Problem 35 - Prealgebra

Question

Find each product. Use an area model if necessary. $$1 \frac{3}{7}\left(-9 \frac{4}{5}\right)$$

Step-by-Step Solution

Answer

The product is \(-14\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For \(1 \frac{3}{7}\), multiply the whole number \(1\) by the denominator \(7\), and then add the numerator \(3\). Thus, \(1 \frac{3}{7}\) becomes \(\frac{10}{7}\). Similarly, convert \(-9 \frac{4}{5}\). Multiply the whole number \(-9\) by the denominator \(5\), and add the numerator \(4\). This becomes \(\frac{-49}{5}\). Now we have the expression \(\frac{10}{7} \times \frac{-49}{5}\).

Step 2: Multiply the Fractions

To multiply two fractions, multiply the numerators together and the denominators together. Multiply \(10 \times -49 = -490\) for the numerator, and \(7 \times 5 = 35\) for the denominator. This results in the fraction \(\frac{-490}{35}\).

Step 3: Simplify the Fraction

To simplify \(\frac{-490}{35}\), find the greatest common divisor (GCD) of \(490\) and \(35\). Both numbers are divisible by \(35\). Dividing the numerator \(-490\) by \(35\) gives \(-14\), and dividing the denominator \(35\) by \(35\) gives \(1\). Hence, \(\frac{-490}{35} = -14\).

Step 4: Interpret the Final Answer

Since the final result is a whole number, the product of \(1 \frac{3}{7}\) and \(-9 \frac{4}{5}\) is \(-14\).

Key Concepts

Understanding Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This makes their value equal to or greater than one. Here's why they matter:

• Improper fractions are more straightforward for calculations involving multiplication or division, avoiding the conversion back and forth with mixed numbers.
• In our problem, the mixed numbers have been converted to improper fractions for ease. For example, converting \(1 \frac{3}{7}\) into \(\frac{10}{7}\).
• This conversion involves multiplying the whole number by the denominator and adding the numerator, resulting in a numerator like in \(10\) for our example.

There's no need to shy away from improper fractions; they are just a different way to perceive quantities greater than one.

Using an Area Model

The area model is a visual representation that helps in understanding multiplication of larger numbers or fractions. It's like breaking down numbers into smaller chunks:

• Normally used in simpler scenarios, it involves drawing a rectangle where one side reflects one number and the other side reflects the second number.
• Each section of the rectangle then represents a partial multiplication product, which totals up to the final answer.
• Even though the exercise didn't primarily rely on this, understanding and drawing it helps in building a better mental image of how fractions multiply.

This model turns complex numeric multiplication into straightforward visual comprehension, reducing computation errors and increasing students' confidence.

The Role of Mixed Numbers

Mixed numbers are part of day-to-day communication with fractions, composed of a whole number and a proper fraction. They provide a clearer expression of quantities:

• In our daily life, mixed numbers are often more intuitive than improper fractions for understanding and communicating amounts.
• For example, \(1 \frac{3}{7}\) denotes one whole item plus a fraction, which gives a clear image of what the quantity represents, unlike the abstract \(\frac{10}{7}\).
• However, for calculations, mixed numbers need converting into improper fractions, as in the exercise, for easier arithmetic handling.

While they offer clear representation, during calculations, always switch them to improper fractions to simplify operations.