Problem 21 - Basic College Mathematics

Question

Divide and simplify. \(12 \div \frac{3}{2}\)

Step-by-Step Solution

Answer

8

Step 1: Rewrite the Division as Multiplication

Rewrite the division problem using multiplication by the reciprocal of the divisor. Reciprocal of \( \frac{3}{2} \) is \( \frac{2}{3} \). So, \( 12 \div \frac{3}{2} \) becomes \( 12 \times \frac{2}{3} \).

Step 2: Simplify the Multiplication

Multiply 12 by the reciprocal of \( \frac{3}{2} \), which is \( \frac{2}{3} \). This gives: \[ 12 \times \frac{2}{3} = \frac{12 \times 2}{3} \].

Step 3: Perform the Multiplication and Simplification

Multiply the numbers in the numerator: \( 12 \times 2 = 24 \). Simplify the fraction by dividing the numerator by the denominator: \[ \frac{24}{3} = 8 \].

Key Concepts

Reciprocal

To divide one fraction by another, we use a helpful concept called the 'reciprocal.' A reciprocal is simply what you get when you flip a fraction upside down.
For example, the reciprocal of \( \frac{3}{2} \) is \( \frac{2}{3} \). This means the numerator (top number) becomes the denominator (bottom number), and vice versa.

• Reciprocals are essential when changing a division problem into a multiplication problem.
• Always find the reciprocal of the divisor—the number you are dividing by—to continue solving.

Multiplication

Once you have the reciprocal of the fraction, you replace the division sign with a multiplication sign. This is because dividing by a fraction is the same as multiplying by its reciprocal.
In our example, \( 12 \div \ \frac{3}{2} \) becomes \( 12 \times \ \frac{2}{3} \). Note how the operation switches from division to multiplication.

• Multiplication is often easier to handle than division, especially with fractions.
• Make sure to carefully replace the operations to avoid confusion.

Simplification

Simplification makes math problems easier to handle and understand.
After multiplying, you might get a fraction that can be reduced to its simplest form.
In our step-by-step example, multiplying \( 12 \times \ \frac{2}{3} \) results in \( \frac{24}{3} \).
Here, you simplify by dividing the numerator (24) by the denominator (3), giving \( 8 \).

• Always check if the fraction can be reduced by finding common factors for the numerator and denominator.
• This makes the number easier to understand and work with in further calculations.

Fraction

A fraction represents a part of a whole.
It consists of two numbers: the numerator (top number) and the denominator (bottom number). For example, in \( \frac{3}{2} \), 3 is the numerator, and 2 is the denominator.

• Fractions can sometimes look tricky, but understanding their components and how to manipulate them can make solving problems much simpler.
• Operations such as addition, subtraction, multiplication, and division can be applied to fractions just like whole numbers.