Problem 46 - Prealgebra

Question

Find each quotient. $$\frac{4 z}{w} \div \frac{7 y z}{w}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{4}{7y} \).

Step 1: Rewrite the Division as Multiplication

When dividing by a fraction, you multiply by its reciprocal. Thus, we rewrite the original division problem: \[\frac{4z}{w} \div \frac{7yz}{w} = \frac{4z}{w} \times \frac{w}{7yz}.\]

Step 2: Cancel Common Factors

We can cancel the common term \( w \) in the numerator of the first fraction and the denominator of the second fraction.\[\frac{4z}{w} \times \frac{w}{7yz} = \frac{4z}{1} \times \frac{1}{7yz}.\]

Step 3: Simplify the Multiplication

Multiply across the numerators and the denominators:\[\frac{4z}{1} \times \frac{1}{7yz} = \frac{4z \times 1}{1 \times 7yz} = \frac{4z}{7yz}.\]

Step 4: Further Simplify

The \( z \) term in the numerator and denominator can be canceled:\[\frac{4z}{7yz} = \frac{4}{7y}.\]

Key Concepts

Fraction Division

Dividing fractions might seem tricky at first, but it becomes straightforward once you understand the basic rule: to divide by a fraction, multiply by its reciprocal. For example, if you have a problem like \( \frac{a}{b} \div \frac{c}{d} \), you rewrite it as \( \frac{a}{b} \times \frac{d}{c} \). You simply "flip" the second fraction and change the operation to multiplication.This works because you're considering how many of the divisor fraction fit into the dividend fraction. By multiplying by the reciprocal, you're essentially finding this relationship in a simpler way. So, always remember:

• Change the division mark to multiply
• Use the reciprocal of the divisor

This rule is fundamental in all fraction divisions, making problems more manageable and easy to handle.

Simplifying Fractions

Simplifying fractions is an important concept that helps to solve math problems more efficiently by reducing fractions to their simplest form. This means rewriting a fraction so that the numerator and denominator have no common factors other than 1.For instance, if we take \( \frac{8}{12} \), both 8 and 12 can be divided by 4, which is their greatest common divisor (GCD). Dividing both the numerator and the denominator by 4, we get \( \frac{2}{3} \). This fraction cannot be simplified further because 2 and 3 share no common divisors except for 1.To simplify any fraction:

• Find the greatest common divisor of the numerator and the denominator
• Divide both by this greatest common divisor

Simplifying fractions makes them easier to work with and often reveals a more intuitive insight into the problem.

Reciprocal

The concept of reciprocal is critical in fraction division and many areas of mathematics. A reciprocal of a number is simply another number which, when multiplied with the original number, results in 1.For any non-zero fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). You basically switch the numerator and the denominator.It's important to remember that:

• The reciprocal of a whole number \( n \) is \( \frac{1}{n} \)
• The reciprocal of a reciprocal returns you to the original number

The reciprocal is useful because it transforms division problems into multiplication problems, making calculations easier. This process of switching helps solve equations and perform operations requiring division by a fraction.