Problem 90 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer in simplest form. $$ \frac{1}{6} \div \frac{2}{3} $$

Step-by-Step Solution

Answer

\frac{1}{4}

Step 1: Rewrite the Division as Multiplication

The division of fractions can be rewritten as the multiplication by the reciprocal of the divisor. This means rewriting \( \frac{1}{6} \div \frac{2}{3} \) as \( \frac{1}{6} \times \frac{3}{2} \). The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).

Step 2: Multiply the Fractions

Multiply the numerator with the numerator and the denominator with the denominator. \n\n\( \frac{1}{6} \times \frac{3}{2} = \frac{1 \times 3}{6 \times 2} \) \n\nThis simplifies to \( \frac{3}{12} \)

Step 3: Simplify the Fraction

Reduce the fraction \( \frac{3}{12} \) to simplest form. The highest common factor of 3 and 12 is 3, so divide the numerator and the denominator by 3. This gives the answer in simplest form: \( \frac{1}{4} \)

Key Concepts

Simplest Form

Understanding the concept of 'simplest form' in fractions is crucial for making mathematical operations more manageable. A fraction is said to be in its simplest form when the numerator and denominator are both at their smallest possible values while still maintaining the same ratio. To achieve this, one must find the greatest common divisor (GCD) that both the numerator and the denominator share. Then, divide both by this number.

For example, if we have the fraction \(\frac{3}{12}\), we notice that both 3 and 12 can be divided by 3. By doing so, we obtain \(\frac{3 \/ 3}{12 \/ 3} = \frac{1}{4}\), which is the fraction in its simplest form. In this instance, 3 is the GCD of 3 and 12. The act of simplifying not only makes the numbers easier to work with but also aids in the visualization and comparison of different fractions.

Multiplying Fractions

When it comes to multiplying fractions, the concept is more straightforward than other operations with fractions. To multiply two fractions, simply multiply the numerators together to find the new numerator, and multiply the denominators together to find the new denominator. After this, you might need to simplify the fraction.

Let's say we multiply \(\frac{1}{6}\) by \(\frac{3}{2}\). According to the rule, that would be \(\frac{1 \times 3}{6 \times 2} = \frac{3}{12}\). Afterwards, you would check if the fraction can be simplified into its simplest form, which, in our case, simplifies to \(\frac{1}{4}\), following the process discussed in the 'Simplest Form' section.

Reciprocal of a Fraction

The reciprocal of a fraction, often called the multiplicative inverse, is simply flipping the numerator and the denominator of the fraction. In other words, if you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\), assuming that neither a nor b is zero. This concept is particularly important when dividing fractions, as the division can be turned into multiplication by the reciprocal.

In our exercise, the division of \(\frac{1}{6}\) by \(\frac{2}{3}\) is converted into multiplication by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This switch from division to multiplication simplifies the operation and utilizes the straightforward method for multiplying fractions.

Simplifying Fractions

Simplifying fractions, a crucial skill in mathematics, goes hand in hand with understanding simplest form. To simplify a fraction, we seek to reduce it to its most basic form by eliminating any common factors shared by the numerator and the denominator. This often involves knowing basic multiplication and division facts.

Using our exercise as an example, after multiplying \(\frac{1}{6}\) and \(\frac{3}{2}\), we got \(\frac{3}{12}\). Simplifying is needed to achieve the simplest form. Since 3 is a common factor, divide both the numerator and the denominator by 3 to get \(\frac{1}{4}\). It's important to always check if a fraction can be simplified further, as it might not be in its simplest form immediately after performing an operation.