Problem 33 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ \frac{-18}{\frac{3}{8}} $$

Step-by-Step Solution

Answer

The quotient of \( \frac{-18}{\frac{3}{8}} \) is -48.

Step 1: Understanding Division by Fraction

To divide by a fraction, it's easier to convert the division operation into a multiplication operation by using the reciprocal of the denominator fraction. Reciprocal of a fraction is obtained by swapping its numerator and denominator.

Step 2: Find the Reciprocal of the Denominator Fraction

The given denominator is \( \frac{3}{8} \). Its reciprocal will be \( \frac{8}{3} \).

Step 3: Convert Division Operation into Multiplication Operation

Replace the division of -18 by \( \frac{3}{8} \) with multiplication of -18 and \( \frac{8}{3} \). This results in the equation \( -18 * \frac{8}{3} \).

Step 4: Solve the Multiplication Equation

The result of multiplying -18 and \( \frac{8}{3} \) gives -48.

Key Concepts

Reciprocal of a Fraction

Understanding the concept of the reciprocal is pivotal when it comes to dividing fractions. Essentially, the reciprocal of a fraction is a way of flipping a fraction over. Imagine you have a fraction \( \frac{a}{b} \), where \( a \) is the numerator (the top number) and \( b \) is the denominator (the bottom number). To find the reciprocal, you simply switch these numbers, so the reciprocal of this fraction would be \( \frac{b}{a} \). It's like turning the fraction upside down.

Why do we care about reciprocals? Whenever you divide by a fraction, it is the same as multiplying by its reciprocal. This handy swap simplifies the process of division, circumventing direct division by fractions, which can be more difficult. This transformative step in finding the quotient of fractions is integral for successful division.

Multiplication of Fractions

When tackling the multiplication of fractions, it's quite straightforward: you multiply the numerators with each other and the denominators with each other. For instance, if you are multiplying \( \frac{a}{b} \) by \( \frac{c}{d} \) the resulting product would be \( \frac{a \times c}{b \times d} \). It's that simple—no common denominators or other complications as seen in addition or subtraction of fractions.

Things get interesting when you're dealing with whole numbers and fractions, or negative values. If you need to multiply a whole number by a fraction, you can convert the whole number into a fraction by giving it a denominator of 1. With negative signs, just remember that a negative times a positive is a negative, and a negative times a negative is a positive.

Converting Division to Multiplication

Converting division to multiplication when working with fractions is a game changer. It leverages the concept of the reciprocal. Here's how it works: whenever you come across a division problem with a fraction, like \( a \div \frac{b}{c} \), you can convert it to multiplication by flipping the divisor (the fraction after the division symbol) to its reciprocal, and then multiply. So it becomes \( a \times \frac{c}{b} \).

This maneuver is known in mathematics as 'multiplying by the reciprocal' and it transforms a potentially confusing fraction division problem into a simpler multiplication problem. It's a vital strategy and makes solving fraction problems much more manageable. By using this method, you can solve complex fraction division with ease.