Problem 77 - Prealgebra

Question

Multiply or divide as indicated. \(\frac{7}{6} \cdot \frac{12}{7}\)

Step-by-Step Solution

Answer

The result is 2.

Step 1: Set up the multiplication of fractions

First, recognize that when you multiply fractions, you multiply the numerators together and the denominators together. Write down the fractions and their multiplication: \( \frac{7}{6} \times \frac{12}{7} \).

Step 2: Simplify Before Multiplying

Check for any possible simplifications before performing the multiplication. Notice that the \(7\) in the numerator of the first fraction and the \(7\) in the denominator of the second fraction can cancel each other out.

Step 3: Simplify the Fractions

After canceling the \(7\)'s, you are left with \(\frac{1}{6} \times \frac{12}{1}\), which simplifies to \(\frac{12}{6}\).

Step 4: Perform the Division

Now, divide \(12\) by \(6\) to simplify the fraction. This results in \(2\).

Step 5: Conclude the Calculation

The result of multiplying \( \frac{7}{6} \cdot \frac{12}{7} \) simplifies to \(2\).

Key Concepts

Fraction Simplification

Simplifying fractions is a crucial step in performing fraction operations like multiplication and division. It refers to reducing the fraction to its simplest form. For example, in the exercise given, identifying common factors in the numerator and denominator can vastly facilitate solving the problem. When a fraction is simplified, it's much easier to understand and work with. To simplify a fraction:

• Determine the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

In a lot of cases, as seen in the original exercise, simplification might involve recognizing that certain terms in different fractions can cancel each other out before any actual multiplication takes place. It's a good habit to look for simplifications before conducting other operations. This can save time and inticate a clearer understanding of the problem.

Multiplying Numerators and Denominators

Once you've simplified the fractions, if necessary, you can proceed with multiplication. The basic rule for multiplying fractions is straightforward: multiply the numerators together and then multiply the denominators together.Given two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), multiplying them involves:

• Calculating the product of the numerators: \( a \times c \).
• Calculating the product of the denominators: \( b \times d \).
• Writing the new fraction as \( \frac{a \times c}{b \times d} \).

In the original exercise with \(\frac{7}{6}\) and \(\frac{12}{7}\), once you simplify the fractions, you multiply the numerators \(1\) and \(12\), and the denominators, in this simplified example, \(6\) and \(1\), resulting in \(\frac{12}{6}\).While it sounds simple, keeping these steps separate can help avoid errors, especially with more complex fractions.

Cancelling Terms in Fractions

One of the key insights to tackle fraction multiplication is cancelling terms across numerators and denominators that share common factors. This makes the process far more efficient and less error-prone.In any fraction multiplication scenario:

• Look for common factors in the numerator of one fraction and the denominator of another.
• Cancel these common factors out, which simplifies your calculations significantly.

In the example \(\frac{7}{6} \cdot \frac{12}{7}\), the \(7\) in the numerator of the first fraction and the \(7\) in the denominator of the second can be cancelled early on, simplifying the challenge before any multiplication.This step of cancelling not only reduces the numbers involved but also builds a deeper understanding of how fraction multiplication can sometimes be transformed into simpler arithmetic operations. It’s a smart strategy that is very valuable when dealing with larger fractions.