Problem 51 - Algebra A Combined Function

Question

Perform the indicated operation. \(\frac{30}{-2}\)

Step-by-Step Solution

Answer

The result of \( \frac{30}{-2} \) is -15.

Step 1: Identify the operation

The problem asks for the division of the two numbers: 30 and -2. The division operation is indicated by the fraction bar in \( \frac{30}{-2} \).

Step 2: Divide the numbers

Perform the division by dividing the numerator (30) by the denominator (-2). Divide 30 by -2 to get the quotient.

Step 3: Simplify the result

When dividing 30 by -2, compute \( 30 \div (-2) = -15 \). The quotient is -15 because a positive number divided by a negative number results in a negative number.

Key Concepts

Negative Numbers Division

Dividing negative numbers can seem tricky at first, but it's actually quite simple. A basic rule to remember is that when you divide a positive number by a negative number, the result is always a negative quotient. Likewise, dividing a negative number by a positive number will also yield a negative result. This is because positive and negative numbers are considered opposite in sign.
Therefore, the operation becomes straightforward:

• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative
• Negative ÷ Negative = Positive
• Positive ÷ Positive = Positive

Understanding these rules helps you predict the sign of the quotient even before completing the math. In our example, \(\frac{30}{-2}\), the quotient will be negative because you are dividing a positive number (30) by a negative number (-2).

Numerator and Denominator

In division expressed as a fraction, such as \(\frac{30}{-2}\), the top number is called the numerator, and the bottom number is the denominator.
The numerator (30 in our example) represents how many parts you have, while the denominator (-2) indicates into how many parts you want to divide the numerator.
Think of the fractional bar as a division sign. When you see \(\frac{30}{-2}\), it simply tells you to divide 30 by -2.
Understanding the role of the numerator and denominator helps clearly frame division problems:

• Numerator \(\Rightarrow\) Quantity to Divide
• Denominator \(\Rightarrow\) Dividing Factor

Another key point is that if the negative sign is with the denominator, as in this case, the result will be negative, as explained in the previous section.

Quotient Calculation

After understanding the signs and parts of a fraction, the next step is calculating the quotient. The quotient is the result you get when you divide the numerator by the denominator. In simple terms, it is the answer to the division problem.
For the calculation, you perform a simple division: \[30 \div (-2)\]Calculate as you normally would, ignoring the signs initially:

• Divide 30 by 2, which gives 15

Now, reintroduce the signs. As division involves a positive number (30) and a negative number (-2), the quotient becomes negative. Thus:

• The final quotient is -15

The key takeaways are:

• Divide the numbers as usual
• Apply the rule of signs to determine if the quotient is positive or negative