Problem 60 - Prealgebra

Question

Find each product. Write in simplest form. $$-\frac{5}{12} \cdot 1 \frac{1}{7}$$

Step-by-Step Solution

Answer

The product is \(-\frac{10}{21}\).

Step 1: Convert Mixed Number to Improper Fraction

The mixed number is \(1 \frac{1}{7}\). To convert it to an improper fraction, multiply the whole number (1) by the denominator (7) and add the numerator (1). Thus, \(1 \times 7 + 1 = 8\). So, \(1 \frac{1}{7}\) becomes \(\frac{8}{7}\).

Step 2: Multiply the Fractions

Now, multiply the fractions \(-\frac{5}{12}\) and \(\frac{8}{7}\). Use the formula for multiplying fractions: \(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\).This gives, \[ -\frac{5}{12} \cdot \frac{8}{7} = \frac{-5 \cdot 8}{12 \cdot 7} = \frac{-40}{84} \].

Step 3: Simplify the Fraction

To simplify \(\frac{-40}{84}\), find the greatest common divisor (GCD) of 40 and 84.Both 40 and 84 are divisible by 4, which is their GCD.Divide the numerator and the denominator by 4: \[ \frac{-40 \div 4}{84 \div 4} = \frac{-10}{21} \].So, the simplest form is \(\frac{-10}{21}\).

Key Concepts

Understanding Improper Fractions

Improper fractions are a specific type of fraction where the numerator is greater than or equal to the denominator. This means the fraction represents a value greater than or equal to one. For instance, in the fraction \( \frac{8}{7} \), 8 is the numerator, and 7 is the denominator.
When dealing with improper fractions:

• They often come from converting mixed numbers to a single fraction format.
• Improper fractions are crucial when performing operations such as multiplication, as they allow for consistent application of arithmetic rules.
• They're particularly useful for calculations because they eliminate the complexities associated with mixed numbers.

By understanding how to identify and work with improper fractions, you can simplify many steps in arithmetic problems, especially those involving multiplication or calculation.

Transitioning Between Mixed Numbers and Improper Fractions

Mixed numbers combine a whole number with a fraction, like \( 1 \frac{1}{7} \). They're often used in everyday contexts where the value isn't a whole number, but a fraction of it is needed.
To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the resulting product to the numerator.
• Place this sum over the original denominator.

For example, converting \( 1 \frac{1}{7} \) involves multiplying 1 (the whole number) by 7 (the denominator) and adding 1 (the numerator). This results in the improper fraction \( \frac{8}{7} \).
Conversely, you can convert an improper fraction back to a mixed number by dividing the numerator by the denominator, where the quotient is the whole number, and the remainder is the new numerator of the fraction.

Simplifying Fractions: A Key Skill

Simplifying fractions is about reducing the fraction to its lowest terms. This process makes fractions easier to read and use, particularly in arithmetic operations like multiplication or division.
Here's how to simplify:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.
• This results in the simplest form of the fraction, where no number but 1 can evenly divide both the numerator and denominator again.

Consider simplifying \( \frac{-40}{84} \). To do this, recognize that both 40 and 84 are divisible by 4, which is the GCD. Dividing both by 4 gives \( \frac{-10}{21} \), the simplest form.
Practicing this skill helps in advancing mathematical proficiency and is especially useful for algebraic operations.