Problem 34 - Prealgebra

Question

Multiply. $$7 \frac{1}{7} \times 2 \frac{1}{3}$$

Step-by-Step Solution

Answer

The product is \(16 \frac{2}{3}\).

Step 1: Convert Mixed Numbers to Improper Fractions

To solve this problem, we first need to convert the mixed numbers to improper fractions. The mixed number \(7 \frac{1}{7}\) can be written as an improper fraction by multiplying the whole number (7) by the denominator (7) and adding the numerator (1). This gives:\[7 \frac{1}{7} = \frac{7 \times 7 + 1}{7} = \frac{49 + 1}{7} = \frac{50}{7}\]Similarly, convert \(2 \frac{1}{3}\):\[2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}\]

Step 2: Multiply the Improper Fractions

Now that we have improper fractions, we can multiply them. Multiply the numerators together and the denominators together:\[\frac{50}{7} \times \frac{7}{3} = \frac{50 \times 7}{7 \times 3}\]This simplifies to:\[\frac{350}{21}\]

Step 3: Simplify the Resulting Fraction

To simplify \(\frac{350}{21}\), we need to find the greatest common divisor (GCD) of 350 and 21. The GCD of 350 and 21 is 7. Divide both the numerator and the denominator by their GCD:\[\frac{350 \div 7}{21 \div 7} = \frac{50}{3}\]

Step 4: Convert the Improper Fraction to a Mixed Number

Convert \(\frac{50}{3}\) to a mixed number by dividing the numerator by the denominator:Divide 50 by 3, which gives 16 with a remainder of 2. This can be expressed as:\[16 \frac{2}{3}\]

Key Concepts

Improper Fractions

Improper fractions are a bit different from what we're used to. An improper fraction has a larger numerator (top number) than its denominator (bottom number). Basically, it's saying that you've got more parts than the fractional size. For example, \(\frac{50}{7}\) means you have fifty parts, each of the size one-seventh. It's super important to know how to convert mixed numbers—which combine whole numbers and fractions—into these improper fractions.

To convert, multiply the whole number by the fraction's denominator and then add the numerator. The outcome is your new numerator, and your denominator remains unchanged. So, \(7 \frac{1}{7}\) becomes \(\frac{50}{7}\). This step is essential for operations like multiplication because you can't multiply mixed numbers directly, only improper fractions.

Greatest Common Divisor

The greatest common divisor, or GCD for short, is a key concept when you're trying to simplify fractions. It's the largest number that divides both the numerator and the denominator perfectly, without leaving a remainder. Finding the GCD helps in bringing fractions down to their simplest form, making them easier to work with and understand.

Here's how to find it efficiently:

• List the factors of both numbers.
• Identify the largest factor common to both lists.

For instance, to simplify \(\frac{350}{21}\), we find that 7 is the largest number that can divide both 350 and 21 cleanly. Once you identify GCD as 7, you divide both the numerator and the denominator by this number to simplify the fraction to \(\frac{50}{3}\).

Simplifying Fractions

Simplifying fractions makes them more manageable. When a fraction is expressed in its simplest form, it's easier to work with in calculations and understand conceptually. Simplifying is often the final step achieved using the GCD.

Once you identify the GCD, divide both the numerator and denominator by this number. Simplifying \(\frac{350}{21}\) with GCD 7, gives you \(\frac{50}{3}\). Remember, a simplified fraction is not just tidier; it's essential for further steps like converting back to a mixed number for more accurate results in practical situations.

Simplicity in fractions helps avoid errors in larger calculations and makes comprehension effortless. Simplifying fractions is a fundamental skill, and practicing it frequently will help you ace math problems with ease and confidence.