Problem 88 - Prealgebra

Question

Find each quotient. (Divide) $$1 \div \frac{1}{3}$$

Step-by-Step Solution

Answer

The quotient of \( 1 \div \frac{1}{3} \) is \( 3 \).

Step 1: Understand Division by a Fraction

To divide a whole number by a fraction, you need to multiply the whole number by the reciprocal of the fraction. The reciprocal of \( \frac{1}{3} \) is \( 3 \) because when you flip the numerator and denominator, \( \frac{1}{3} \) becomes \( \frac{3}{1} \), which is \( 3 \) as a whole number.

Step 2: Multiply by Reciprocal

Instead of dividing \( 1 \) by \( \frac{1}{3} \), multiply \( 1 \) by \( 3 \), which is the reciprocal of \( \frac{1}{3} \). So the equation becomes: \[ 1 \times 3 = 3 \]

Step 3: Calculate the Product

Multiply the whole number by the reciprocal to find the quotient. In this case, \( 1 \times 3 = 3 \). Therefore, the quotient of \( 1 \div \frac{1}{3} \) is \( 3 \).

Key Concepts

Understanding Reciprocals

A reciprocal is a concept that helps simplify complex arithmetic, especially when dealing with fractions. To find the reciprocal of a fraction, you simply flip the numerator and the denominator. What does this mean? For a fraction like \( \frac{1}{3} \), the reciprocal is \( \frac{3}{1} \).

• This reversal swaps the top and bottom parts of the fraction.
• In essence, the reciprocal turns the division into a multiplication.
• This step is vital when dividing by fractions.

Understanding and finding reciprocals become particularly necessary when you need to divide whole numbers by fractions, because, with the reciprocal, division transforms into a simpler multiplication problem.

The Role of Multiplication in Division by Fractions

Multiplication is central when dividing by fractions. Instead of performing a traditional division operation, you multiply by the reciprocal of the divisor.

• This conversion simplifies the operation significantly and eliminates potential confusion.
• Since multiplying by a whole number is often more straightforward, determining the reciprocal allows you to use multiplication instead.
• This method changes questions like \( 1 \div \frac{1}{3} \) into \( 1 \times 3 \).

This switch from division to multiplication makes solving the problem direct and efficient. The result gives you the same quotient as division, making the process both practical and expedient.

Exploring Whole Numbers in Division Context

Whole numbers are numbers without fractions or decimals, like 1, 2, 3, and so on. When dividing whole numbers by fractions, one often initially thinks of complex procedures. However, this task becomes manageable using reciprocals and multiplication.

• For example, with \( 1 \div \frac{1}{3} \), the whole number \( 1 \) stays intact as we work with the fraction's reciprocal.
• Set the operation as \( 1 \times 3 \), highlighting how a simple multiplication ensues.
• Thus, such scenarios teach that whole numbers maintain their form and simplify fraction actions.

Whole numbers simplify division tasks when reciprocals enter the method, often leading to straightforward solutions without fraction results.