Problem 85 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$9 \frac{1}{3} \cdot \frac{9}{16} \cdot 1 \frac{1}{3}$$

Step-by-Step Solution

Answer

The product is 7.

Step 1: Convert Mixed Numbers to Improper Fractions

First, let's convert the mixed numbers into improper fractions. For \(9 \frac{1}{3}\), the improper fraction is calculated as \((9 \times 3) + 1 = 28\), giving \(\frac{28}{3}\).For \(1 \frac{1}{3}\), the improper fraction is \((1 \times 3) + 1 = 4\), giving \(\frac{4}{3}\).

Step 2: Set Up the Multiplication

Now, we set up the multiplication of the three fractions:\[\frac{28}{3} \cdot \frac{9}{16} \cdot \frac{4}{3}\]

Step 3: Multiply the Numerators

Multiply all the numerators together:\[28 \cdot 9 \cdot 4 = 1008\]

Step 4: Multiply the Denominators

Multiply all the denominators together:\[3 \cdot 16 \cdot 3 = 144\]

Step 5: Form the Product Fraction

Now, we form the fraction using the products of the numerators and the denominators:\[\frac{1008}{144}\]

Step 6: Simplify the Fraction

Find the greatest common divisor (GCD) of 1008 and 144, which is 144. Divide both the numerator and denominator by 144:\[\frac{1008 \div 144}{144 \div 144} = \frac{7}{1}\]

Step 7: Express as a Whole Number

Since \(\frac{7}{1}\) is equivalent to 7, the final answer is 7.

Key Concepts

Mixed Numbers

Mixed numbers can be a little confusing at first, but with practice, they become straightforward to work with. A mixed number consists of a whole number and a proper fraction. In the exercise, we saw numbers like \(9 \frac{1}{3}\) and \(1 \frac{1}{3}\). Here:

• \(9\) and \(1\) are the whole numbers.
• \(\frac{1}{3}\) is the fraction part.

To work with these in calculations, mixed numbers are often converted to improper fractions. This conversion is essential because improper fractions are easier to multiply. To convert, multiply the whole number by the fraction's denominator, then add the numerator. This becomes the new numerator, giving us an improper fraction.

Improper Fractions

Improper fractions sound odd because they have numerators that are larger than their denominators. For example, the improper fractions we got from the mixed numbers earlier were \(\frac{28}{3}\) and \(\frac{4}{3}\). Here's why they're useful:

• They make multiplication and division of fractions more straightforward than using mixed numbers right away.
• An improper fraction basically tells us how many whole parts we have and some leftovers.

To find these improper fractions, remember to convert the mixed number using the method described earlier. Ensure your calculations are right to achieve accurate results, and don't forget: the improper fractions eventually need to be simplified to give the final answer.

Simplifying Fractions

Simplifying fractions is about reducing the fraction to its smallest form. The goal is to make it as simple as possible to understand and communicate. Once we've multiplied fractions, as in the exercise, we had \(\frac{1008}{144}\). This fraction forms one last essential step in solving fraction problems: simplifying.

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In our example, \(144\) was the GCD for both \(1008\) and \(144\), simplifying \(\frac{1008}{144}\) to \(\frac{7}{1}\), which is 7. This step is crucial: it neatly wraps up your calculation and offers the simplest expression of your answer.