Problem 86 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$4 \frac{1}{2} \div 2 \frac{4}{7}$$

Step-by-Step Solution

Answer

The result of the division is \(1 \frac{3}{4}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert both mixed numbers into improper fractions before performing division. For \(4 \frac{1}{2}\), multiply the whole number 4 by the denominator 2 and add the numerator 1: \(4 \times 2 + 1 = 9\), making it \( \frac{9}{2} \). For \(2 \frac{4}{7}\), multiply the whole number 2 by the denominator 7 and add the numerator 4: \(2 \times 7 + 4 = 18\), making it \( \frac{18}{7} \).

Step 2: Division and Using Reciprocals

Divide \(\frac{9}{2}\) by \(\frac{18}{7}\) by multiplying by the reciprocal of the second fraction. The reciprocal of \(\frac{18}{7}\) is \(\frac{7}{18}\). Thus, \(\frac{9}{2} \div \frac{18}{7} = \frac{9}{2} \times \frac{7}{18}\).

Step 3: Multiply the Fractions

Multiply the numerators and the denominators. So, \(\frac{9}{2} \times \frac{7}{18}\) results in \(\frac{9 \times 7}{2 \times 18} = \frac{63}{36}\).

Step 4: Simplify the Fraction

Simplify \(\frac{63}{36}\) by finding the greatest common divisor (GCD) of 63 and 36, which is 9. Divide both the numerator and the denominator by 9: \(\frac{63 \div 9}{36 \div 9} = \frac{7}{4}\).

Step 5: Convert Improper Fraction to Mixed Number

To express \(\frac{7}{4}\) as a mixed number, divide 7 by 4. The quotient is 1 and the remainder is 3. This gives \(1 \frac{3}{4}\).

Key Concepts

Mixed Numbers

Mixed numbers can seem tricky, but they’re simpler than they look. A mixed number is a combination of a whole number and a proper fraction. For example, in the mixed number \(4 \frac{1}{2}\), 4 is the whole number while \(\frac{1}{2}\) is the fraction part.

• To tackle problems with mixed numbers, the first step is often to convert them into improper fractions.
• This simplifies calculations, especially when dealing with division or multiplication.

For instance, \(4 \frac{1}{2}\) was transformed into an improper fraction by multiplying the whole number (4) by the denominator of the fraction (2), and then adding the numerator (1). That gave us \(\frac{9}{2}\). This conversion allows us to handle the entire number as a single fraction.

Improper Fractions

An improper fraction is simply a fraction where the numerator is greater than or equal to the denominator. This means the fraction is equal to or greater than one. They are extremely useful in mathematical operations because they remove the complexity of dealing with both wholes and parts separately.

• For example, converting \(2 \frac{4}{7}\) into an improper fraction involves a similar process as described earlier. By multiplying 2 (whole number) by 7 (denominator), and then adding 4 (numerator), we get \(\frac{18}{7}\).
• Improper fractions make it seamless to multiply and divide, as noted in the given problem.

Once you have all your numbers in improper fraction form, you can proceed to multiplication or division with ease.

Reciprocals

The concept of reciprocals is a cornerstone in dividing fractions. A reciprocal of a fraction is created by swapping its numerator and denominator. For instance, the reciprocal of \(\frac{18}{7}\) is \(\frac{7}{18}\).

• This reversal is what allows us to perform division through multiplication.
• When dividing fractions, we multiply by the reciprocal instead. It's a universal trick for transforming division problems into multiplication problems.

This method simplifies calculations, reducing the risk of errors and making the process more intuitive. In our exercise, dividing \(\frac{9}{2}\) by \(\frac{18}{7}\) becomes a multiplication: \(\frac{9}{2} \times \frac{7}{18}\).

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. It is an essential skill in mathematics, ensuring answers are presented neatly and concisely.

• The simplest form of \(\frac{63}{36}\) requires finding the greatest common divisor (GCD) of 63 and 36, which is 9.
• By dividing both numerator and denominator by 9, we achieve the reduced fraction \(\frac{7}{4}\).

Simplicity is aesthetically pleasing and practical; it reduces the complexity and heightens the clarity of mathematical expressions.

Division of Fractions

Dividing fractions might initially seem daunting, but it becomes straightforward with a few simple steps. The core idea is to turn the problem into a multiplication task.

• Start by converting any mixed numbers into improper fractions.
• Then, invert the divisor (take its reciprocal).
• Proceed by multiplying the dividend by this reciprocal.

This elegant trick works consistently and streamlines the division process. As a final step, remember to simplify your result, and when needed, convert any improper fractions back into mixed numbers for clarity. For example, \(\frac{7}{4}\) simplifies further to the mixed number \(1 \frac{3}{4}\). It all circles back to presenting your final answer in the simplest, most understandable form.