Problem 52 - Prealgebra

Question

Find each quotient. Write in simplest form. $$\frac{2}{3} \div \frac{1}{3}$$

Step-by-Step Solution

Answer

The quotient is 2.

Step 1: Understand the Operation

We need to find the quotient of two fractions, \( \frac{2}{3} \) divided by \( \frac{1}{3} \). This means we are looking for how many times \( \frac{1}{3} \) fits into \( \frac{2}{3} \).

Step 2: Apply the Definition of Division for Fractions

To divide by a fraction, we multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of \( \frac{1}{3} \) is \( \frac{3}{1} \) or simply 3.

Step 3: Multiply the Fractions

Now, we multiply \( \frac{2}{3} \) by the reciprocal of \( \frac{1}{3} \). So, the operation becomes:\[\frac{2}{3} \times \frac{3}{1} = \frac{2 \times 3}{3 \times 1}\]

Step 4: Simplify the Expression

Calculate the multiplication:\[\frac{2 \times 3}{3 \times 1} = \frac{6}{3}\]Now, divide both the numerator and the denominator by their greatest common divisor, which is 3, to simplify the fraction to its lowest terms:\[\frac{6 \div 3}{3 \div 3} = \frac{2}{1} = 2\]

Step 5: Verify the Simplification

Check that the resulting number \( 2 \) is in its simplest form. Since it is a whole number with a denominator of 1, we have already achieved the simplest form.

Key Concepts

Reciprocal of a Fraction

The reciprocal of a fraction is a very useful concept, especially when you are dealing with division involving fractions. Let's break it down!

To find the reciprocal, you simply flip the numerator and the denominator. For example, the reciprocal of \( \frac{1}{3} \) is \( \frac{3}{1} \), which can also be written as just 3. This flipping lets you transform division into multiplication, a much simpler operation!

Reciprocals are also essential in checking division problems, as using them converts a complex fraction division problem into a straightforward multiplication problem. Mastery of this skill can significantly speed up and simplify your calculations!

Simplifying Fractions

Simplifying fractions is all about getting a fraction into its simplest form, where the numerator and the denominator cannot be divided by any common number except for 1.

Consider the example \( \frac{6}{3} \). You can simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD), which in this case is 3:

• The numerator: \( 6 \div 3 = 2 \)
• The denominator: \( 3 \div 3 = 1 \)

Therefore, \( \frac{6}{3} \) simplifies to \( \frac{2}{1} \) or just 2.

This process of simplification makes it easier to identify equivalent fractions and to ensure your answers are neat and tidy. Always remember, a fraction is fully simplified when no larger common factors exist other than 1.

Multiplying Fractions

Multiplying fractions might seem challenging at first, but it's actually quite straightforward. Let's get into it!

When you multiply fractions, you multiply the numerators together to get the new numerator, and do the same with the denominators. Imagine multiplying \( \frac{2}{3} \) by \( \frac{3}{1} \). This is how you do it:

• Numerator: \( 2 \times 3 = 6 \)
• Denominator: \( 3 \times 1 = 3 \)

So, \( \frac{2}{3} \times \frac{3}{1} = \frac{6}{3} \) .

After multiplying, you should always check if the fraction can be simplified. In this case, \( \frac{6}{3} \) simplifies to 2. Thus, multiplying fractions is simple so long as you remember to multiply straight across and simplify your result whenever possible.