Problem 51 - Prealgebra

Question

Find each product. Write in simplest form. $$2 \frac{1}{2} \cdot\left(-\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

The product is \(\frac{-25}{12}\), and it is already in simplest form.

Step 1: Convert Mixed Number to Improper Fraction

The mixed number \(2 \frac{1}{2}\) can be converted to an improper fraction. First, multiply the whole number 2 by the denominator 2: \(2 \times 2 = 4\). Next, add the numerator 1: \(4 + 1 = 5\). Therefore, \(2 \frac{1}{2} = \frac{5}{2}\).

Step 2: Multiply the Fractions

Now multiply the improper fraction \(\frac{5}{2}\) by the fraction \(-\frac{5}{6}\). The product of two fractions \(\frac{a}{b} \cdot \frac{c}{d}\) is computed as \(\frac{a \times c}{b \times d}\). Therefore, \(\frac{5}{2} \cdot -\frac{5}{6} = \frac{5 \times -5}{2 \times 6} = \frac{-25}{12}\).

Step 3: Simplify the Fraction

The fraction \(\frac{-25}{12}\) is already in simplest form because 25 and 12 do not have any common prime factors other than 1. Thus, the fraction cannot be simplified further.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fractional part. For instance, when you see something like \(2 \frac{1}{2}\), it contains the whole number 2 and the fraction \(\frac{1}{2}\). Before you can use mixed numbers in multiplication or division, you must convert them into improper fractions. This makes it easier to perform operations like multiplying fractions. To convert a mixed number into an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fractional part. For example, in \(2 \frac{1}{2}\), multiply 2 (the whole number) by 2 (the denominator): \(2 \times 2 = 4\).
• Add the result to the numerator of the fractional part. Continuing with our example, add 4 to the numerator 1: \(4 + 1 = 5\).
• This result becomes the new numerator, with the original denominator remaining the same. So, \(2 \frac{1}{2}\) becomes \(\frac{5}{2}\).

This conversion is crucial for performing operations correctly.

Improper Fractions

Improper fractions have numerators larger than their denominators. For example, \(\frac{5}{2}\) is improper because 5 is greater than 2. While they may look unusual, improper fractions are very useful, especially when multiplying fractions.To multiply fractions, you multiply the numerators and denominators. For instance, with \(\frac{5}{2} \cdot -\frac{5}{6}\), you perform these multiplications:

• Multiply the numerators: \(5 \times -5 = -25\).
• Multiply the denominators: \(2 \times 6 = 12\).

Thus, the product is \(\frac{-25}{12}\). Improper fractions make calculations straightforward, allowing you to handle complex mathematical operations with ease.

Simplifying Fractions

Once you have the result of a fraction, like \(\frac{-25}{12}\), you should always check if it can be simplified. Simplifying fractions means reducing them to their simplest form, where the numerator and denominator share no common factors except 1.For the fraction \(\frac{-25}{12}\), we need to:

• Determine the greatest common factor (GCF) of the numerator and denominator. Here, the GCF is 1 as 25 and 12 do not share any prime factors other than 1.
• Since there's no common factor other than 1, the fraction \(\frac{-25}{12}\) is already in its simplest form and cannot be further simplified.

It's important to simplify fractions to their simplest form for easier interpretation and cleaner results in calculations.