Problem 74 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{5}{6} \cdot 10$$

Step-by-Step Solution

Answer

The product is \( 8 \frac{1}{3} \).

Step 1: Write the expression as a multiplication of fractions

First, write the number 10 as a fraction. Any whole number can be written as a fraction by placing it over 1. So, the expression becomes \( \frac{5}{6} \times \frac{10}{1} \).

Step 2: Multiply the numerators

Multiply the numerators of the fractions together. Here it's 5 multiplied by 10, which equals 50: \[ 5 \times 10 = 50 \]

Step 3: Multiply the denominators

Multiply the denominators of the fractions together. Here, it's 6 multiplied by 1, which equals 6: \[ 6 \times 1 = 6 \]

Step 4: Write the result as a fraction

Now, write the result of the numerator over the result of the denominator. This gives you the fraction: \[ \frac{50}{6} \]

Step 5: Simplify the fraction

To simplify \( \frac{50}{6} \), find the greatest common divisor (GCD) of 50 and 6. The GCD is 2. Divide both numerator and denominator by 2:\[ \frac{50 \div 2}{6 \div 2} = \frac{25}{3} \]

Step 6: Convert improper fraction to mixed number (if needed)

Since \( \frac{25}{3} \) is an improper fraction, convert it to a mixed number. Divide 25 by 3:25 ÷ 3 = 8 remainder 1, so the mixed number is:\[ 8 \frac{1}{3} \]

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator, which is the top number, is greater than or equal to the denominator, the bottom number. For example, in the fraction \(\frac{25}{3}\), 25 is greater than 3, making it an improper fraction.

It's important to understand improper fractions because they can often arise from multiplication or division of fractions. They represent quantities greater than one whole. In the context of fraction multiplication, you might start with regular fractions, but the result can be an improper fraction, just like in our example where \(\frac{5}{6} \times \frac{10}{1}\) becomes \(\frac{50}{6}\), and eventually \(\frac{25}{3}\).

When dealing with improper fractions, you might need to convert them into mixed numbers for better understanding, but it's crucial first to ensure any fraction is simplified.

Mixed Numbers

Mixed numbers are combinations of a whole number and a fraction. They offer a way to express improper fractions in a more understandable form. For instance, the improper fraction \(\frac{25}{3}\) can be converted into the mixed number \(8\frac{1}{3}\).

To convert an improper fraction to a mixed number, simply divide the numerator by the denominator. The quotient becomes the whole number part, and any remainder forms the new numerator for the fractional part. The original denominator remains the same. This allows you to see both the whole and fractional portions of the number.

• For \(\frac{25}{3}\), divide 25 by 3 to get 8, with a remainder of 1.
• This becomes \(8\frac{1}{3}\), meaning 8 whole parts and one third of another part.

Understanding mixed numbers can aid in visualizing fractions, especially when adding or subtracting them in your calculations.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. You do this by dividing the numerator and the denominator by their greatest common divisor (GCD), the largest number that divides both evenly.

For example, the fraction \(\frac{50}{6}\) can be simplified by identifying the GCD. Here, the GCD of 50 and 6 is 2. By dividing both the numerator and denominator by 2, you get \(\frac{25}{3}\). Simplifying helps in reducing fractions to a form that is easier to understand and work with.

• Identify the GCD of the numerator and denominator.
• Divide both by the GCD.
• Rewrite the fraction with the result.

Always ensuring fractions are simplified makes arithmetic operations much simpler and the results more clear, especially when dealing with multiples or comparisons.