Problem 6 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$8 \div\left(-\frac{3}{4}\right)$$

Step-by-Step Solution

Answer

The quotient is \( -\frac{32}{3} \).

Step 1: Identify the Dividend and Divisor

In the expression \( 8 \div \left(-\frac{3}{4}\right) \), the number 8 is the dividend and \( -\frac{3}{4} \) is the divisor.

Step 2: Find the Reciprocal of the Divisor

The reciprocal of a number is found by exchanging the numerator and the denominator. Therefore, the reciprocal of \( -\frac{3}{4} \) is \( -\frac{4}{3} \).

Step 3: Replace Division with Multiplication

Change the division operation to multiplication by replacing the divisor with its reciprocal. So, the expression \( 8 \div \left(-\frac{3}{4}\right) \) becomes \( 8 \times \left(-\frac{4}{3}\right) \).

Step 4: Multiply the Numbers Together

Multiply 8 by \( -\frac{4}{3} \). First multiply the numerators: \( 8 \times -4 = -32 \). Then divide by the denominator: \( \frac{-32}{3} \).

Step 5: Simplify the Result

The fraction \( \frac{-32}{3} \) is already in its simplest form, as the numerator 32 and the denominator 3 have no common factors other than 1.

Key Concepts

Reciprocal

When we talk about the reciprocal of a number, we mean a particular version of the number where you flip its numerator and denominator. Imagine you have a fraction like \( \frac{a}{b} \). The reciprocal would be \( \frac{b}{a} \). For whole numbers like 5, you can think of them as \( \frac{5}{1} \), making the reciprocal \( \frac{1}{5} \).

Reciprocals play a crucial role in division, especially with fractions. Here is why:

• When you divide by a fraction, you're actually multiplying by its reciprocal.
• Finding reciprocals allows you to transform a division problem into a multiplication problem, which is often easier to handle.

In our original exercise, the reciprocal of the divisor \(-\frac{3}{4}\) was found to be \(-\frac{4}{3}\), making the arithmetic straightforward.

Multiplication of Fractions

Multiplication of fractions might sound tricky at first, but it’s quite systematic. When you multiply two fractions like \( \frac{a}{b} \times \frac{c}{d} \), you follow a simple rule: multiply the numerators together and the denominators together.

Here’s a step-by-step guide on how to do it:

• Multiply the Numerators: Simply multiply the top numbers (numerators) of the fractions. For example, in our exercise, 8 was multiplied by -4 to get -32.
• Multiply the Denominators: For whole numbers like 8, consider them as fractions like \( \frac{8}{1} \). Hence, multiply the denominator of the fraction (in this case, 1) by the other denominator (3), resulting in 3.

Combining the results, you'll find the product \( \frac{-32}{3} \), which leads us to how this product can be simplified.

Simplifying Fractions

Simplifying fractions involves making a fraction as simple as possible. A fraction is simplified when the numerator and denominator have no other common factors except 1.

Here's how you can simplify fractions when necessary:

• Find the Greatest Common Factor (GCF): Check for the highest number that can divide both the numerator and denominator without leaving a remainder.
• Divide Both Terms: Use the GCF to divide both the numerator and denominator to reduce the fraction.
• Simplifications in Extreme Cases: Sometimes, such as in our exercise with \( \frac{-32}{3} \), a fraction is already in its simplest form. Here, the GCF of 32 and 3 is 1.

This simplification ensures that the fraction is easy to interpret and use, maintaining its mathematical equivalence.