.

multiplying fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. Let’s take the problem \(\frac{8}{9} \times \frac{27}{4}\) as an example. Before jumping into multiplication, it's always a good idea to simplify first to make the process easier.
Here's how simplifying works:

• Check if any numerator and denominator can be divided by the same number.
• In this case, divide both 8 and 4 by 4, and 27 and 9 by 9.
• This gives us \(\frac{2}{1} \times \frac{3}{1}\).

After simplifying, you simply multiply across:

• Numerators: \(2 \times 3 = 6\)
• Denominators: \(1 \times 1 = 1\)

So, \(\frac{6}{1}\) simplifies to 6. Simplifying before multiplying reduces complex steps and potential errors, making it a smooth task.