Problem 30 - Algebra 1

Question

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

Step-by-Step Solution

Answer

-48

Step 1: Identify the Dividend and Divisor

The dividend is -18 and the divisor is \(\frac{3}{8}\). We need to divide -18 by \(\frac{3}{8}\).

Step 2: Conversion to Multiplication

The division \(-18 \div \frac{3}{8}\) can be rewritten as multiplication by the reciprocal of \(\frac{3}{8}\). The reciprocal of \(\frac{3}{8}\) is \(\frac{8}{3}\). So the problem becomes \(-18 * \frac{8}{3}\).

Step 3: Perform the Multiplication

Multiply -18 and \(\frac{8}{3}\) to get the quotient. \(-18 * \frac{8}{3} = -48\). So, \(-18 \div \frac{3}{8} = -48\).

Key Concepts

Reciprocals

In the world of fractions and division, the concept of reciprocals is crucial. Think of a reciprocal as a fraction's upside-down cousin. For any non-zero fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). By swapping the numerator and denominator, you create the reciprocal. This transformation is essential when dividing by fractions. Instead of dividing, you can multiply by the reciprocal which simplifies calculations considerably.
Here's how it works:

• Find the reciprocal of a fraction by flipping its numerator and denominator.
• Use this reciprocal for multiplication.

This technique is used because multiplying is often easier than dividing, especially when fractions are involved. Understanding and applying reciprocals can make these tasks more straightforward.

Multiplication

Multiplication is a fundamental math operation that combines quantities or numbers. When multiplying fractions, it's a matter of simple arithmetic: multiply the numerators and then multiply the denominators.
Here's how you multiply fractions:

• Multiply the numerators together to find the new numerator.
• Multiply the denominators together to find the new denominator.

In our exercise, we used multiplication to solve a division problem by using a reciprocal.
The equation \(-18 \cdot \frac{8}{3} = -48\) is the result of multiplying a whole number by a fraction. This showcases the ease of multiplication, once you have converted the division into a multiplication using the reciprocal of the divisor. Multiplying by fractions might seem tricky at first, but with practice, it becomes an easy-to-use tool in solving problems.

Rational Numbers

Rational numbers are like mathematical building blocks. These numbers can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). They include both positive and negative numbers. Any arithmetic performed on rational numbers results in another rational number, making them versatile and essential for solving diverse mathematical problems.
Rational numbers:

• Include fractions, whole numbers, and integers.
• Are closed under addition, subtraction, multiplication, and division (except by zero).

In our exercise, \(-18\) and \(\frac{3}{8}\) are both rational numbers. When performing division, particularly with fractions, understanding that the result, in this case \(-48\), is also a rational number underlines their consistency and reliability in math operations.