Problem 60 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$ \frac{1}{1000} \div \frac{1}{100} $$

Step-by-Step Solution

Answer

\( \frac{1}{10} \)

Step 1: Understand the Division of Fractions

When dividing two fractions, use the rule: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). This means that dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Identify the Fractions

The given problem is \( \frac{1}{1000} \div \frac{1}{100} \). Here, \( \frac{1}{1000} \) is the first fraction (the dividend) and \( \frac{1}{100} \) is the second fraction (the divisor).

Step 3: Find the Reciprocal of the Divisor

The reciprocal of \( \frac{1}{100} \) is found by swapping the numerator and the denominator, which gives \( \frac{100}{1} \).

Step 4: Multiply by the Reciprocal

Replace the division operation with multiplication by the reciprocal: \( \frac{1}{1000} \times \frac{100}{1} \).

Step 5: Perform the Multiplication

Multiply the numerators together and the denominators together: \( \frac{1 \times 100}{1000 \times 1} = \frac{100}{1000} \).

Step 6: Simplify the Fraction

Simplify \( \frac{100}{1000} \) by dividing both the numerator and the denominator by their greatest common factor, which is 100. This simplifies to \( \frac{1}{10} \).

Key Concepts

Understanding the Reciprocal in Fraction Division

When you come across division of fractions, the concept of the reciprocal plays a vital role. A reciprocal of a fraction is simply created by swapping its numerator and denominator. This means turning the fraction upside down. For example, the reciprocal of \( \frac{1}{100} \) is \( \frac{100}{1} \). Reciprocals are important because dividing by a fraction is the same process as multiplying by its reciprocal.

• Swapping numerator and denominator gives the reciprocal.
• Using the reciprocal transforms division into multiplication.

This makes operations with fractions easier and more intuitive to perform, as multiplication is often simpler than division.

Multiplication of Fractions Made Simple

When multiplying fractions, the process is straightforward compared to other operations. Instead of complex procedures, you multiply across the tops (numerators) and then do the same across the bottoms (denominators). In the given example, \( \frac{1}{1000} \times \frac{100}{1} \), it's about multiplying the numerators: \( 1 \times 100 \) and the denominators: \( 1000 \times 1 \). This results in \( \frac{100}{1000} \).

• Multiply numerators together.
• Multiply denominators together.

This method can be done in a single step without worrying about finding a common denominator, making it a quick and efficient operation.

Simplifying Fractions: Your Final Step

Once you've multiplied the fractions, you often end up with a fraction that can be simplified. Simplifying a fraction involves reducing it to its simplest form, where the numerator and the denominator are as small as possible, while still retaining the same value. For instance, with \( \frac{100}{1000} \), you find the greatest common factor (GCF), which is 100 in this case. Divide both numerator and denominator by this number, leading to the simplified fraction of \( \frac{1}{10} \).

• Find the GCF of the numerator and denominator.
• Divide both by their GCF to simplify.

This will ensure all operations with the fraction are easier later down the line and help in understanding equivalent fractions easily.