Problem 57 - Elementary Algebra

Question

Perform the operations. $$ 50 \div(-5) \div(-10) $$

Step-by-Step Solution

Answer

The result is 1.

Step 1: Perform First Division

Start by performing the first division operation. Divide 50 by -5. \[50 \div (-5) = -10 \] This means that when 50 is divided by -5, the result is -10.

Step 2: Perform Second Division

Now, take the result from Step 1 and perform the second division operation. Divide -10 by -10. \[-10 \div (-10) = 1 \] This indicates that when -10 is divided by -10, the result is 1.

Key Concepts

Understanding the Division Order of Operations

In mathematics, the order in which you perform operations is crucial to obtaining the correct result. This concept is often encapsulated in the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In our original exercise, 50 \( \div(-5) \div(-10) \), both divisions need to be carried out before any other operations, like addition or subtraction, if they existed. This is because division is intended to be computed from left to right.

• First, you divide 50 by -5.
• Then, take that result and divide it by -10.

Breaking down the calculation in this order ensures that you arrive at the correct outcome of 1. This step-by-step approach maintains the integrity of mathematical solutions and avoids errors.

Working with Negative Numbers

Dealing with negative numbers can sometimes be confusing, but understanding them is crucial to performing operations correctly. Negative numbers are simply the opposite of positive numbers. In division, when you divide a positive number by a negative one, or vice versa, the result is always a negative number. For instance, in our example:

• 50 divided by -5 results in -10. Here, a positive number divided by a negative number yields a negative result.

When dividing two negative numbers, the result is positive. This is because two negatives cancel each other out:

• -10 divided by -10 produces a positive 1.

Being diligent with signs (positive or negative) during division helps avoid calculation mistakes.

A Closer Look at Mathematical Operations

Mathematics relies on certain basic operations such as addition, subtraction, multiplication, and division. In the context of the exercise, division takes center stage. Division is essentially repeated subtraction, and understanding this can make the concept more intuitive. For example, dividing 50 by 5 can be thought of as finding out how many times 5 can be subtracted from 50 until what remains is zero.

• This concept applies similarly when negative numbers are involved, only the signs of the results change as we’ve discussed.
• Understanding each operation and its impact is pivotal in solving problems involving multiple operations.

Mathematical operations are building blocks to more complex problem-solving scenarios and understanding their order and effect is key to solving exercises accurately.