Problem 23 - Algebra 1

Question

Find the quotient. $$-35 \div(-70)$$

Step-by-Step Solution

Answer

The quotient of -35 divided by -70 is 0.5.

Step 1: Identify the values

Identify the values for the division operation. Here, the dividend is -35 and the divisor is -70.

Step 2: Perform the division

Perform the division operation. Remember, the division of two negative numbers is a positive number. The two negative signs effectively cancel out, so divide 35 by 70 normally.

Step 3: Simplify if necessary

Simplify the result if necessary. In this case, the outcome is already in its simplest form, as no further simplification is possible.

Key Concepts

Negative Numbers

Negative numbers have a unique place in mathematics. They can be confusing at first, but understanding them is crucial. A negative number is any number less than zero, marked by a minus sign (\(-\)). For example, \(-35\) and \(-70\) are negative numbers.
Negative numbers play a significant role in various mathematical operations, especially in division. When two negative numbers are divided, the result is positive. You can think of the negatives canceling each other out. For instance, the division of \(-35\) and \(-70\) yields a positive result. Just count the negatives: \(-- = +\).
Keep in mind these useful tips:

• An odd number of negative signs results in a negative product or quotient.
• An even number of negative signs results in a positive outcome.
• This rule helps simplify complex equations by focusing on the sign patterns first.

Division Operation

Division is a fundamental concept in mathematics. It involves splitting a number into equal parts. The number you are dividing is the dividend, while the number by which you divide is the divisor. In the exercise example, \(-35\) is the dividend and \(-70\) is the divisor.
When dividing with negative numbers, it’s important to focus on the signs, as they affect the outcome. Negative divided by negative results in a positive quotient.

• To perform division, calculate as if dealing with positive numbers.
• Then apply the sign rule: two negatives make a positive.

This method simplifies complex problems by reducing them to basic division facts, reminding you to always address the signs both before and after computations.

Simplification in Division

Simplification in division refers to reducing fractions or outcomes to their simplest form. It's the final step to ensure the result is as straightforward as possible. When there is nothing more to reduce, you know the fraction is in its simplest form.
In the division of \(-35\) by \(-70\), simplification shows that the quotient is in its simplest form right from the start. The factors between numerators and denominators are common, but cannot reduce \(1/2\) any further because it is already simplified.

• Check divisibility to find common factors.
• Reduce the fraction if there are common factors other than 1.
• A fraction is simplest when you can no longer divide the numerator and the denominator by the same number other than 1.

Simplification makes calculations manageable and results clear, avoiding unnecessary complexity.