Problem 119 - Prealgebra

Question

Multiply or divide as indicated. $$\frac{7}{6} \div \frac{7}{12}$$

Step-by-Step Solution

Answer

The result of \(\frac{7}{6} \div \frac{7}{12}\) is 2.

Step 1: Understand the Division of Fractions

When dividing fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is found by swapping its numerator and denominator.

Step 2: Find the Reciprocal

The given fractions are \(\frac{7}{6}\) and \(\frac{7}{12}\). The reciprocal of \(\frac{7}{12}\) is \(\frac{12}{7}\), which is obtained by swapping the numerator and denominator of \(\frac{7}{12}\).

Step 3: Multiply the Fractions

Now, multiply \(\frac{7}{6}\) by the reciprocal of \(\frac{7}{12}\), which is \(\frac{12}{7}\). This calculation is given by: \[ \frac{7}{6} \times \frac{12}{7} \]

Step 4: Simplify the Expression

In the multiplication \(\frac{7}{6} \times \frac{12}{7}\), the sevens in the numerator and the denominator cancel each other out. Thus, the product becomes ​\[ \frac{12}{6} \].

Step 5: Final Simplification

Divide 12 by 6 to simplify the expression:\[ \frac{12}{6} = 2 \]. Thus, the result of the division is 2.

Key Concepts

Division of Fractions

Division of fractions might sound challenging, but it becomes much easier once you understand the concept behind it. When we divide one fraction by another, we actually turn the division into multiplication. Here's how it works:

• Take the first fraction as it is.
• Find the reciprocal of the second fraction.
• Multiply the first fraction by this reciprocal.

By turning division into multiplication, we often find it easier to handle fractions, because multiplying is generally more straightforward than dividing. In the given exercise, this means that instead of dividing \( \frac{7}{6} \) by \( \frac{7}{12} \), we multiply \( \frac{7}{6} \) by the reciprocal of \( \frac{7}{12} \). It's like flipping the second fraction and changing the operation.

Reciprocal

The reciprocal is a key concept in fractions, especially when it comes to division. If you think of a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \). In simple terms, you're swapping the numerator and the denominator.Why does this matter? In fraction division, the reciprocal is crucial because it allows us to convert division problems into multiplication problems. For instance, in our exercise, we had to find the reciprocal of \( \frac{7}{12} \) which is \( \frac{12}{7} \). This reciprocal is then used to multiply with the first fraction \( \frac{7}{6} \) allowing the division to be performed as multiplication.

Fraction Simplification

Simplifying fractions is about reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. In our exercise, while multiplying \( \frac{7}{6} \times \frac{12}{7} \), the number 7 appears in both the numerator and the denominator. This allows us to cancel out the 7s, simplifying the fraction to \( \frac{12}{6} \).To simplify the fraction completely, we divide 12 by 6. Since 6 is a factor of 12, the division gives us 2. Thus, \( \frac{12}{6} \) simplifies to 2. Simplification is a crucial step to ensure our answers are clear and concise, making them easier to understand and work with in further calculations.