Problem 30 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$12 \div \frac{6}{7} \cdot 7$$

Step-by-Step Solution

Answer

The quotient is 98.

Step 1: Identify the Divisor

The problem is to find the quotient of the expression \(12 \div \frac{6}{7} \cdot 7\). First, identify the divisor, which is the fraction \(\frac{6}{7}\).

Step 2: Find the Reciprocal of the Divisor

The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). Therefore, the reciprocal of \(\frac{6}{7}\) is \(\frac{7}{6}\).

Step 3: Replace Division with Multiplication

Replace the division operation \(\div \frac{6}{7}\) with multiplication by its reciprocal \(\times \frac{7}{6}\). The expression now reads: \(12 \times \frac{7}{6} \cdot 7\).

Step 4: Reorder the Expression

Reorder the expression to make multiplication straightforward: \((12 \cdot 7) \times \frac{7}{6}\).

Step 5: Multiply Numerators and Denominators

First, compute the multiplication outside the fraction: \(12 \cdot 7 = 84\). Now multiply this with \(\frac{7}{6}\), calculating the numerator and denominator separately: \(84 \times 7 = 588\) and the denominator is just \(6\).

Step 6: Simplify the Result

Divide the numerator by the denominator: \(\frac{588}{6} = 98\). So, the quotient of the given expression is 98.

Key Concepts

Reciprocal of a Fraction

A reciprocal of a fraction is an essential concept when it comes to division involving fractions. To find the reciprocal of a fraction, you simply swap its numerator and denominator. This means that if you have a fraction like \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
This simple flip is crucial because it helps turn division into multiplication, making calculations much easier. Consider the step where we deal with \(\frac{6}{7}\); its reciprocal is \(\frac{7}{6}\), which allows us to replace the division operation with multiplication in our problem.
Understanding how to find and apply the reciprocal of a fraction is foundational in not just division problems but also in simplifying complex equations involving fractions.

Simplifying Expressions

Simplifying expressions is all about breaking down complicated mathematical expressions into their simplest form. This process involves several steps, including reducing fractions and reordering terms.
In our exercise, the expression changed from division to multiplication by the reciprocal, resulting in \(12 \times \frac{7}{6} \cdot 7\). Rearranging this to \((12 \cdot 7) \times \frac{7}{6}\) simplifies the task further.

• This step-by-step approach minimizes errors and streamlines calculations.
• It allows for focusing on easier parts of the expression one step at a time.

Through simplification, we aim to find the cleanest path to the solution, which ultimately helps achieve clarity and accuracy in solving mathematical problems.

Multiplication of Fractions

Understanding the multiplication of fractions involves a straightforward process: both numerators are multiplied together, and both denominators are multiplied together.
For a case like in our problem, after simplifying the expression, we calculate \(12 \cdot 7 = 84\). This result is then used to multiply by the fractional part, \(\frac{7}{6}\):

• The formula for multiplying is \(84 \times \frac{7}{6} = \frac{588}{6}\).
• We further simplify \(\frac{588}{6}\) to derive the final answer, which is 98.

By breaking it down step by step, students can readily apply this knowledge to similar problems, reinforcing their understanding of how multiplication works with fractions.