Problem 32 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$4 \cdot \frac{7}{6} \div 7$$

Step-by-Step Solution

Answer

The quotient is \( \frac{2}{3} \).

Step 1: Rewrite Division as Multiplication

Begin by rewriting the division of fractions as multiplication. The problem is to find the quotient of \( 4 \cdot \frac{7}{6} \div 7 \). To do this, replace the division by multiplication with the reciprocal of the divisor. Thus, the problem becomes \( 4 \cdot \frac{7}{6} \times \frac{1}{7} \).

Step 2: Multiply the Fractions

Next, perform the multiplication of the fractions. Multiply the numerators together and the denominators together: \( (4 \times 7) \) and \( (6 \times 7) \). This results in \( \frac{4 \times 7 \times 1}{6 \times 7} \).

Step 3: Simplify the Expression

Simplify the fraction by canceling out the common factor. The number 7 in the numerator can be canceled with the 7 in the denominator, resulting in \( \frac{4 \times 1}{6} \).

Step 4: Simplify Further

Simplify the remaining expression. This becomes \( \frac{4}{6} \). Reduce \( \frac{4}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives \( \frac{2}{3} \).

Key Concepts

Understanding the Reciprocal

A reciprocal is simply the flipped version of a fraction. For example, if you have a fraction like \( \frac{7}{6} \), its reciprocal would be \( \frac{6}{7} \). This means you switch the numerator and the denominator.
Reciprocals are crucial when it comes to dividing fractions. Instead of dividing directly, which can be tricky, you can take the reciprocal of the divisor (the number you're dividing by) and multiply.
This transforms a division problem involving fractions into an easier multiplication problem. For instance, if you're given \( 4 \cdot \frac{7}{6} \div 7 \), you turn \( 7 \) into its reciprocal, \( \frac{1}{7} \), and multiply: \( 4 \cdot \frac{7}{6} \times \frac{1}{7} \). It’s a handy trick to simplify your work by following this mix of reversing and multiplying steps.
When a whole number like 7 is involved, remember it can be considered as \( \frac{7}{1} \) and its reciprocal is \( \frac{1}{7} \). Thus, applying the reciprocal simplifies the division.

Simplifying Fractions

Simplifying fractions is all about making them as simple as possible, while still being equivalent to the original. To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator.
For example, if you have the fraction \( \frac{4}{6} \), you simplify it by dividing both 4 and 6 by their largest common factor. Here, it's 2.
Perform the division:

• 4 divided by 2 equals 2
• 6 divided by 2 equals 3

This results in the simplified fraction \( \frac{2}{3} \).
By reducing fractions, you make them easier to work with in future calculations. It might also help you see relationships between numbers more clearly. In our exercise, after multiplying and simplifying step by step, we ended up with \( \frac{4}{6} \), which then simplified to \( \frac{2}{3} \). This makes the result cleaner and easier to understand.

Multiplying Fractions

Multiplying fractions might seem daunting, but it's quite straightforward once you get the hang of it. When multiplying two fractions together, you multiply across both the numerators and the denominators.
For instance, if you have to multiply \( \frac{7}{6} \) by \( \frac{1}{7} \), you simply do the following:

• Multiply the numerators: 7 times 1 equals 7
• Multiply the denominators: 6 times 7 equals 42

This results in the fraction \( \frac{7}{42} \). Always remember to look for common factors in the fraction afterward for possible simplifications, just like you do after all arithmetic operations involving fractions.
In our problem, when multiplying \( 4 \cdot \frac{7}{6} \times \frac{1}{7} \), the method of multiplying straight across gives us a fraction that can then be simplified. Keeping the operation of multiplication clear and ordered makes solving fraction problems a breeze.