Problem 76 - Algebra 1: Concepts and Skills

Question

Simplify. $$ \frac{1}{2} \cdot \frac{1}{3} \div \frac{1}{4} \cdot \frac{1}{5} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{2}{15} \)

Step 1: Convert Division Operation to Multiplication

The first step is to convert the division operation into a multiplication one, by taking the reciprocal of the fraction to be divided. \( \frac{1}{2} \cdot \frac{1}{3} \div \frac{1}{4} \cdot \frac{1}{5} \) changes to \( \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{4}{1} \cdot \frac{1}{5} \)

Step 2: Perform the Multiplication

The multiplication of the fractions can be done by carrying out multiplication of the numerators together and denominators together. Thereby, \( \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{4}{1} \cdot \frac{1}{5} \) simplifies to \( \frac{1 \cdot 1 \cdot 4 \cdot 1}{2 \cdot 3 \cdot 1 \cdot 5} = \frac{4}{30} \)

Step 3: Simplify the Result

The last step is to simplify the fraction. \( \frac{4}{30} \) simplifies to \( \frac{2}{15} \), by dividing both the numerator and denominator by 2

Key Concepts

Fraction Division

When dividing fractions, it's important to transform the division operation into a more manageable form. Typically, this is done by converting the division into multiplication.
This is achieved using the reciprocal of the fraction that follows the division sign. For example, in the exercise \( \frac{1}{2} \cdot \frac{1}{3} \div \frac{1}{4} \cdot \frac{1}{5} \), the division \( \div \frac{1}{4} \) is changed to a multiplication \( \cdot \frac{4}{1} \).
By making these changes, the division of fractions can be approached similarly to multiplication, which is generally simpler to handle.

Reciprocal

The reciprocal of a fraction is essentially flipping the numerator and the denominator. For any fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
In simple terms:

• The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
• The reciprocal of \( \frac{5}{2} \) is \( \frac{2}{5} \).

Understanding reciprocals is crucial when dividing fractions because it allows us to replace the division operation with multiplication. In the original exercise, when \( \frac{1}{4} \) is the divisor, its reciprocal \( \frac{4}{1} \) is used instead.
Therefore, finding the reciprocal simplifies many fraction operations and makes calculations more straightforward.

Simplification of Fractions

Simplification of fractions is the process of reducing a fraction to its simplest form. This means that the numerator and the denominator have no common factors other than 1.
Completing the simplification involves looking for the greatest common divisor (GCD) of both the numerator and the denominator, and dividing them by that number. In our simplified result of the exercise, we obtained \( \frac{4}{30} \).

• We noticed both 4 (numerator) and 30 (denominator) can be divided by 2 (their GCD).
• Dividing both by 2 gives us \( \frac{2}{15} \), the simplest form.

Simplified fractions make it easier to compare and perform additional operations on them. By reducing fractions, numbers become more manageable, further enhancing calculation clarity and accuracy.