Problem 24 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$\frac{2}{9} \div \frac{1}{4}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{8}{9} \).

Step 1: Understand Division of Fractions

To divide fractions, we change the division problem into a multiplication problem by multiplying by the reciprocal of the divisor. For the problem \( \frac{2}{9} \div \frac{1}{4} \), this means multiplying \( \frac{2}{9} \) by the reciprocal of \( \frac{1}{4} \), which is \( \frac{4}{1} \).

Step 2: Multiply Fractions

Now multiply the numerators together and the denominators together: \( \frac{2}{9} \times \frac{4}{1} = \frac{2 \times 4}{9 \times 1} = \frac{8}{9} \).

Step 3: Simplify the Result

Since \( \frac{8}{9} \) is already in its simplest form (there are no common factors other than 1), this is our final result for the division.

Key Concepts

Reciprocal

When we talk about the reciprocal of a fraction, we refer to flipping the numerator and the denominator. For example, the reciprocal of \(\frac{1}{4}\) is \(\frac{4}{1}\). Understanding how to find and use reciprocals is crucial for dividing fractions because instead of dividing, we can multiply by the reciprocal.
Imagine you're dividing by a number—it's essentially the same as multiplying by its reciprocal. This is a handy trick because it allows us to avoid the often tricky operation of division, and instead use multiplication, which can be easier to calculate and conceptualize.
Key points about reciprocals:

• To find a reciprocal, swap the numerator and denominator.
• The reciprocal of a fraction times the fraction itself always equals 1, for instance, \(\frac{1}{4} \times \frac{4}{1} = 1\).

Grasping the concept of reciprocals using this method will make the task of dividing fractions much simpler.

Multiplying Fractions

Multiplying fractions is straightforward once you understand the basic steps. If you have two fractions, each with a numerator (the top number) and a denominator (the bottom number), you simply multiply the numerators together to get the new numerator, and the denominators together for the new denominator.
Let's say you are multiplying \(\frac{2}{9}\) by \(\frac{4}{1}\), like in the exercise. Here's how you do it:

• Multiply the numerators: \(2 \times 4 = 8\).
• Multiply the denominators: \(9 \times 1 = 9\).
• Combine these to form the new fraction: \(\frac{8}{9}\).

This process gives you your new fraction quickly and easily. There are no complicated operations or additional steps required beyond this basic multiplication.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factor other than 1. This means there's nothing more to divide out, making the fraction the most compact it can be.
To simplify a fraction, you must find the greatest common divisor (GCD) of the numerator and the denominator. For example, if you have the fraction \(\frac{8}{9}\), you would check if there are any numbers greater than 1 that both 8 and 9 can be divided by.
In the case of \(\frac{8}{9}\),:

• Both numbers share no common factors other than 1.
• This means the fraction is already as simple as it gets.

Simplifying fractions helps to easily visualize and work with them, making further calculations or comparisons much less complicated.