Problem 56 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ \frac{-50}{25} $$

Step-by-Step Solution

Answer

\(-2\)

Step 1: Understand the Problem

The task is to divide \(-50\) by \(25\).

Step 2: Perform the Division

Divide \(-50\) by \(25\). Start with the division like you would with positive numbers: \ \ \[ \frac{-50}{25} = -2 \]

Step 3: Check the Quotient

Multiply the quotient \(-2\) by the divisor \(25\): \ \ \[ -2 \times 25 = -50 \] This matches the original numerator, confirming the quotient is correct.

Step 4: Conclusion

Since the division results in an integer and no undefined expressions appear, the quotient is \(-2\).

Key Concepts

Division with Negative Numbers

Dividing with negative numbers follows the same basic rules as division with positive numbers. The key difference is handling the signs.
When you divide a negative number by a positive number, the result is negative. Conversely, dividing a positive number by a negative one also results in a negative quotient.
For instance, dividing -50 by 25, as in our example, gives us:
\[\frac{-50}{25} = -2\]
Similarly, dividing 50 by -25 yields:
\[\frac{50}{-25} = -2\]
Remember these steps:

• Divide the absolute values as usual.
• Determine the sign of the result by noting if the signs of the numerator and divisor are different (result is negative) or the same (result is positive).

Checking Division Results

Ensuring the accuracy of division results is crucial. You can verify your answer by multiplying the quotient by the divisor.
In our exercise, after dividing -50 by 25, we get a quotient of -2. To check:
Multiply the quotient (\text{-2}) by the divisor (25):
\[-2 \times 25 = -50\]
This equals the original numerator, -50, confirming our division was correct.
These steps for verification are handy:

• After dividing, multiply the quotient by the original divisor.
• If the multiplication gives you the original numerator, your division is correct.

This simple method helps avoid mistakes and ensures confidence in your results.

Integer Division

Integer division refers to dividing numbers where the quotient is also an integer. No fractions or decimals result from this operation.
For example, in the given exercise, \[\frac{-50}{25} = -2\], the quotient is -2, which is an integer.
It is important to watch out for cases where division might not be straightforward:

• If the division of two integers doesn't result in an exact whole number, then the quotient isn't an integer division result.
• An undefined division occurs when you try to divide by zero or perform other non-permissible operations.

Always ensure that your final quotient represents a valid integer, confirming clear and precise results in division problems, keeping fractions or remainders for other contexts.