Problem 82 - Algebra 1

Question

find the quotient. $$ 3 \div \frac{1}{4} $$

Step-by-Step Solution

Answer

The quotient of \(3 \div \frac{1}{4}\) is 12.

Step 1: Identify the dividend and divisor

The problem asks for the quotient of \(3 \div \frac{1}{4}\). Here 3 is the dividend and \(\frac{1}{4}\) is the divisor.

Step 2: Change division operation to multiplication by reciprocal

Replace the division operation with multiplication by the reciprocal of the divisor. The reciprocal of a fraction is obtained by swapping its numerator and denominator, hence reciprocal of \(\frac{1}{4}\) is \(4\). The operation now becomes \(3 \times 4\).

Step 3: Perform the multiplication

Multiplying 3 and 4 gives a product of 12. Therefore, \(3 \div \frac{1}{4} = 12\) .

Key Concepts

Understanding the Reciprocal

The reciprocal of a number is simply a way of flipping a fraction. To find the reciprocal of a fraction, you switch the numerator and denominator. For instance, the reciprocal of \(\frac{1}{4}\) is \(4\) or \(\frac{4}{1}\). This concept is vital in division of fractions because instead of dividing by a fraction, you multiply by its reciprocal. This effectively simplifies the process of dividing fractions.

• In the case of whole numbers like 3, the reciprocal is obtained by placing the number over 1. So, the reciprocal would be \(\frac{1}{3}\).
• Every number (except zero) has a reciprocal that when multiplied by the number results in 1.
• The reciprocal helps turn division into a multiplication problem, which tends to be easier to handle.

Keep in mind that the reciprocal is only defined for non-zero numbers. The idea of reciprocals dates back to the ancient Egyptians who used fractions heavily in their calculations.

Multiplication in Division of Fractions

When dealing with division by fractions, we transform the operation to multiplication. This transformation requires the use of the divisor's reciprocal. For example, the problem \(3 \div \frac{1}{4}\) changes to \(3 \times 4\) once the reciprocal of \(\frac{1}{4}\) (which is 4) is introduced.

• To perform multiplication, multiply the numerators together and the denominators together.
• If you start with a whole number like 3, think of it as \(\frac{3}{1}\) to make the multiplication easier.
• By changing the operation and using the reciprocal, we're turning a division problem into a simpler multiplication task.

Multiplication is generally easier than division because it uses the concept of repeated addition. In division of fractions using multiplication, the whole process becomes straightforward and simple to follow.

Defining the Quotient

A quotient is the result you obtain when one number is divided by another. In our example of dividing 3 by \(\frac{1}{4}\), we used multiplication to find the quotient.
The process involved changing the division into multiplication by incorporating the reciprocal, then performing the multiplication itself.
As a result of this multiplication, we found that the quotient is 12.

• The quotient gives us a measure of how many times the divisor fits into the dividend.
• Using reciprocals and multiplication ensures the calculation is accurate and efficient.
• The final answer to our division problem, the quotient, reflects the simplicity gained by converting the problem into a familiar operation.

Hence, understanding the concept of a quotient as an outcome or result of division helps clarify not only this problem but also provides a foundation for solving more complex fractional equations in the future.