Problem 33 - Algebra 1

Question

Simplify the expression. $$6 t \div\left(-\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

-12t

Step 1: Identify the reciprocal of the denominator

The reciprocal of \(-\frac{1}{2}\) is \(-2\). Remember that the reciprocal of a fraction is obtained by swapping the numerator and denominator.

Step 2: Replace division operation with multiplication

Replace the division operation in the expression \(6t \div -\frac{1}{2}\) with multiplication, this gives us \(6t \times -2\).

Step 3: Perform the multiplication operation

Multiply 6 and -2 and keep the \(t\) attached. The result is \(-12t\)

Key Concepts

Reciprocal

Understanding the concept of a reciprocal is crucial in mathematics, especially when simplifying expressions. The reciprocal of a number is simply one divided by that number. For a fraction, this means swapping the numerator and the denominator.
For instance, the reciprocal of \(-\frac{1}{2}\) is \(-2\). This is because when you flip \(-\frac{1}{2}\), it becomes \(-2\).
If you multiply a number by its reciprocal, the result is always 1. This property is useful in various mathematical operations, such as division, where finding reciprocals can simplify complex calculations.

Division to Multiplication

Sometimes, division in expressions can be simplified by converting it into a multiplication problem. This technique involves using the reciprocal of the divisor.
For the expression \(6t \div -\frac{1}{2}\), we replace division by multiplying by the reciprocal. Thus, division by \(-\frac{1}{2}\) becomes multiplication by \(-2\).
This switch is beneficial because multiplication is generally more straightforward and less error-prone, especially when dealing with complex fractions or variables.

Multiplication of Integers

Once division is converted into multiplication, the next step is to carry out the multiplication of integers. In the expression \(6t \times -2\), simply multiply the integers 6 and -2.
Performing this multiplication, we get \(-12\), and since we initially had the variable \(t\) attached, the simplified expression becomes \(-12t\).
Remember, when multiplying integers, combining a positive and a negative number always results in a negative product. Attention to these signs ensures accuracy in your final answer.