Problem 79 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$3 \frac{1}{2} \div \frac{7}{2}$$

Step-by-Step Solution

Answer

The result is 1.

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(3 \frac{1}{2}\) to an improper fraction. Recall that to convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator, placing the result over the original denominator.\[3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}\]

Step 2: Set Up Division Problem

Now, set up the division problem. Since both \(3 \frac{1}{2}\) and \(\frac{7}{2}\) are \(\frac{7}{2}\), the expression now looks like: \[\frac{7}{2} \div \frac{7}{2}\]

Step 3: Apply the Inversion and Multiplication Rule

In division involving fractions, convert the division into multiplication by inverting the second fraction. Multiply the first fraction by the reciprocal of the second fraction. \[\frac{7}{2} \div \frac{7}{2} = \frac{7}{2} \times \frac{2}{7}\]

Step 4: Multiply the Fractions

Multiply the numerators together and the denominators together.\[\frac{7 \times 2}{2 \times 7} = \frac{14}{14}\]

Step 5: Simplify the Fraction

Lastly, simplify \(\frac{14}{14}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which in this case is 14.\[\frac{14 \div 14}{14 \div 14} = \frac{1}{1} = 1\]

Step 6: Final Answer

The result of \(3 \frac{1}{2} \div \frac{7}{2}\) is \(1\).

Key Concepts

Mixed Numbers

A mixed number is a number that includes both a whole number and a fractional part. For instance, in the mixed number \(3 \frac{1}{2}\), 3 is the whole number, and \(\frac{1}{2}\) is the fractional part. Mixed numbers are useful for representing values greater than one in a way that's easy to visualize, as they show both complete units and a part-left-over.

• To convert a mixed number to an improper fraction, you perform the operation \(\text{(whole number)} \times \text{(denominator)} + \text{(numerator)}\). Place the result over the original denominator. For example, \(3 \frac{1}{2} = \frac{7}{2}\).

Converting mixed numbers to improper fractions is especially helpful when you need to perform operations such as multiplication or division, where fractions are used. Working with improper fractions simplifies calculations.

Improper Fractions

Improper fractions feature a numerator that is larger than or equal to their denominator. This means the numeric value of the fraction is either one or greater than one, making it unique compared to proper fractions, where the numerator is always smaller.

• For example, the fraction \(\frac{7}{2}\) is improper because the numerator (7) is greater than the denominator (2).

Improper fractions are advantageous in calculations because they streamline arithmetic processes.
During operations, such as division, having all numbers in a similar form like improper fractions provides consistency, allowing calculations to be more straightforward.Sometimes, after completing a calculation, it's helpful to convert an improper fraction back to a mixed number to easily interpret or present the result.

Division of Fractions

Dividing fractions might seem tricky at first, but there's a simple rule to follow that makes it more straightforward: you multiply by the reciprocal.
Here's how you do it:

• First, take the fraction you're dividing by and flip it. This new fraction is called the reciprocal. For instance, the reciprocal of \(\frac{7}{2}\) is \(\frac{2}{7}\).
• Then multiply the first fraction by this reciprocal. In our example, \(\frac{7}{2} \div \frac{7}{2} = \frac{7}{2} \times \frac{2}{7}\).

Once set up,
multiply the numerators together and the denominators together: \(\frac{7 \times 2}{2 \times 7} = \frac{14}{14}\).
The final step involves simplifying, if necessary.When done right, division by fractions can be just as easy as multiplication. Each time, watch out for simplifying fractions to make the simplest form possible.