Problem 19 - Fundamentals of Mathematics

Question

Determine each of the following quotients and then write a rule for this type of division. $$1 \div \frac{5}{2}$$

Step-by-Step Solution

Answer

\( 1 \div \frac{5}{2} = \frac{2}{5} \); multiply by the reciprocal.

Step 1: Understanding Division by Fraction

When dividing by a fraction, division is modified into multiplication by the reciprocal of that fraction. For example, dividing by \( \frac{5}{2} \) means multiplying by its reciprocal \( \frac{2}{5} \).

Step 2: Identify the Reciprocal

Identify the reciprocal of the fraction \( \frac{5}{2} \), which is obtained by swapping the numerator and the denominator. Thus, the reciprocal is \( \frac{2}{5} \).

Step 3: Perform the Multiplication

Change the division problem into a multiplication problem using the reciprocal. Thus, \( 1 \div \frac{5}{2} \) becomes \( 1 \times \frac{2}{5} \).

Step 4: Multiply

To multiply \( 1 \times \frac{2}{5} \), multiply the numerators together and the denominators together: \( 1 \times 2 = 2 \) and \( 1 \times 5 = 5 \). Therefore, \( 1 \times \frac{2}{5} = \frac{2}{5} \).

Step 5: Rule for Division by Fractions

The rule for dividing by a fraction is to multiply by its reciprocal. Thus, \( a \div \frac{b}{c} = a \times \frac{c}{b} \).

Key Concepts

Reciprocal of a Fraction

When dealing with fractions, a key concept to understand is the reciprocal. The reciprocal of a fraction is what you get when you flip the fraction upside down. More specifically, you swap the numerator (the top number) and the denominator (the bottom number). For example, if you have the fraction \( \frac{5}{2} \), its reciprocal would be \( \frac{2}{5} \).

The important role of a reciprocal comes into play when you're trying to divide by a fraction. Instead of dividing, we actually multiply by the reciprocal of that fraction. Knowing how to find and use the reciprocal can make working with fractions much easier. So, just remember: for \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).

Finding reciprocals is simple once you get the hang of it and can simplify many mathematical problems.

Multiplication by Reciprocal

Multiplication by the reciprocal is a fundamental way to handle division with fractions. It turns an otherwise complicated division problem into a more straightforward multiplication one.

Let's consider an example: dividing \(1\) by \(\frac{5}{2}\). Instead of dividing, you can multiply by the reciprocal of \(\frac{5}{2}\), which is \(\frac{2}{5}\).

This transforms the problem from \(1 \div \frac{5}{2}\) into \(1 \times \frac{2}{5}\). Multiplying fractions requires multiplying the numerators and denominators separately, so here you get \(1 \times 2\) over \(1 \times 5\), resulting in \(\frac{2}{5}\).

The concept of multiplying by the reciprocal streamlines the process, and this method can be applied to divide any fractions, just by flipping the divisor and converting the operation to multiplication.

Mathematical Operations

Mathematical operations consist of processes such as addition, subtraction, multiplication, and division. Each operation follows certain rules that allow us to solve problems accurately.

When working with fractions, these basic operations can sometimes seem challenging, especially division. However, understanding key tactics like division by multiplication with reciprocals simplifies the process. For instance, when confronted with division by a fraction, as seen in \(1 \div \frac{5}{2}\), you can switch gears to multiplication using the reciprocal \(\frac{2}{5}\).

This not only helps in finding quick solutions but also reinforces how interconnected mathematical operations are. Recognizing these connections makes tackling similar problems more intuitive and helps solidify your grasp of mathematical concepts.