Problem 18 - Technical Mathematics with Calculus

Question

Divide and reduce. Try some by calculator. $$3 \frac{7}{8} \div 2$$

Step-by-Step Solution

Answer

\(3 \frac{7}{8} \div 2 = \frac{31}{16}\) or as a mixed number, \(1 \frac{15}{16}\).

Step 1: Convert Mixed Number to Improper Fraction

Convert the mixed number to an improper fraction by multiplying the whole number by the denominator of the fraction and adding the numerator. For the mixed number \(3 \frac{7}{8}\), the improper fraction is obtained by computing \(3 \times 8 + 7 = 31\). So, the mixed number is equivalent to the improper fraction \(\frac{31}{8}\).

Step 2: Convert Division to Multiplication

Convert the division problem to a multiplication problem by finding the reciprocal of the divisor. In this case, the divisor is 2, whose reciprocal is \(\frac{1}{2}\). Now, the equation \(3 \frac{7}{8} \div 2\) becomes \(\frac{31}{8} \times \frac{1}{2}\).

Step 3: Multiply the Fractions

Multiply the numerators and then the denominators of the two fractions to find the product. Numerators: \(31 \times 1 = 31\). Denominators: \(8 \times 2 = 16\). So the product of the fractions is \(\frac{31}{16}\).

Step 4: Reduce the Fraction (if necessary)

In this case, \(\frac{31}{16}\) cannot be reduced because 31 and 16 have no common factors other than 1. Therefore, the simplified form of the fraction is \(\frac{31}{16}\). If desired, you can also express it as a mixed number, which would be \(1 \frac{15}{16}\).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is larger than or equal to the denominator (the bottom number). This can initially seem counterintuitive because we often think of fractions as representing numbers smaller than one, but improper fractions are simply another way to represent numbers that are greater than or equal to one.

For example, when we look at the mixed number from our exercise, we have the whole part, 3, and the fractional part, \( \frac{7}{8} \). To convert it into an improper fraction, we multiply the whole number by the fraction's denominator and add the numerator. The formula is:
\[ \text{Improper fraction} = (\text{Whole number} \times \text{Denominator}) + \text{Numerator} \].

So, for \( 3 \frac{7}{8} \), we perform the calculation \( 3 \times 8 + 7 \) to get \( 31 \). This tells us the mixed number is equivalent to the improper fraction \( \frac{31}{8} \), with 31 as the new numerator and 8 as the denominator. Improper fractions are a vital step in operations like division because they allow us to work with whole numbers and fractions in a unified manner.

Multiplicative Inverse

The multiplicative inverse of a number is what you multiply that number by to get the number 1. In the case of fractions, it's what turns a division problem into a multiplication one. The multiplicative inverse, also known as the reciprocal, essentially 'flips' a fraction.

For any non-zero number \(a\), the multiplicative inverse is \( \frac{1}{a} \). For a fraction, the multiplicative inverse is created by flipping the numerator and denominator. If we have \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \).

In our exercise, the divisor 2 (which can also be written as \(\frac{2}{1}\)) has a multiplicative inverse of \( \frac{1}{2} \). By multiplying by this inverse, we turn our division problem into a simpler multiplication problem. This is an essential step because it changes the operation from division (which can be more complex with fractions) to multiplication, which follows a straightforward method: simply multiply across the numerators and denominators.

Fraction Multiplication

Multiplying fractions is simpler than you might think. Unlike adding or subtracting fractions, you don't need to find a common denominator. Instead, you multiply the numerators together and the denominators together. The outcome is a new fraction that might or might not need further simplification.

Let's look at the multiplication step from our example problem: \( \frac{31}{8} \times \frac{1}{2} \). When we multiply the numerators, 31 and 1, we get 31. When we multiply the denominators, 8 and 2, we get 16. Consequently, the answer is \( \frac{31}{16} \), a new fraction.

Reducing Fractions

If the resulting fraction from your multiplication can be reduced (simplified), you do so by finding the greatest common divisor (GCD) of its numerator and denominator. Simplifying makes your fraction easier to understand and work with. In our problem, \( \frac{31}{16} \) is already in its simplest form because 31 and 16 have no common factors other than 1. The simplicity of fraction multiplication makes solving division problems involving mixed numbers much more accessible once you've converted them to improper fractions.