Problem 54 - Algebra 1

Question

$$ 3 \frac{2}{3} \cdot 3 $$

Step-by-Step Solution

Answer

The result of the multiplication \(3 \frac{2}{3} \cdot 3\) is 33.

Step 1: Convert mixed number to improper fraction

Firstly, convert the mixed number \(3 \frac{2}{3}\) to an improper fraction. To do this, multiply the whole number by the denominator of the fraction and add the numerator. This will give the numerator of the improper fraction. The denominator remains as is. In this case, the result is: \(3 \cdot 3 + 2 = 11\), so the mixed number converts to \(\frac{11}{3}\).

Step 2: Multiplication with integer

Now, multiply this improper fraction \(\frac{11}{3}\) with the integer 3. This can be done as follows: \(\frac{11}{3} \cdot 3 = \frac{11 \cdot 3}{3}\). Because the same number (3) appears in the numerator and denominator, they cancel out given term as result.

Step 3: Simplify product

Finally, simplify the expression \(\frac{11 \cdot 3}{3}\) to get the final answer. Multiplication in the numerator leads to 33. As mentioned before, the 3 from the numerator and the 3 from the denominator cancel out, leading to a final result of 33.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator. These types of fractions are crucial when working with mixed numbers, especially in operations like addition or multiplication.To convert a mixed number into an improper fraction, you need to:

• Multiply the whole number part by the denominator of the fractional part.
• Add the numerator of the fractional part to the result of the multiplication.
• Place this sum over the original denominator to obtain the improper fraction.

For example, in the case of converting the mixed number \(3 \frac{2}{3}\):
- Multiply 3 (the whole number) by 3 (the denominator): \(3 \times 3 = 9\)
- Add the numerator of the fraction part, 2: \(9 + 2 = 11\)
- Thus, the improper fraction is \(\frac{11}{3}\).
This improper fraction represents the same quantity as the mixed number, making calculations uniform and often simpler.

Simplification

Simplification is a process used to make fractions or expressions as simple as possible. When you simplify, you reduce the complexity of a fraction but not its value. For fractions, it means reducing the numerator and denominator to their smallest exit value where they have no common factor other than 1. In multiplication involving improper fractions, simplification can often occur during multiplication or division.
Let’s take a step-by-step simplification:

• You first perform multiplication in the numerator.
• Check if the numerator and denominator share any common factors, and divide them by the greatest common factor.

As seen in the example, when multiplying \(\frac{11}{3} \times 3\):
- Multiply to get \(\frac{33}{3}\).
- The numbers, 33 and 3, share a common factor 3.
- Divide both the numerator and denominator by 3: \(\frac{33 \div 3}{3 \div 3} = 11\).
This gives you the simplified result, 11, from the original multiplication, showing how powerful simplification is to get clean results.

Mixed Numbers Conversion

Mixed numbers combine whole numbers and fractions. Converting them to improper fractions is common in mathematical operations because improper fractions are easier to multiply or divide.To convert from a mixed number, the process is:

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

This step turns a mixed number into an improper fraction, which allows for straightforward calculations.
Take \(3 \frac{2}{3}\) as an example:
- Multiply the whole number part (3) by the denominator of the fraction (3): \(3 \times 3 = 9\).
- Add the product to the numerator (2): \(9 + 2 = 11\).
- The improper fraction is \(\frac{11}{3}\).
Return to a mixed number when necessary such as after division. Here, you would reverse the process to see how many whole parts fit into the improper fraction.