Problem 17 - Fundamentals of Mathematics

Question

Determine each of the following quotients and then write a rule for this type of division. $$1 \div \frac{3}{8}$$

Step-by-Step Solution

Answer

Quotient is \( \frac{8}{3} \). Rule: Divide by a fraction by multiplying by its reciprocal.

Step 1: Understand the Problem

We have to divide the number 1 by the fraction \( \frac{3}{8} \). This is written as \( 1 \div \frac{3}{8} \).

Step 2: Rewrite the Division as Multiplication

Reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). Thus, \( 1 \div \frac{3}{8} \) can be rewritten as \( 1 \times \frac{8}{3} \).

Step 3: Perform the Multiplication

Now multiply the number 1 by the reciprocal of \( \frac{3}{8} \). So, \( 1 \times \frac{8}{3} = \frac{8}{3} \).

Step 4: Write the General Rule

When dividing by a fraction, multiply by its reciprocal. If you have \( a \div \frac{b}{c} \), it is equivalent to \( a \times \frac{c}{b} \).

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is key to mastering fraction division. The reciprocal comes into play when you need to divide fractions. But what is a reciprocal exactly? It's quite simple. The reciprocal of a fraction is created by swapping the numerator and the denominator. For instance, if you have a fraction \( \frac{3}{8} \), its reciprocal would be \( \frac{8}{3} \).

• Remember, reciprocals are like mirror images of fractions, flipping the top and bottom parts of the fraction.
• This concept is crucial because when you divide by a fraction, you actually multiply by its reciprocal.

Recognizing and calculating reciprocals allows you to transform complex division problems into more straightforward multiplication tasks. This technique simplifies the process and ensures accurate results every time.

Fraction Division

Dividing fractions might seem tricky at first, but once you understand the process, it becomes much easier. The key is using the reciprocal of the fraction you're dividing by. Here's how it works:

• When you divide by a fraction, you're asking, 'how many times does this fraction fit into the number I'm dividing?' But rather than solving this directly, we transform division into multiplication.

To divide fractions:1. Find the reciprocal of the fraction you're dividing by—flip its numerator and denominator.2. Change the division into multiplication, then multiply with the reciprocal you just found.For example, dividing 1 by \( \frac{3}{8} \) becomes multiplying 1 by \( \frac{8}{3} \).This method consistently yields the correct result and turns a complex operation into a simple one.

Multiplication of Fractions

Though it seems division and multiplication of fractions are worlds apart, they are actually closely related through the concept of reciprocals. Once you've converted a division problem into a multiplication one using the reciprocal, multiplying fractions becomes the focus. Multiplying fractions is straightforward:

• Multiply the numerators (top numbers) together.
• Then multiply the denominators (bottom numbers) together.
• Simplify the resulting fraction if possible.

In our example, after converting division into multiplication, you multiply \( 1 \times \frac{8}{3} \). The multiplication results in \( \frac{8}{3} \). The simplicity of this process shows how interconnected division and multiplication are and why understanding both is essential for mastering fractions.