Problem 65 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ -4.8 \div 1.2 $$

Step-by-Step Solution

Answer

The quotient is \-4.\

Step 1 - Identify Dividend and Divisor

In the given problem, the number \(-4.8\) is the dividend, and the number \(1.2\) is the divisor.

Step 2 - Perform the Division

Divide the dividend by the divisor. \[ \frac{-4.8}{1.2} = -4 \]

Step 3 - Verify the Quotient

Multiply the quotient by the divisor to check the result. \[ -4 \times 1.2 = -4.8 \] The multiplication confirms that the division is correct.

Step 4 - State if the Quotient is Undefined

Since the division \(-4.8 \div 1.2\) resulted in a defined quotient, the quotient is not undefined.

Key Concepts

Dividend and Divisor Identification

Before performing a division of decimals, it's essential to know the terms 'dividend' and 'divisor'. The dividend is the number you are dividing. In our example, \( -4.8 \) is the dividend.
The divisor is the number you are dividing by. Here, \( 1.2 \) is the divisor.
Knowing which number is which helps in setting up the division problem correctly.

Performing Division

After identifying the dividend and the divisor, the next step is to perform the division. This can be done using a calculator or manually.
In our case, we divide \( -4.8 \) by \( 1.2 \). The calculation is as follows:

\[ \frac{-4.8}{1.2} = -4 \]
The division operation gives us \( -4 \) as the quotient.

Verifying Quotient

It's always a good idea to verify the quotient to ensure the division was done correctly.
We can do this by multiplying the quotient by the divisor.
If the product equals the original dividend, our division is correct.
Here is the verification process for our example:

\[ -4 \times 1.2 = -4.8 \]
The multiplication \( -4 \times 1.2 \) equals the original dividend \( -4.8 \), confirming the quotient is correct.

Defined and Undefined Quotients

Finally, it's crucial to understand when a quotient is defined or undefined.
A quotient is defined when a number can be divided by another without any issues. In other words, when dividing, if both numbers are real and the divisor is not zero, the quotient is defined.
For \( -4.8 \) divided by \( 1.2 \), the quotient \( -4 \) is defined because both numbers are real and the divisor (\( 1.2 \)) is not zero.
On the other hand, a quotient is undefined if you attempt to divide by zero. For instance, \( a \div 0 \) (where \( a \) is any real number) is undefined.