Problem 70 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The result of the operation \(3 \div \frac{1}{4}\) is 12.

Step 1: Identify the Dividend and the Divisor

In the given problem, 3 is the dividend (the number to be divided) and \( \frac{1}{4} \) is the divisor (the number by which the dividend is to be divided).

Step 2: Find the Reciprocal of the Divisor

The reciprocal of a fraction is obtained by switching the numerator and the denominator. The reciprocal of \( \frac{1}{4} \) is \( 4/1 = 4 \).

Step 3: Multiply the Dividend by the Reciprocal of the Divisor

After obtaining the reciprocal of the divisor, you perform the operation of multiplication. It is \( 3 \times 4 = 12 \).

Key Concepts

Multiplication of Fractions

Multiplying fractions is a fundamental skill you need when working with divisions involving fractions. To multiply two fractions, you take the numerator (top number) of the first fraction and multiply it by the numerator of the second. Then, do the same with the denominators (bottom numbers).
You get a fraction for the result. So, for example, if you multiply \( \frac{2}{3} \times \frac{4}{5} \), you get:

• multiply the numerators: \( 2 \times 4 = 8 \)
• multiply the denominators: \( 3 \times 5 = 15 \)

The resulting fraction is \( \frac{8}{15} \).
When multiplying whole numbers by fractions, like in our problem where 3 is multiplied by the reciprocal of \( \frac{1}{4} \), treat the whole number as a fraction by placing it over 1.
Therefore, \( 3 \times \frac{4}{1} \) becomes \( \frac{3}{1} \times \frac{4}{1} \). Multiply numerators and denominators to get \( \frac{12}{1} = 12 \).
This confirms the importance of multiplication of fractions techniques in simplifying complex fraction problems.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is crucial for solving division problems involving fractions. The reciprocal is simply flipping the nominator and denominator. So, if you have a fraction like \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
This concept is often used to turn a division problem into a multiplication problem, which is usually more straightforward to solve.
In the exercise example, to divide by \( \frac{1}{4} \), you first convert \( \frac{1}{4} \) into its reciprocal, which is \( 4 \) or \( \frac{4}{1} \). This way, dividing by \( \frac{1}{4} \) becomes multiplying by 4.

• The reciprocal is used because division by a fraction equals multiplication by its reciprocal.
• This makes calculations easier and faster by eliminating complex division.

Remember, this is always a flip and multiply process, making division manageable.

Arithmetical Operations

Arithmetical operations include the basic operations of addition, subtraction, multiplication, and division. When dealing with fractions, focusing on multiplication and division is particularly important.
These operations maintain balance and equality by transforming division into an easier multiplying task using reciprocals.
Consider the division problem \( 3 \div \frac{1}{4} \). Initially, it seems daunting, but when you transform it to \( 3 \times 4 \), it becomes simple.

• Division becomes multiplication by altering the divisor to its reciprocal.
• The operation's aim is to provide a simpler solution path, ensuring accurate results without altering the underlying numerical relationships.

Thus, grasping these arithmetical tools not only simplifies the calculations but connects the operations seamlessly to provide the correct answer.