Problem 47 - Prealgebra

Question

Find each quotient. $$\frac{12 t}{3 s} \div \frac{3 s}{12 t}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{16t^2}{s^2} \).

Step 1: Write the Division of Fractions

The problem given is a division of two fractions: \( \frac{12t}{3s} \div \frac{3s}{12t} \). Division of fractions is equivalent to multiplying by the reciprocal. Therefore, we rewrite the expression as \( \frac{12t}{3s} \times \frac{12t}{3s} \).

Step 2: Multiply the Numerators and Denominators

Next, multiply the numerators: \( 12t \times 12t = 144t^2 \). Then, multiply the denominators: \( 3s \times 3s = 9s^2 \). So, the product of the fractions is \( \frac{144t^2}{9s^2} \).

Step 3: Simplify the Fraction

To simplify the fraction \( \frac{144t^2}{9s^2} \), divide both the numerator and the denominator by 9: \( \frac{144 \div 9 t^2}{9 \div 9 s^2} \). This results in \( \frac{16t^2}{s^2} \).

Step 4: Conclusion: State the Final Answer

Hence, the simplified form of the original division is \( \frac{16t^2}{s^2} \).

Key Concepts

Division of Fractions

Dividing fractions might seem a bit tricky at first, but with practice, it becomes straightforward. Whenever you see a division involving fractions, recall that dividing by a fraction is the same as multiplying by its reciprocal. This means instead of trying to directly divide the fractions, you can turn the division into multiplication.
For example, in the exercise given: \( \frac{12t}{3s} \div \frac{3s}{12t} \), we don't struggle with direct division. Instead, we flip the second fraction (\( \frac{3s}{12t} \)) to get its reciprocal \( \frac{12t}{3s} \).
This transformation changes our operation to multiplication: \( \frac{12t}{3s} \times \frac{12t}{3s} \). Don't forget that flipping the second fraction is crucial in getting this step right. Now, you're all set to move forward with multiplying!

Reciprocal

The concept of a reciprocal is fundamental when dealing with the division of fractions. Understanding what a reciprocal is helps simplify complex fraction problems. A reciprocal is what you multiply a number by to get the value of 1. Essentially, you swap the numerator and the denominator.
For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). This means for any fraction, just flip the numbers above and below the fraction bar.
In the exercise, we turned \( \frac{3s}{12t} \) into \( \frac{12t}{3s} \) so we could multiply instead of divide. This makes it a lot easier to compute. Using reciprocals is a great strategy for simplifying calculations with fractions.

Simplifying Fractions

Once you've turned division into multiplication, you often end up with a more complex fraction called the product of the numerators and denominators. Simplifying these fractions is the final, crucial step.
Let's consider the expression \( \frac{144t^2}{9s^2} \) from the solution. Simplifying involves dividing the numerator and denominator by their greatest common divisor (GCD), here it's 9.

• First, divide 144 by 9, which results in 16.
• Next, the denominator 9 divided by 9 equals 1, effectively simplifying the process.

This gives us the simplified fraction \( \frac{16t^2}{s^2} \). Simplifying fractions makes them easier to understand and use in subsequent calculations, and ensures you have the most straightforward expression possible.