Problem 44 - Algebra 1

Question

Simplify the expression. $$\frac{42 t}{-14 z} \div \frac{-6}{7 t}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{7 t^2}{2 z} \)

Step 1: Arrangement

Firstly, rearrange the given expression as a multiplication problem where the second fraction flips (reciprocal) and changes the division to multiplication.So, \(\frac{42 t}{-14 z} \div \frac{-6}{7 t}\) becomes \(\frac{42 t}{-14 z} * \frac{7 t}{-6}\)

Step 2: Perform Multiplication

Next, perform the multiplication across the numerators and the denominators separately.So, \(\frac{42 t}{-14 z} * \frac{7 t}{-6}\) becomes \(\frac{42 t * 7 t}{-14 z * -6}\)

Step 3: Simplify Multiplication

Simplify the terms in both the numerator and the denominator to obtain \(\frac{294 t^2}{84 z}\)

Step 4: Simplify the fraction

Reduce the fraction to its simplest form by dividing both numerator and denominator by their greatest common divisor, which is 42. Thus, \(\frac{294 t^2}{84 z}\) simplifies further to \(\frac{7 t^2}{2 z}\)

Key Concepts

Fraction Division

Dividing fractions might seem tricky, but it becomes easier once you learn the steps. Instead of directly dividing, you convert the division into a multiplication problem by using the reciprocal. A reciprocal is simply flipping a fraction upside down. For example, the reciprocal of \(\frac{-6}{7t}\) is \(\frac{7t}{-6}\). By flipping, the problem now involves multiplication instead of division, which is often simpler to handle.

• Flipping the divisor: Change \(\frac{-6}{7t}\) to \(\frac{7t}{-6}\).
• Change division to multiplication: \(\frac{42t}{-14z} \div \frac{-6}{7t}\) becomes \(\frac{42t}{-14z} \times \frac{7t}{-6}\).

This transformation makes it possible to apply straightforward multiplication rules to solve the expression.

Multiplication of Fractions

Once division is transformed into multiplication, the following step is straightforward. Multiplying fractions involves multiplying the numerators together and the denominators together. This means multiplying the top numbers across and then doing the same with the bottom numbers.

• Numerator multiplication: Multiply \(42t\) by \(7t\) to get \(294t^2\).
• Denominator multiplication: Multiply \(-14z\) by \(-6\) to get \(84z\).

This results in a new fraction \(\frac{294t^2}{84z}\). It’s crucial to handle signs properly in these steps. Negative * Negative = Positive, so calculating the denominator will generate a positive result.

Greatest Common Divisor

After performing the multiplication, it's important to simplify the fraction by reducing it to its simplest form. To do this, you find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can divide both numbers without leaving a remainder.

• Identify the GCD for the fraction \(\frac{294t^2}{84z}\). Here, GCD of 294 and 84 is 42.
• Divide both the numerator and the denominator by 42.

This results in \(\frac{7t^2}{2z}\). Simplifying fractions ensures that the expression is presented in its most elementary form, making it easier to interpret and use further.

Reciprocal in Algebra

Understanding the concept of a reciprocal is essential in algebra, especially when dealing with fractions. The reciprocal of a number or fraction is essentially flipping the fraction numerator and denominator. This switch is especially useful in division of fractions, where it turns the problem into a more manageable multiplication task.

• The reciprocal of \(a/b\) is \(b/a\).
• For negative numbers or expressions, the sign remains with the numerator when flipped.

In our problem, changing the division \(\frac{-6}{7t}\) to multiplication by its reciprocal \(\frac{7t}{-6}\) simplifies calculations and leads to solving the expression efficiently. Understanding this concept helps streamline many processes in algebraic manipulations.