Problem 39 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{7}{8} \div \frac{7}{8}$$

Step-by-Step Solution

Answer

The value of \(\frac{7}{8} \div \frac{7}{8}\) is 1.

Step 1: Understand Division of Fractions

When dividing fractions, we generally multiply by the reciprocal of the divisor. The divisor here is \(\frac{7}{8}\).

Step 2: Find the Reciprocal

The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). Therefore, the reciprocal of \(\frac{7}{8}\) is \(\frac{8}{7}\).

Step 3: Change Division to Multiplication

Change the division problem \(\frac{7}{8} \div \frac{7}{8}\) into a multiplication problem by multiplying the dividend by the reciprocal of the divisor. Thus, it becomes \(\frac{7}{8} \times \frac{8}{7}\).

Step 4: Multiply the Fractions

Multiply the numerators and the denominators: \(\frac{7 \times 8}{8 \times 7}\). This equals \(\frac{56}{56}\).

Step 5: Simplify the Fraction

Since \(\frac{56}{56}\) is equal to 1, the division problem simplifies to 1.

Key Concepts

Reciprocal

In mathematics, a crucial concept is the *reciprocal*. You might wonder, what exactly is a reciprocal? It's simpler than you think! A reciprocal of a fraction flips the numerator and the denominator.
Imagine you have a fraction like \( \frac{7}{8} \). Its reciprocal would be \( \frac{8}{7} \).

• Numerator becomes the denominator.
• Denominator becomes the numerator.

This idea of flipping fractions is essential when dividing fractions. Why? Because dividing by a fraction is the same as multiplying by its reciprocal. If you have \( \frac{a}{b} \div \frac{c}{d} \), this changes to \( \frac{a}{b} \times \frac{d}{c} \). The entire operation hinges on the simplicity and ease of finding reciprocals.

Fraction Multiplication

Multiplying fractions might sound tricky, but it's very straightforward! When you multiply fractions, you multiply their numerators and their denominators. Let's look at an example:
If you have: \[ \frac{7}{8} \times \frac{8}{7} \]Here's how you do it:

• Multiply the numerators: \( 7 \times 8 = 56 \).
• Multiply the denominators: \( 8 \times 7 = 56 \).

This multiplication is straightforward, just going step by step. It doesn't matter how complex numbers in numerators or denominators might be; the steps remain the same. And remember, once you've performed the multiplication, you often need to simplify the result!

Simplifying Fractions

Simplifying fractions means finding an equivalent fraction that is as "simple" as possible. When a fraction is simplified, its numerator and denominator have no common factors other than 1. In the context of our problem:
After multiplying a fraction: \[ \frac{56}{56} \]You notice both the numerator and the denominator are identical. Simplifying it becomes easy:

• Since \( 56 \) is the same on top and bottom, you divide both by 56.
• This results in 1: \( \frac{56}{56} = 1 \).

Why does simplification matter? A fraction like \( \frac{56}{56} \) clearly equals 1, a much simpler and more comprehendible result. This concept helps in reducing complexity and making your final answers neat and comprehensible.