Problem 16 - Fundamentals of Mathematics

Question

Determine each of the following quotients and then write a rule for this type of division. $$1 \div \frac{2}{3}$$

Step-by-Step Solution

Answer

The quotient is \(\frac{3}{2}\) or 1.5. Multiply by the reciprocal to divide by a fraction.

Step 1: Understand Division of Fractions

To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator.

Step 2: Rewrite the Division as Multiplication

Rewrite the division operation \(1 \div \frac{2}{3}\) as a multiplication operation by using the reciprocal of \(\frac{2}{3}\). This becomes \(1 \times \frac{3}{2}\).

Step 3: Multiply the Numbers

Multiply the whole number \(1\) by the fraction \(\frac{3}{2}\). The calculation is \(1 \cdot \frac{3}{2} = \frac{3}{2}\).

Step 4: Interpret the Result

The resulting fraction \(\frac{3}{2}\) can be expressed as a decimal (1.5) or left as a fraction. This is the quotient of the original division problem.

Step 5: Write a General Rule

When dividing a number by a fraction, multiply that number by the reciprocal of the fraction. The operation \(a \div \frac{b}{c}\) becomes \(a \times \frac{c}{b}\).

Key Concepts

Understanding Reciprocals

The term "reciprocal" is an important part of dividing fractions. A reciprocal is what you get when you swap the numerator and denominator of a fraction. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\). When dividing by a fraction, you don't really divide; you multiply by the reciprocal instead.

• Finding Reciprocals: To find the reciprocal of a given fraction \(\frac{a}{b}\), simply swap the positions of \(a\) and \(b\), resulting in \(\frac{b}{a}\).
• Why Use Reciprocals: Multiplying by a reciprocal changes the direction of division, making calculations easier and more straightforward.
• An Example: If you need to divide by \(\frac{3}{4}\), instead, you multiply by \(\frac{4}{3}\).

Mastering the concept of reciprocals simplifies many mathematical operations, especially when performing division with fractions.

Multiplication of Fractions

When dealing with fractions, multiplying them is often simpler than dividing. Once you've found the reciprocal of the divisor fraction, the division problem becomes a multiplication problem. This is usually straightforward because multiplication of fractions involves three simple steps:

• Step 1: Multiply the numerators (the numbers on top) of each fraction to get the new numerator.
• Step 2: Multiply the denominators (the numbers on the bottom) to get the new denominator.
• Step 3: Simplify the resulting fraction, if possible, by finding common factors.

For example, in the problem \(1 \div \frac{2}{3}\), you rewrite it as \(1 \times \frac{3}{2}\). You multiply the numerators and the denominators to get \(\frac{3}{2}\). Thus, the division of fractions is neatly transformed into a simpler multiplication task.

Converting Fractions to Decimals

After performing calculations with fractions, such as division or multiplication, you might want to express your result as a decimal. This is often simpler to understand or use in real-world applications. Converting a fraction to a decimal is generally easy:

• Method: Divide the numerator by the denominator.
• Example: For \(\frac{3}{2}\), divide 3 by 2 to get 1.5.
• Usefulness: Converting fractions to decimals can make comparisons and interpretations more intuitive, especially in areas like measurement or finance.

By understanding both the fractional and decimal forms, you increase your mathematical versatility and ease of communication in various contexts.