Problem 58 - Prealgebra

Question

Perform the indicated operations. $$\frac{2}{3} \div \frac{1}{6} \cdot 12$$

Step-by-Step Solution

Answer

The result of the operations is 48.

Step 1: Understand the Expression

The expression to evaluate is \( \frac{2}{3} \div \frac{1}{6} \cdot 12 \). This means you need to divide \( \frac{2}{3} \) by \( \frac{1}{6} \) and then multiply the result by 12.

Step 2: Divide Fractions

To divide \( \frac{2}{3} \) by \( \frac{1}{6} \), multiply \( \frac{2}{3} \) by the reciprocal of \( \frac{1}{6} \). The reciprocal of \( \frac{1}{6} \) is \( 6 \), so the division becomes: \( \frac{2}{3} \times 6 \).

Step 3: Simplify the Division Result

Perform the multiplication: \( \frac{2}{3} \times 6 = \frac{2 \times 6}{3} = \frac{12}{3} \). Simplifying \( \frac{12}{3} \) gives the result of 4.

Step 4: Multiply by 12

Now, take the result from Step 3, which is 4, and multiply it by 12: \( 4 \times 12 = 48 \).

Key Concepts

Multiplying Fractions

Multiplying fractions is a straightforward operation where you multiply the numerators together and the denominators together. For example, when multiplying \( \frac{a}{b} \) by \( \frac{c}{d} \), the result is \( \frac{a \cdot c}{b \cdot d} \). This operation simplifies when dealing with whole numbers, as whole numbers can be written as fractions with a denominator of 1.
For instance, multiplying \( \frac{2}{3} \) by 6 involves treating 6 as \( \frac{6}{1} \), making the multiplication \( \frac{2}{3} \times \frac{6}{1} \) resulting in \( \frac{2 \cdot 6}{3 \cdot 1} = \frac{12}{3} \). After performing the multiplication, it is crucial to simplify the resulting fraction if possible. Here, simplifying \( \frac{12}{3} \) results in 4, since 12 divided by 3 is 4. Simplification makes fractions easier to understand and use in further calculations.

Dividing Fractions

Dividing fractions might seem tricky at first, but it's just a matter of flipping and multiplying. The general rule for dividing by a fraction is to multiply by its reciprocal.
Take, for example, dividing \( \frac{2}{3} \) by \( \frac{1}{6} \). You find the reciprocal of \( \frac{1}{6} \), which is 6 or \( \frac{6}{1} \), and then multiply \( \frac{2}{3} \) by this reciprocal. This looks like \( \frac{2}{3} \times 6 \), following the same multiplying principles, resulting in \( \frac{12}{3} \) which simplifies to 4.
Steps to Divide Fractions:

• Identify the reciprocal of the divisor (the second fraction).
• Multiply the dividend (first fraction) by this reciprocal.
• Simplify the resulting fraction if necessary.

Remember this method every time you need to divide fractions, as it can significantly simplify your calculations.

Reciprocal

The reciprocal of a number, specifically a fraction, is what you multiply the number by to get a product of 1. For a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Understanding the concept of a reciprocal is essential, especially in operations involving division of fractions.
Using the reciprocal lets you convert division problems into multiplication problems, which are often simpler to solve.
Finding the Reciprocal:

• For fractions: flip the numerator and denominator. So, the reciprocal of \( \frac{2}{5} \) is \( \frac{5}{2} \).
• For whole numbers: consider the whole number as a fraction with a denominator of 1 and then flip it. Thus, the reciprocal of 4 is \( \frac{1}{4} \).

Recognizing how to quickly find and apply reciprocals helps solve fraction division tasks efficiently.