Problem 34 - Algebra 1

Question

Simplify the expression. $$58 z \div\left(-\frac{2}{5}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-145z\).

Step 1: Understand the expression

The original expression is \(58z \div\left(-\frac{2}{5}\right)\), where the operation being performed is division. In mathematics, division by a fraction is equivalent to multiplication by its reciprocal.

Step 2: Identify the reciprocal

The reciprocal of \(-\frac{2}{5}\) is \(-\frac{5}{2}\). So the next step is to convert the division operation to multiplication by using the reciprocal of \(-\frac{2}{5}\). This changes the expression to \(58z \cdot \left(-\frac{5}{2}\right)\)

Step 3: Perform the multiplication

Multiplying \(58z\) by \(-\frac {5}{2}\) gives the final simplified expression as \(-145z\).

Key Concepts

Division by Fractions

To divide by a fraction, you actually multiply by its reciprocal. This might sound confusing at first, but once you get the hang of it, it's quite straightforward.
If you see an expression like \(58z \div \left(-\frac{2}{5}\right)\), consider the fraction \(-\frac{2}{5}\). Instead of performing division directly, swap the operation to multiplication using the reciprocal of the fraction.
This change will transform the expression to \(58z \cdot \left(-\frac{5}{2}\right)\). By understanding this principle, you'll be able to simplify complex-looking division problems with ease!
Here are some quick steps to help you remember:

• Find the reciprocal of the fraction you're dividing by.
• Change the division sign to multiplication.
• Multiply as usual.

Reciprocals

A reciprocal is simply what you multiply a number by to get 1. For fractions, you flip the numerator and the denominator. So, the reciprocal of \(-\frac{2}{5}\) is \(-\frac{5}{2}\). This little trick is quite useful when you're dealing with division problems.
Let's explore this further:

• A fraction \(\frac{a}{b}\) has a reciprocal \(\frac{b}{a}\).
• The product of a number and its reciprocal is always 1.
• To find the reciprocal of a fraction with a negative sign, like \(-\frac{2}{5}\), ensure you keep the sign with the reciprocal, making it \(-\frac{5}{2}\).

Understanding reciprocals makes it simple to switch from division to multiplication, helping to simplify expressions quickly.

Multiplication in Algebra

When multiplying algebraic expressions, each part of the expression must be addressed. Take \(58z \cdot \left(-\frac{5}{2}\right)\) as an example. Here’s how to approach it:
First, multiply the coefficients (numbers without the variable). Multiply 58 by \(-\frac{5}{2}\). Remember, multiplying positive by negative gives a negative result.
You can break it down like this:

• Multiply 58 by 5 to get 290.
• Divide 290 by 2 to get 145.

Since we have a negative fraction, the result is \(-145z\).
Keeping track of each step helps in avoiding mistakes. This method can be applied to any algebraic multiplication, ensuring clarity and accuracy in your results.