Problem 59 - Prealgebra

Question

Find each product. Write in simplest form. $$-2 \frac{4}{5} \cdot \frac{3}{8}$$

Step-by-Step Solution

Answer

The simplest form of the product is \(-\frac{9}{20}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(-2\frac{4}{5}\) into an improper fraction. Multiply the whole number by the denominator and add the numerator: \(-2 \times 5 + 4 = -10 + 4 = -6\). Therefore, the improper fraction is \(-\frac{6}{5}\).

Step 2: Multiply the Fractions

Next, multiply the two fractions: \(-\frac{6}{5} \cdot \frac{3}{8}\). Multiply the numerators: \(-6 \times 3 = -18\). Multiply the denominators: \(5 \times 8 = 40\). This gives \(-\frac{18}{40}\).

Step 3: Simplify the Fraction

To simplify \(-\frac{18}{40}\), find the greatest common divisor (GCD) of 18 and 40, which is 2. Divide both the numerator and the denominator by 2: \(-\frac{18 \div 2}{40 \div 2} = -\frac{9}{20}\).

Step 4: Solution Confirmation

Finally, confirm that \(-\frac{9}{20}\) is in its simplest form by ensuring that 9 and 20 have no common factors other than 1. This confirms the fraction is simplified.

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator is larger than the denominator. This means the fraction is greater than or equal to one. When dealing with mixed numbers, which are numbers that have both a whole part and a fractional part, it's often necessary to convert them to improper fractions, especially for multiplication or division.

To convert a mixed number to an improper fraction, multiply the whole number part by the denominator. Then, add the numerator to the result. The original denominator remains the same.

For example, to convert \(-2\frac{4}{5}\) to an improper fraction:

• Multiply the whole number (-2) by the denominator (5): \(-2 imes 5 = -10\).
• Add the numerator (4) to this product: \(-10 + 4 = -6\).
• Thus, \(-2\frac{4}{5}\) becomes \(-\frac{6}{5}\).

Simplifying Fractions

Simplifying fractions means reducing a fraction to its simplest form, in which the numerator and denominator have no common factor other than 1. This makes calculations easier and results more comprehensible.

To simplify a fraction, first find the greatest common divisor (GCD) of the numerator and the denominator. Divide both the numerator and the denominator by this number.

For instance, after multiplying to get \(-\frac{18}{40}\), determine the GCD:

• The factors of 18 are 1, 2, 3, 6, 9, and 18.
• The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40.
• The greatest common factor is 2.

So, dividing both parts by 2 gives \(-\frac{9}{20}\), thus simplifying the fraction.

Mixed Numbers

Mixed numbers are numbers that consist of a whole number and a fraction. They are commonly encountered in everyday situations where sums extend beyond simple fractions, like measuring ingredients for a recipe.

For example, \(2\frac{4}{5}\) is a mixed number: it consists of the whole number 2 and the fraction \(\frac{4}{5}\). Mixed numbers can be easier to grasp in practical applications compared to improper fractions.

However, when performing mathematical operations like multiplication and division, working with improper fractions often becomes necessary to streamline calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide both the numerator and the denominator of a fraction without any remainder. Identifying the GCD enables us to simplify fractions effectively.

To find the GCD, you can employ a method known as listing common factors or using the Euclidean algorithm. For example, when seeking the GCD of 18 and 40, the shared factors are 1 and 2—so the GCD is 2.

The process involves breaking down numbers into their prime factors or iteratively dividing until a remainder of zero is reached, thus ensuring the fraction is presented in its simplest form.