Problem 8 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$3 \frac{1}{2} \cdot 2 \frac{1}{6}$$

Step-by-Step Solution

Answer

The product is \(7 \frac{7}{12}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. A mixed number like \(3 \frac{1}{2}\) can be converted by multiplying the whole number (3) by the denominator (2) and adding the numerator (1), giving \(\frac{(3 \cdot 2) + 1}{2} = \frac{7}{2}\). Similarly, \(2 \frac{1}{6}\) becomes \(\frac{(2 \cdot 6) + 1}{6} = \frac{13}{6}\).

Step 2: Multiply the Improper Fractions

Now, multiply the two improper fractions obtained. The product of two fractions \(\frac{a}{b} \cdot \frac{c}{d}\) is \(\frac{a \cdot c}{b \cdot d}\). Thus, \(\frac{7}{2} \cdot \frac{13}{6} = \frac{7 \cdot 13}{2 \cdot 6} = \frac{91}{12}\).

Step 3: Simplify the Improper Fraction to a Mixed Number

Convert the improper fraction \(\frac{91}{12}\) into a mixed number by dividing 91 by 12. This gives 7 with a remainder of 7, or \(7 \frac{7}{12}\). This is our simplified mixed number.

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a fraction. It's like having some complete pies and a portion of another pie when visualizing. They are often seen in baking and everyday situations. For example, if you have 3 whole pies and half of another pie, you can write that as \(3 \frac{1}{2}\). The process of working with them in math often involves converting them first, especially in operations like multiplication. This makes calculations easier.
To convert a mixed number to an improper fraction:

• Multiply the whole number by the fraction's denominator.
• Add the resulting product to the fraction's numerator.
• Write the result over the original denominator.

For instance, in \(3 \frac{1}{2}\):

• First, multiply 3 (whole number) by 2 (denominator) to get 6.
• Then, add 1 (numerator) to get a total of 7.
• This gives us an improper fraction of \(\frac{7}{2}\).

Improper Fractions

Understanding improper fractions is crucial when dealing with fractions. An improper fraction has a numerator larger than its denominator, which means it represents more than a whole unit. For example, \(\frac{13}{6}\) suggests there are 13 parts, while each whole is divided into 6 parts. Such fractions are particularly useful in multiplication, as they simplify the process significantly.
When multiplying, improper fractions make calculations straightforward. Instead of dealing with mixed numbers, which can be cumbersome, their improper counterparts allow us to use simple multiplication rules:

• Multiply the numerators together.
• Multiply the denominators together.
• Write the result as a fraction of these two products.

For example, multiplying \(\frac{7}{2}\) by \(\frac{13}{6}\) is simply multiplying 7 by 13 and 2 by 6, resulting in \(\frac{91}{12}\).

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, or converting improper fractions to mixed numbers for ease of understanding. It is an important skill in mathematics to make numbers more intuitive and easier to read.
When you simplify, there are a couple of steps involved, especially after performing a multiplication of fractions. If the result is an improper fraction, it often needs simplifying to a mixed number by division:

• Divide the numerator by the denominator.
• The quotient becomes the whole number, while the remainder forms the new numerator over the original denominator.

In our example, \(\frac{91}{12}\):

• Dividing 91 by 12 gives a quotient of 7 (as the whole number) and a remainder of 7.
• Thus, the simplified mixed number is \(7 \frac{7}{12}\).

This transformation makes it clearer how many whole units and fractions of a unit we have, which is a key part of better understanding and using numbers in practice.